Rend. Sem. Mat. Univ. Pol. Torino
Vol. 58, 3 (2000)
Partial Diff. Operators
D. Kapanadze – B.-W. Schulze
BOUNDARY VALUE PROBLEMS ON
MANIFOLDS WITH EXITS TO INFINITY
Abstract. We construct a new calculus of boundary value problems with the transmission property on a non-compact smooth manifold with boundary and conical
exits to infinity. The symbols are classical both in covariables and variables. The
operators are determined by principal symbol tuples modulo operators of lower
orders and weights (such remainders are compact in weighted Sobolev spaces).
We develop the concept of ellipticity, construct parametrices within the algebra
and obtain the Fredholm property. For the existence of Shapiro-Lopatinskij elliptic boundary conditions to a given elliptic operator we prove an analogue of the
Atiyah-Bott condition.
1. Introduction
Elliptic differential (and pseudo-differential) boundary value problems are particularly simple
on either a compact smooth manifold with smooth boundary or on a non-compact manifold
under local aspects, e.g., elliptic regularity or parametrix constructions. This concerns pseudodifferential operators with the transmission property, cf. Boutet de Monvel [3], or Rempel and
Schulze [16], with ellipticity of the boundary data in the sense of a pseudo-differential analogue
of the Shapiro-Lopatinskij condition. An essential achievement consists of the algebra structure
of boundary value problems and of the fact that parametrices of elliptic operators can be expressed within the algebra. There is an associated boundary symbol algebra that can be viewed
as a parameter-dependent calculus of pseudo-differential operators on the half-axis, the inner
normal to the boundary (with respect to a chosen Riemannian metric).
The global calculus of pseudo-differential boundary value problems on non-compact or
non-smooth manifolds is more complicated. In fact, it is only known in a number of special
situations, for instance, on non-compact smooth manifolds with exits to infinity, modelled near
the boundary by an infinite half-space, cf. the references below, and then globally generated by
charts with a specific behaviour of transition maps. Pseudo-differential boundary value problems are also studied on manifolds with singularities, e.g., conical singularities by Schrohe and
Schulze [20], [21] or edge and corner singularities by Rabinovich, Schulze and Tarkhanov [14],
[15]. An anisotropic theory of boundary value problems on an infinite cylinder and parabolicity
are studied in Krainer [12]. Moreover, essential steps for an algebra of operator-valued symbols
for manifolds with edges may be found in Schrohe and Schulze [22], [23], [24]. The latter theory
belongs to the concept of operator algebras with operator-valued symbols with a specific twisting
in the involved parameter spaces, expressed by strongly continuous groups of isomorphisms in
those spaces. The calculus of pseudo-differential operators based on symbols and Sobolev spaces
with such twistings was introduced in Schulze [25] in connection with pseudo-differential operators on manifolds with edges, cf. also the monograph [26]. This is, in fact, also a concept to
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D. Kapanadze – B.-W. Schulze
establish algebras of boundary value problems; the corresponding theory is elaborated in [27]
for symbols that have not necessarily the transmission property. The case with the transmission
property is automatically included, except for the aspect of types in Green and trace operators in
Boutet de Monvel’s operators; a characterization in twisted operator-valued symbol terms is contained in Schulze [28] and also in [27]. An application of the edge pseudo-differential calculus
for boundary value problems is the crack theory that is treated in a new monograph of Kapanadze
and Schulze [10], cf. also the article [29]. An essential tool for this theory are pseudo-differential
boundary value problems on manifolds with exits to infinity.
The main purpose of the present paper is to single out a convenient subalgebra of a global
version of Boutet de Monvel’s algebra on a smooth manifold with exits to infinity. Such a calculus with general non-classical symbols (without the edge-operator machinery) has been studied
by Schrohe [18], [19]. We are interested in classical symbols both in covariables and variables.
This is useful in applications (e.g., in edge boundary value problems, or crack theory), where
explicit criteria for the global ellipticity of boundary conditions are desirable. Our approach
reduces all symbol information of the elliptic theory to a compact subset in the space of covariables and variables, though the underlying manifold is not compact. Moreover, we derive a
new topological criterion for the existence of global boundary conditions satisfying the ShapiroLopatinskij condition, when an elliptic interior symbol with the transmission condition is given.
This is an analogue of the Atiyah-Bott condition, well-known for the case of compact smooth
manifolds with smooth boundary, cf. Atiyah and Bott [1] and Boutet de Monvel [3]. Note that
our algebra can also be regarded as a special calculus of pseudo-differential operators on a manifold with edges that have exits to infinity. The edge here is the boundary and the model cone (of
the wedge) the inner normal. Some ideas of our theory seem to generalize to the case of edges
in general, though there are also essential differences. The main new point in general is that
the transmission property is to be dismissed completely. A theory analogous to the present one
without the transmission property would be of independent interest. New elements that appear
in this context are smoothing Mellin and Green operators with non-trivial asymptotics near the
boundary. Continuity properties of such operators “up to infinity” are studied in Seiler [32], cf.
also [31].
Acknowledgements: The authors are grateful to T. Krainer of the University of Potsdam for
useful remarks to the manuscript.
2. Pseudo-differential operators with exit symbols
2.1. Standard material on pseudo-differential operators
First we recall basic elements of the standard pseudo-differential calculus as they are needed for
the more specific structures in boundary value problems below.
Let S µ (U × n ) for µ ∈ and U ⊆ m open denote the space of all a(x, ξ ) ∈ C ∞ (U ×
n ) that satisfy the symbol estimates
α β
(1)
Dx Dξ a(x, ξ ) ≤ chξ iµ−|β|
for all α ∈ m , β ∈ n and all x ∈ K for arbitrary K U , ξ ∈
1
c(α, β, K ) > 0; hξ i = 1 + |ξ |2 2 .
n , with constants c =
Moreover, let S (µ) (U ×( n \0)) be the space of all f ∈ C ∞ (U ×( n \ 0)) with the property
f (x, λξ ) = λµ f (x, ξ ) for all λ ∈ + , (x, ξ ) ∈ U × ( n \ 0). Then we have χ(ξ )S (µ) (U ×
( n \ 0)) ⊂ S µ (U × n ) for any excision function χ(ξ ) ∈ C ∞ ( n ) (i.e., χ(ξ ) = 0 for |ξ | < c0 ,
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Boundary value problems
µ
χ(ξ ) = 1 for |ξ | > c1 for certain 0 < c0 < c1 ). We then define Scl (U × n ) to be the subspace
of all a(x, ξ ) ∈ S µ (U × n ) such that there are elements a(µ− j ) (x, ξ ) ∈ S (µ− j ) (U × ( n \ 0)),
P
j ∈ , with a(x, ξ ) − Nj=0 χ(ξ )a(µ− j ) (x, ξ ) ∈ S µ−(N+1) (U × n ) for all N ∈ . Symbols
µ
in Scl (U × n ) are called classical. The functions a(µ− j ) (uniquely determined by a) are called
the homogeneous components of a of order µ − j , and we call
σψ (a)(x, ξ ) := a(µ) (x, ξ )
the homogeneous principal symbol of order µ (if the order µ is known by the context, otherwise
µ
we also write σψ instead of σψ ). We do not repeat here all known properties of symbol spaces,
such as the relevant Fréchet topologies, asymptotic sums, etc., but tacitly use them. For details we
refer to standard expositions on pseudo-differential analysis, e.g., Hörmander [9] or Treves [33],
or to the more general scenario with operator-valued symbols below, where scalar symbols are a
special case.
Often we have m = 2n, U = × for open  ⊆ n . In that case symbols are also denoted
by a(x, x 0 , ξ ), (x, x 0 ) ∈  × . The Leibniz product between symbols a(x, ξ ) ∈ S µ ( × n ),
b(x, ξ ) ∈ S ν ( × n ) is denoted by #, i.e.,
a(x, ξ )#b(x, ξ ) ∼
X 1 Dξα a(x, ξ ) ∂xα b(x, ξ )
α!
α
Dx = 1i ∂∂x , . . . , 1i ∂∂x , ∂x = ∂∂x , . . . , ∂∂x
. If notation or relations refer to both clasn
n
1
1
sical or non-classical elements, we write (cl) as subscript. In this sense we define the spaces of
classical or non-classical pseudo-differential operators to be
n
o
µ
µ
L (cl) () = Op (a) : a(x, x 0 , ξ ) ∈ S(cl)  ×  × n .
(2)
Here, Op is the pseudo-differential action, based on the Fourier transform F = Fx→ξ in n , i.e.,
RR i(x−x 0 )ξ
Op (a)u(x) =
e
a(x, x 0 , ξ )u(x 0 )dx 0 d̄ξ , d̄ξ = (2π)−n dξ . As usual, this is interpreted
in the sense of oscillatory integrals, first for u ∈ C0∞ (), and then extended to more general
distribution spaces. Recall that L −∞ () = ∩µ∈ L µ () is the space of all operators with
kernel in C ∞ ( × ).
It will be also important to employ parameter-dependent variants of pseudo-differential operators, with parameters λ ∈ l , treated as additional covariables. We set
o
n
µ
µ
L cl ; l = Op (a)(λ) : a(x, x 0 , ξ, λ) ∈ S(cl)  ×  × n+l
,
ξ,λ
µ
µ
using the fact that a(x, x 0 , ξ, λ) ∈ S(cl)  ×  × n+l implies a(x, x 0 , ξ, λ0 ) ∈ S(cl) ( ×  ×
n ) for every fixed λ ∈ l . In particular, we have L −∞ ; l = l , L −∞ () with the
0
identification L −∞ () ∼
= C ∞ ( × ), and l , E being the Schwartz space of E-valued
functions.
Concerning distribution spaces, especially Sobolev spaces, we employ here the usual notan with the standard scalar product.
tion. L 2 ( n ) is the
functions
in
space0 of nsquare integrable
n
s
n
s
2
Then H ( ) = u ∈ ( ) : hξ i û(ξ ) ∈ L
ξ , s ∈ , is the Sobolev space of smoothness s ∈ , û(ξ ) = (Fx→ξ u)(ξ ). Analogous spaces make sense on a C ∞ manifold X. Let
us assume in this section that X is closed and compact. Let Vect(X) denote the set of all complex C ∞ vector bundles on X and H s (X, E), E ∈ Vect(X), the space of all distributional
304
D. Kapanadze – B.-W. Schulze
µ
sections in E of Sobolev smoothness s ∈ . Furthermore, define L (cl) X; E, F; l for µ ∈ ,
E, F ∈ Vect(X), to be the set of all parameter-dependent pseudo-differential operators A(λ)
(with local classical or non-classical symbols) on X, acting between spaces of distributional
sections, i.e.,
A(λ) : H s (X, E) −→ H s−µ (X, F) , λ ∈ l .
µ
For l = 0 we simply write L (cl) (X; E, F). The homogeneous principal symbol of order µ of an
µ
operator A ∈ L (cl) (X; E, F) will be denoted by σψ (A) (or σψ (A)(x, ξ ) for (x, ξ ) ∈ T ∗ X \ 0)
which is a bundle homomorphism
σψ (A) : π ∗ E −→ π ∗ F
for
π : T ∗ X \ 0 −→ X .
µ
Similarly, for A(λ) ∈ L cl X; E, F; l we have a corresponding parameter-dependent homogeneous principal symbol of order µ that is a bundle homomorphism π ∗ E → π ∗ F for
π : T ∗ X × l \ 0 −→ X (here, 0 indicates (ξ, λ) = 0).
2.2. Operators with the transmission property
Boundary value problems on a smooth manifold with smooth boundary will be formulated for
operators with the transmission property with respect to the boundary. We will employ the
transmission property in its simplest version for classical symbols.
µ
µ
Let S(cl)  × + × n = a = ã |× × n : ã(x, ξ ) ∈ S(cl) ( × × n ) , where
+
µ
 ⊆ n−1 is an open set, x = (y, t) ∈  × , ξ = (η, τ ). Moreover, define Scl ( ×
µ
to be the subspace of all a(x, ξ ) ∈ Scl ( × × n ) such that
× n )tr
Dtk Dηα a(µ− j ) (y, t, η, τ ) − (−1)µ− j a(µ− j ) (y, t, −η, −τ ) = 0
(3)
on the set {(y, t, η, τ ) ∈  × × n : y ∈ ,
k ∈ , α ∈ n−1
t = 0, η = 0, τ ∈ \ 0}, for all
µ
µ
and all j ∈ . Set Scl  × + × n tr = a = ã |× × n : ã(x, ξ ) ∈ Scl ( × × n )tr .
+
µ
µ
Symbols in Scl ( × × n )tr or in Scl  × + × n tr are said to have the transmission
property with respect to t = 0.
µ
Pseudo-differential operators with symbols a ∈ Scl  × + × n tr are defined by the rule
Op+ (a)u(x) = r+ Op (ã)e+ u(x) ,
(4)
µ
where ã ∈ Scl ( × × n )tr is any extension of a to  × and e+ is the operator of extension
of restriction from  × to  × + .
by zero from  × + to  × , while r+ is the operator
µ
As is well-known, Op+ (a) for Scl  × + × n tr induces a continuous operator
(5)
Op+ (a) : C0∞  × + −→ C ∞  × +
(that is independent of the choice of the extension ã) and extends to a continuous operator
Op+ (a) : [ϕ]H s ( × + ) −→ H s−µ ( × + )
for arbitrary ϕ ∈ C0∞  × + and s ∈ , s > − 21 . Here, for simplicity, we assume  ⊂ n−1
to be a domain with smooth boundary; then H s ( × + ) = H s ( n )|× + . Moreover, if E is
a Fréchet space that is a (left) module over an algebra A, [ϕ]E for ϕ ∈ A denotes the closure of
{ϕe : e ∈ E} in E.
(6)
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Boundary value problems
2.3. Calculus on a closed manifold with exits to infinity
A further important ingredient in our theory is the calculus of pseudo-differential operators on
a non-compact smooth manifold with conical exits to infinity. The simplest example is the Euclidean space n . It can be viewed as a local model for the general case.
The global pseudo-differential calculus in n with weighted symbols and weighted Sobolev
spaces has been introduced by Parenti [13] and further developed by Cordes [4]. The case
of manifolds with exits to infinity has been investigated by Schrohe [17]. The substructure
with classical (in covariables and variables) symbols is elaborated in Hirschmann [8], see also
Schulze [27], Section 1.2.3. In Section 2.4 below we shall develop the corresponding operator
valued calculus with classical symbols.
Let S µ;δ ( n × n ) =: S µ;δ for µ, δ ∈ denote the set of all a ∈ C ∞ nx × nξ that
satisfy the symbol estimates
α β
(7)
Dx Dξ a(x, ξ ) ≤ chξ iµ−|β| hxiδ−|α|
for all α, β ∈ n , (x, ξ ) ∈ 2n , with constants c = c(α, β) > 0.
This space is Fréchet in a canonical way. Like for standard symbol spaces we have natural
embeddings of spaces for different µ, δ. Moreover, asymptotic sums can be carried out in these
spaces when the orders in one group of variables x and ξ , or in both variables tend to −∞. Basic
notions and results in this context may be found in [30], Section 2.4. Recall that
\
S µ;δ n × n = n × n =: S −∞;−∞ n × n .
µ,δ∈
We are interested in symbols that are classical both in ξ and in x. To this end we introduce some
further notation. Set
(µ) Sξ = a(x, ξ ) ∈ C ∞ n × n \ 0 : a(x, λξ ) = λµ a(x, ξ )
for all λ > 0, (x, ξ ) ∈ n × n \ 0
(δ)
and define analogously the space Sx by interchanging the role of x and ξ . Moreover, we set
(µ;δ)
Sξ ;x
= a(x, ξ ) ∈ C ∞
n \ 0 ×
for all λ > 0, τ > 0, (x, ξ ) ∈
(µ);δ
n \ 0 : a(λx, τ ξ ) = λδ τ µ a(x, ξ )
n \ 0 ×
n \ 0 .
(µ)
It is also useful to have Sξ ;cl defined to be the subspace of all a(x, ξ ) ∈ Sξ such that
x
δ ( n ) where S n−1 = {ξ ∈ n : |ξ | = 1} (clearly, cl means
a(x, ξ )||ξ |=1 ∈ C ∞ S n−1 , Scl
x
x
µ;(δ)
that symbols are classical in x with x being treated as a covariable), and Scl ;x is defined in an
ξ
analogous manner, by interchanging the role of x and ξ .
[µ]
Let Sξ defined to be the subspace of all a(x, ξ ) ∈ C ∞ ( n × n ) such that there is a
c = c(a) with
a(x, λξ ) = λµ a(x, ξ ) for all λ ≥ 1 , x ∈ n , |ξ | ≥ c .
In an analogous manner we define Sx[δ] by interchanging the role of x and ξ . Clearly, for every
[µ]
µ
(µ)
µ
a(x, ξ ) ∈ Sξ there is a unique element σψ (a) ∈ Sξ with a(x, ξ ) = σψ (a)(x, ξ ) for all
(x, ξ ) ∈ n × n with |ξ | ≥ c for a constant c = c(a) > 0. Analogously, for every b(x, ξ ) ∈
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D. Kapanadze – B.-W. Schulze
(δ)
[δ]
Sx there is a unique σeδ (b) ∈ Sx
|x| ≥ c for some c = c(b) > 0.
n ×
with b(x, ξ ) = σeδ (b)(x, ξ ) for all (x, ξ ) ∈
n with
[µ]
µ;[δ]
Set S µ;[δ] = S µ;δ ∩ Sx[δ] , S [µ];δ = S µ;δ ∩ Sξ . Let Scl
be the subspace of all a(x, ξ ) ∈
ξ
[µ−k]
[δ]
µ;[δ]
S
such that there are elements ak (x, ξ ) ∈ Sξ
∩ Sx , k ∈ , with
a(x, ξ ) −
N
X
ak (x, ξ ) ∈ S µ−(N+1);δ
k=0
for all N ∈ . Clearly, the remainders automatically belong to S µ−(N+1);[δ] . Moreover, define
µ;δ
Scl to be the subspace of all a(x, ξ ) ∈ S µ;δ such that there are elements ak (x, ξ ) ∈ S [µ−k];δ ,
ξ
k ∈ , with
N
X
a(x, ξ ) −
ak (x, ξ ) ∈ S µ−(N+1);δ
k=0
[µ];δ
for all N ∈ . By interchanging the role of x and ξ we obtain analogously the spaces Scl
µ;δ
Scl .
x
µ;δ
D EFINITION 1. The space Scl
ξ;x
x
and
( n × n ) of classical (in ξ and x) symbols of order (µ; δ)
is defined to be the set of all a(x, ξ ) ∈ S µ;δ ( n × n ) such that there are sequences
[µ−k];δ
ak (x, ξ ) ∈ Scl
x
, k∈
and
µ;[δ−l]
bl (x, ξ ) ∈ Scl
ξ
, l∈,
such that
a(x, ξ ) −
N
X
µ−(N+1);δ
ak (x, ξ ) ∈ Scl
x
and
a(x, ξ ) −
N
X
µ;δ−(N+1)
bl (x, ξ ) ∈ Scl
ξ
l=0
k=0
for all N ∈ .
[µ];δ
R EMARK 1. It can easily be proved that Scl
x
µ;δ
Scl ( n × n ).
µ;[δ]
µ;δ
⊂ Scl
ξ;x
, Scl
ξ
µ;δ
⊂ Scl
ξ;x
µ;δ
, where Scl =
ξ;x
ξ;x
µ;δ
The definition of Scl
ξ;x
µ−k
σψ
µ;δ
: Scl
ξ;x
µ−k
gives rise to well-defined maps
(µ−k);δ
−→ Sξ ;cl
x
µ−k
, k∈
and
µ;δ
µ;(δ−l)
σeδ−l : Scl −→ Scl ;x
ξ;x
ξ
, l∈,
namely σψ (a) = σψ (ak ), σeδ−l (a) = σeδ−l (bl ), with the notation of Definition 1. From
µ−k
the definition we also see that σψ (a) is classical in x of order δ and σeδ−l (a) is classical in ξ
µ−k
of order µ. So we can form the corresponding homogeneous components σeδ−l σψ (a) and
µ−k
µ−k δ−l
µ−k δ−l
σe (a) =:
σe (a) in x and ξ , respectively. Then we have σeδ−l σψ (a) = σψ
σψ
µ−k;δ−l
σψ,e
(a) for all k, l ∈ .
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Boundary value problems
µ;δ
For a(x, ξ ) ∈ Scl
ξ;x
( n × n ) we set
µ
σe (a) := σeδ (a) ,
σψ (a) := σψ (a) ,
and define
µ;δ
σψ,e (a) := σψ,e (a)
σ (a) = σψ (a), σe (a), σψ,e (a) .
µ;δ
µ−1;δ−1 n
R EMARK 2. a(x, ξ ) ∈ Scl ( n × n ) and σ (a) = 0 implies a(x, ξ ) ∈ Scl
( ×
ξ;x
ξ;x
n ). Moreover, from σ (a) we can recover a(x, ξ ) mod S µ−1;δ−1 ( n ×
clξ;x
n ) by setting
a(x, ξ ) = χ(ξ )σψ (a)(x, ξ ) + χ(x) σe (a)(x, ξ ) − χ(ξ )σψ,e (a)(x, ξ ) ,
where χ is any excision function in
µ;(δ)
n . More generally, let p (x, ξ ) ∈ S (µ);δ , p (x, ξ ) ∈
e
ψ
ξ ;clx
(µ;δ)
Scl ;x and pψ,e (x, ξ ) ∈ Sξ ;x be arbitrary elements with σe ( pψ ) = σψ ( pe ) = pψ,e . Then
ξ
µ;δ
a(x, ξ ) = χ(ξ ) pψ (x, ξ ) + χ(x) pe (x, ξ ) − χ(ξ ) pψ,e (x, ξ ) ∈ Scl ( n × n ), and we have
ξ;x
σψ (a) = pψ , σe (a) = pe , σψ,e (a) = pψ,e .
E XAMPLE 1. Let us consider a symbol of the form
a(x, ξ ) = ω(x)b(x, ξ ) + (1 − ω(x))x −m
X
x α aα (ξ )
|α|≤m
with a cut-off function ω in n i.e., ω ∈ C0∞ ( n ), ω = 1 in a neighbourhood of the origin and
µ
µ
symbols b(x, ξ ) ∈ Scl ( n × n ), aα (ξ ) ∈ Scl ( n ), |α| ≤ m (in the notation of Section 2.1).
µ;0
Then we have a(x, ξ ) ∈ Scl ( n × n ), where
ξ;x
σψ (a)(x, ξ )
σe (a)(x, ξ )
=
=
ω(x)σψ (b)(x, ξ ) + (1 − ω(x))x −m
X
X
x α σψ (a)α (ξ ) ,
|α|≤m
aα (ξ ) ,
|α|=m
σψ,e (x, ξ )
=
X
σψ (aα )(ξ ) .
|α|=m
Let us now pass to spaces of global pseudo-differential operators in n . We formulate some
relations both for the classical and non-classical case and indicate it by subscript (clξ ;x ) at the
spaces of symbols and (cl) at the spaces of operators. Set
n
o
µ;δ
µ;δ
L (cl) n = Op (a) : a(x, ξ ) ∈ S(cl ) n × n ,
ξ;x
cf. (2.1). As it is well-known Op induces isomorphisms
µ;δ
µ;δ
(8)
Op : S(cl ) n × n −→ L (cl)
ξ;x
n
T
for all µ, δ ∈ . Recall that L −∞;−∞ ( n ) = µ,δ∈ L µ;δ ( n ) equals the space of all integral
n
n
operators with kernels in ( × ). Let us form the weighted Sobolev spaces
H s;% n = hxi−% H s n
308
D. Kapanadze – B.-W. Schulze
µ;δ
for s, % ∈ . Then every A ∈ L (cl) ( n ) induces continuous operators
(9)
A : H s;% n −→ H s−µ;%−δ n
for all s, % ∈ . Moreover, A restricts to a continuous operator
(10)
A : n −→ n .
µ;δ
For A ∈ L cl ( n ) we set
σψ (A) = σψ (a) ,
σe (A) = σe (a) ,
σψ,e (A) = σψ,e (a) ,
where a = Op−1 (A), according to relation (8).
R EMARK 3. The pseudo-differential operator calculus globally in n with weighted symbols and weighted Sobolev spaces can be generalized to the case of n × ñ 3 (x, x̃) with differ
β
ent weights for large |x| or |x̃|. Instead of (7) the symbol estimates are Dxα Dx̃α̃ Dξ a(x, x̃, ξ ) ≤
chξ iµ−|β| hxiδ−|α| hx̃iδ̃−|α̃| for all α, α̃, β and (x, x̃) ∈
(α, α̃, β). Such a theory is elaborated in Gerisch [6].
n+ñ , ξ ∈
n+ñ , with constants c =
We now formulate the basic elements of the pseudo-differential calculus on a smooth manifold M with conical exits to infinity, as it is necessary for boundary value problems below. For
simplicity we restrict ourselves to the case of charts that are conical “near infinity”. This is a
special case of a more general framework of Schrohe [17]. Our manifolds M are defined as
unions
k
[
M=K∪
[1 − ε, ∞) × X j
j =1
for some 0 < ε < 1, where X j , j = 1, . . . , k, are closed compact C ∞ manifolds, K is a
compact
smooth manifold with smooth
Skboundary ∂ K that is diffeomorphic to the disjoint union
Sk
X
,
identified
with
{1
−
ε}
×
j =1 X j by a gluing map. On the conical exits to infinity
j =1 j
[1 − ε, ∞) × X j we fix Riemannian metrics of the form dr 2 + r 2 g j , r ∈ [1 − ε, ∞), with
Riemannian metrics g j on X j , j = 1, . . . , k. Moreover, we choose a Riemannian metric on
M that restricts to these metrics on the conical exits. Since X j may have different connected
components we may (and will) assume k = 1 and set X = X 1 .
Let Vect (M) denote the set of all smooth complex vector bundles on M that we represent
over [1, ∞) × X as pull-backs of bundles on X with respect to the canonical projection [1, ∞) ×
X → X. Hermitian metrics in the bundles are assumed to be homogeneous of order 0 with
respect to homotheties along [1, ∞). On M we fix an open covering by neighbourhoods
(11)
U1 , . . . , U L , U L+1 , . . . , U N
with (U1 ∪ . . . U L ) ∩ ([1, ∞) × X) = ∅ and U j ∼
= (1 − ε, ∞) × U 1j , where U 1j j =L+1,...N is
an open covering of X. Concerning charts χ j : U j → V j to open sets V j , j = L + 1, . . . , N,
x ∈ V 1 for certain open sets
we choose them of the form V j = x ∈ n : |x| > 1 − ε, |x|
j
V j1 ⊂ S n−1 (the unit sphere in n ). Transition diffeomorphisms are assumed to be homogeneous
of order 1 in r = |x| for r ≥ 1.
Let us now define weighted Sobolev spaces H s;% (M, E) of distributional sections in E ∈
Vect (M) of smoothness s ∈ and weight % ∈ (at infinity). To this end, let ϕ j ∈ C ∞ (U j ),
309
Boundary value problems
j = L + 1, . . . , N, be a system of functions that are pull-backs χ ∗j ϕ̃ j under the chosen charts
χ j : U j → V j , where ϕ̃ j ∈ C ∞ ( n ), ϕ̃ j = 0 for |x| < 1 − 2ε , ϕ̃ j = 0 in a neighbourhood
x ∈ ∂U 1 , and ϕ̃ (λx) = ϕ̃ (x) for all |x| ≥ 1, λ ≥ 1. In addition we
of x : |x| > 1 − ε, |x|
j
j
j
−1
P
ϕ j on U 1j in such a way that Nj=L+1 ϕ j ≡ 1 for all points
prescribe the values of ϕ̃ j = χ ∗j
in M that correspond to |x| ≥ 1 in local coordinates. Given an E ∈ Vect (M) of fibre dimension
k we choose revitalizations that are compatible with χ j : U j → V j , j = L + 1, . . . , N,
τ j : E|U j → V j × k , and homogeneous of order 0 with respect to homotheties in r ∈ [1, ∞).
s (M, E) in an invariant way by
Then we can easily define H s;% (M, E) as a subspace of Hloc
s;%
n
k
s;%
n
k
requiring (τ j )∗ (ϕ j u) ∈ H ( , ) = H ( ) ⊗
for every L + 1 ≤ j ≤ N, where
(τ j )∗ denotes the push-forward of sections under τ j . Setting
n
(M, E) = proj lim H l;l (M, E) : l ∈ (12)
o
we get a definition of the Schwartz space of sections in E. By means of the chosen Riemannian
metric on M and the Hermitian metric in E we get L 2 (M, E) ∼
= H 0;0 (M, E) with a corresponding scalar product.
µ;δ
Moreover, observe that the operator spaces L (cl) ( n ) have evident (m × k)-matrix valued
µ;δ
µ;δ
variants L (cl) ( n ; k , m ) = L (cl) ( n )⊗ m ⊗ k . They can be localized to open sets V ⊂ n
that are conical in the large (i.e., x ∈ V , |x| ≥ R implies λx ∈ V for all λ ≥ 1, for some
R = R(V ) > 0). Then, given bundles E and F ∈ Vect (M) of fibre dimensions k and m,
respectively, we can invariantly define the spaces of pseudo-differential operators
µ;δ
L (cl) (M; E, F)
on M as subspaces of all standard pseudo-differential operators A of order µ ∈ , acting between
distributional sections in E and F, such that
(i) the push-forwards of ϕ j A ϕ̃ j with respect to the revitalizations of E|U j , F|U j belong to
µ;δ
L (cl) ( n ; k , m ) for all j = L + 1, . . . , N and arbitrary functions ϕ j , ϕ̃ j of the above
kind (recall that “cl” means classical in ξ and x),
(ii) ψ A ψ̃ ∈ L −∞;−∞ (M; E, F) for arbitrary ψ, ψ̃ ∈ C ∞ (M) with supp ψ ∩ supp ψ̃ = ∅
and ψ, ψ̃ homogeneous of order zero for large r (on the conical exits of M).
Here, L −∞;−∞ (M; E, F) is the space of all integral operators on M with kernels in
b π (M, E ∗ ) (integration on M refers to the measure associated with the chosen Rie (M, F)⊗
mannian metric; E ∗ is the dual bundle to E).
Note that the operators A ∈ L µ;δ (M; E, F) induce continuous maps
A : H s;% (M, E) −→ H s−µ;%−δ (M, F)
for all s, % ∈ , and A restricts to a continuous map (M, E) → (M, F).
To define the symbol structure we restrict ourselves to classical operators. First, to A ∈
µ;δ
L cl (M; E, F) we have the homogeneous principal symbol of order µ
(13)
∗ E −→ π ∗ F ,
σψ (A) : πψ
ψ
πψ : T ∗ M \ 0 −→ M .
310
D. Kapanadze – B.-W. Schulze
The exit symbol components of order δ and (µ, δ) are defined near r = ∞ on the conical exit
(R, ∞) × X for any R ≥ 1 − ε. Given revitalizations
(14)
τ j : E|U j −→ V j × k ,
ϑ j : F|U j −→ V j × l ,
of E, F on U j we have the symbols
σe (A j )(x, ξ ) for (x, ξ ) ∈ V j × n ,
σψ,e (A j )(x, ξ ) for (x, ξ ) ∈ V j ×
n \ 0 ,
where A j is the push-forward of A|U j with respect to (14). They behave invariant with respect
to the transition maps and define globally bundle homomorphisms
(15)
(16)
σe (A) : πe∗ E −→ πe∗ F ,
∗ E −→ π ∗ F ,
σψ,e (A) : πψ,e
ψ,e
∧
∧ −→ X ∞ ,
πe : T ∗ M| X ∞
∧
∧ −→ X ∞ .
πψ,e : (T ∗ M \ 0)| X ∞
In this notation X ∞ means the base of [R, ∞) × X “at infinity” with an obvious geometric
meaning (for instance, for M = n we have X ∞ ∼
= S n−1 , interpreted as the manifold that
n
∧
to a compact space at infinity), and X ∞ = + × X ∞ .
completes
µ,δ
An operator A ∈ L cl (M; E, F) is called elliptic if (13), (15) and (16) are isomorphisms.
−µ;−δ
(M; F; E) is called a parametrix of A if P A − I ∈ L −∞;−∞ (M;
An operator P ∈ L cl
−∞;−∞
E, E), A P − I ∈ L
(M; F, F).
µ;δ
T HEOREM 1. Let A ∈ L cl (M; E, F) be elliptic. Then the operator
A : H s;% (M, E) −→ H s−µ;%−δ (M, F)
−µ;−δ
is Fredholm for every s, % ∈ , and there is a parametrix P ∈ L cl
(M; F, E).
2.4. Calculus with operator-valued symbols
As noted in the beginning the theory of boundary value problems can be formulated in a convenient way in terms of pseudo-differential operators with operator-valued symbols. Given a
Hilbert space E with a strongly continuous group {κλ }λ∈ + of isomorphisms, acting on E,
we define the Sobolev space s ( q , E) of E-valued distributions of smoothness s ∈
to
2 1
R 2s −1
q
hηi κ (η)û(η)E dη 2 . Here,
be the completion of ( , E) with respect to the norm
κ(η) := κhηi , and û(η) = (Fy→η u)(η) is the Fourier transform in q . Given an open set
s (, E). Moreover, if E and
s
 ⊆ q there is an evident notion of spaces comp
(, E) and loc
e
E are Hilbert spaces with strongly continuous groups of isomorphisms {κλ }λ∈ + and {κ̃λ }λ∈ + ,
respectively, we define the symbol space
S µ U × q ; E, e
E , µ ∈ , U ⊆ p open, to be the set
of all a(y, η) ∈ C ∞ U × q , E, e
E with E, e
E being equipped with the norm topology
such that
n
o
−1
β
µ−|β|
≤ chηi
κ̃ (η) D αy Dη a(y, η) κ(η)
E,eE
for all α ∈ p , β ∈ q and all y ∈ K for arbitrary K U , η ∈ q , with constants c =
c(α, β, K ) > 0.
E denote the set of all f (y, η) ∈ C ∞ U × ( q \ 0), E, e
E
Let S (µ) U × ( q \ 0); E, e
such that f (y, λη) = λµ κ̃λ f (y, η)κλ−1 for all λ ∈ + , (y, η) ∈ U × ( q \ 0). Furthermore,
µ
E (the space of classical operator-valued symbols of order µ) defined to
let Scl U × q ; E, e
311
Boundary value problems
be the set of all a(y, η) ∈ C ∞ U × q , E, e
E such that there are elements a(µ− j ) (y, η) ∈
E , j ∈ , with
S (µ− j ) U × ( q \ 0); E, e
a(y, η) −
N
X
j =0
χ(η)a(µ− j ) (y, η) ∈ S µ−(N+1) U × q
for all N ∈ (with χ being any excision function in η). Set σ∂ (a)(y, η) := a(µ) (y, η) for the
homogeneous principal symbol of a(y, η) of order µ.
In the case U =  × ,  ⊆ q open, the variables in U will also be denoted by (y, y 0 ).
Similarly to (2) we set
n
o
µ
µ
E = Op (a) : a(y, y 0 , η) ∈ S(cl)  ×  × q ; E, e
L (cl) ; E, e
(17)
E ,
where Op refers to the
on , while the values of amplitude functions are
action in theµy-variables
operators in E, e
E we set σ∂ (A)(y, η) = a(µ) (y, y 0 , η)| y 0 =y , called
E . For A ∈ L cl ; E, e
the homogeneous principal symbol of A of order µ. Every A ∈ L µ ; E, e
E induces continuous
operators
s−µ
s
A : comp
E
(, E) −→ loc , e
for each s ∈ . More details of this kind on the pseudo-differential calculus with operator-valued
symbols may be found in [26], [30]. In particular, all elements of the theory have a reasonable
generalization to Fréchet spaces E and e
E, written as projective limits of corresponding scales of
Hilbert spaces, where the strong continuous actions are defined by extensions or restrictions to
the Hilbert spaces of the respective scales [30], Section 1.3.1. This will tacitly be used below.
Let us now pass to an analogue of the global pseudo-differential
calculus of Section 2.3 with
E for µ, δ ∈ denote the space of all a(y, η) ∈
operator-valued symbols.
Let S µ;δ q × q ; E, e
C ∞ q × q , E, e
E that satisfy the symbol estimates
n
o
−1
β
κ̃ (η) D αy Dη a(y, η) κ(η)
E,eE ≤ chηiµ−|β| hyiδ−|α|
for all α, β ∈ q , (y, η) ∈ 2q , with constants c = c(α, β) > 0. This space is Fréchet, and
again, like for standard symbols, we have generalizations of the structures from the local spaces
to the global ones. Further details are given in [30], [5], see also [31].
We now define operator-valued symbols that are classical both in η and y, where the group
actions on E, e
E are taken as the identities for all λ ∈ + when y is treated as a covariable.
Similarly to the scalar case we set
(µ)
Sη
= a(y, η) ∈ C ∞
q ×
for all λ > 0, (y, η) ∈
(δ) S y = a(y, η) ∈ C ∞
and
(µ;δ)
Sη;y
q \ 0 ,
E, e
E : a(y, λη) = λµ κ̃λ a(y, η)κλ−1
q ×
q \ 0 ,
q \ 0 × q ,
E, e
E : a(λy, η) = λδ a(y, η)
for all λ > 0, (y, η) ∈ q \ 0 × q ,
= a(y, η) ∈ C ∞
q \ 0 ×
for all λ > 0, τ > 0, (y, η) ∈
q \ 0 ,
E, e
E : a(λy, τ η) = λδ τ µ κ̃τ a(y, η)κτ−1
q \ 0 ×
q \ 0 .
312
D. Kapanadze – B.-W. Schulze
(µ);δ
(µ)
Moreover, let Sη;cl defined to be the subspace of all a(y, η) ∈ Sη such that a(y, η)||η|=1 ∈
y
δ
δ
q ; E, e
q ; E, e
E where in Scl
E the spaces E and e
E are endowed with the
C ∞ S q−1 , Scl
y
y
µ;(δ)
identities for all λ ∈ + as the corresponding group actions , and Scl ;y the subspace of all
η
µ
(δ)
q ; E, e
a(y, η) ∈ S y such that a(y, η)||y|=1 ∈ C ∞ S q−1 , Scl
E .
η
[µ]
∞
q × q,
Let Sη defined to be the set of all a(y, η) ∈ C
E, e
E such that there is a
c = c(a) with
a(y, λη) = λµ κ̃λ a(y, η)κλ−1
λ ≥ 1 , y ∈ q , |η| ≥ c .
for all
∞
Similarly, the space S [δ]
y is defined to be the set of all a(y, η) ∈ C
that there is a c = c(a) with
a(λy, η) = λδ a(y, η)
q ×
q,
λ ≥ 1 , |y| ≥ c , η ∈ q .
for all
[µ]
µ
E, eE such
(µ)
Clearly, for every a(y, η) ∈ Sη there is a unique element σ∂ (a) ∈ Sη with a(y, η) =
µ
σ∂ (a)(y, η) for all (y, η) ∈ q × q with |η| ≥ c for a constant c = c(a) > 0. Analogously,
(δ)
δ
δ
for every b(y, η) ∈ S [δ]
y there is a unique σe0 (b) ∈ S y with b(y, η) = σe0 (b)(y, η) for all
[δ]
(y, η) ∈ q × q with |y| ≥ c for some c = c(b) > 0. Set S µ;[δ] = S µ;δ ∩ S y , S [µ];δ =
[µ]
µ;[δ]
S µ;δ ∩ Sη . Moreover, let Scl
η
denote the subspace of all a(y, η) ∈ S µ;[δ] such that there are
PN
[µ−k]
µ−(N+1);δ for all
elements ak (y, η) ∈ Sη
∩ S [δ]
y , k ∈ , with a(y, η) − k=0 ak (y, η) ∈ S
[µ];δ
[µ];δ
N ∈ . Similarly, we define Scl
to be the subspace of all a(y, η) ∈ S
such that there are
y
PN
[µ]
[δ−l]
elements bl (y, η) ∈ Sη ∩ S y
, l ∈ , with a(y, η) − l=0 al (y, η) ∈ S µ;δ−(N+1) for all
N ∈ .
µ;δ
Let Scl defined to be the set of all a(y, η) ∈ S µ;δ such that there are elements ak (y, η) ∈
η
PN
ak (y, η) ∈ S µ−(N+1);δ for all N ∈ .
S [µ−k];δ , k ∈ , satisfying the relation a(y, η) − k=0
µ;δ
Analogously, define Scl to be the set of all a(y, η) ∈ S µ;δ such that there are elements
y
PN
al (y, η) ∈ S µ;δ−(N+1) for
al (y, η) ∈ S µ;[δ−l] , l ∈ , satisfying the relation a(y, η) − l=0
all N ∈ .
[µ]
[µ];δ ∩ S µ;[δ] .
Note that Sη ∩ S [δ]
y ⊂S
µ;δ
D EFINITION 2. The space Scl
η;y
q × q ; E, e
E of classical (in y and η) symbols of order
q × q ; E, e
E such that there are sequences
[µ−k];δ
µ;[δ−l]
, k ∈ , and bl (y, η) ∈ Scl
ak (y, η) ∈ Scl
, l ∈ , with
y
η
(µ; δ) is defined to be the set of all a(y, η) ∈ S µ;δ
a(y, η) −
N
X
µ−(N+1);δ
ak (y, η) ∈ Scl
y
and a(y, η) −
N
X
µ;δ−(N+1)
bl (y, η) ∈ Scl
η
l=0
k=0
for all N ∈ .
[µ];δ
R EMARK 4. We have Scl
y
q ; E, e
E .
µ;δ
⊂ Scl
η;y
µ;[δ]
, Scl
η
µ;δ
⊂ Scl
η;y
µ;δ
µ;δ
, where Scl
= Scl
η;y
η;y
q ×
313
Boundary value problems
µ;δ
µ−k
µ−k
Given a ∈ Scl we set σ∂ (a) = σ∂
η;y
Definition 2. This gives us well-defined maps
µ−k
σ∂
µ;δ
: Scl
η;y
(µ−k);δ
−→ Sη;cl
y
, k∈,
δ−l
(ak ), σeδ−l
0 (a) = σe0 (bl ), with the notation of
µ;δ
µ;(δ−l)
σeδ−l
: Scl −→ Scl ;y
0
η
η;y
, l∈.
µ−k
µ−k δ−l
The corresponding homogeneous components σeδ−l
σ∂ (a) and σ∂
σe0 (a) in y and η,
0
respectively, are compatible in the sense
µ−k
µ−k
µ−k;δ−l
σeδ−l
σ∂ (a) = σ∂
σeδ−l
(a)
0
0 (a) =: σ∂,e0
µ;δ
for all k, l ∈ . For a(y, η) ∈ Scl
η;y
µ
q ×
q ; E, e
E we set
µ;δ
σ∂ (a) = σ∂ (a) , σe0 (a) = σeδ0 (a) , σ∂,e0 (a) = σ∂,e0 (a)
and define
(18)
σ (a) = σ∂ (a), σe0 (a), σ∂,e0 (a) .
µ; δ
q × q ; E, e
E and σ (a) = 0 implies a(y, η) ∈
a(y, η) ∈ Scl
η; y
µ−1;δ−1 q
Scl
× q ; E, e
E . Moreover, if χ is any excision function in q , we have χ(η)σ∂ (a)
η;y
µ−1;δ−1 q
(y, η) + χ(y){σe0 (a)(y, η) − χ(η)σ∂,e0 (a)(y, η)} = a(y, η) mod Scl
× q ; E, e
E .
R EMARK 5.
η;y
(µ);δ
(µ;δ)
µ;(δ)
More generally, let p∂ (y, η) ∈ Sη;cl , pe0 (y, η) ∈ Scl ;y and p∂,e0 (y, η) ∈ Sη;y be arbitrary
η
y
elements with σe0 ( p∂ ) = σ∂ ( pe0 ) = p∂,e0 . Then a(y, η) = χ(η) p∂ (y, η) + χ(y){ pe0 (y, η) −
µ;δ
q × q ; E, e
χ(η) p∂,e0 (y, η)} ∈ Scl
E , and we have σ∂ (a) = p∂ , σe0 (a) = pe0 , σ∂,e0 (a) =
η;y
p∂,e0 .
µ;δ
q × q ; E, e
R EMARK 6. An element a(y, η) ∈ Scl
E is uniquely determined by the
η;y
sequence
o
n
δ− j
µ− j
(19)
σ∂ (a)(y, η), σe0 (a)(y, η)
,
j∈
mod S −∞;−∞
q × q ; E, e
E .
In fact, by the construction of Remark 5 we can form
n
o
µ
µ;δ
a0 (y, η) = χ(η)σ∂ (a)(y, η) + χ(y) σeδ0 (a)(y, η) − χ(η)σ∂,e0 (a)(y, η)
µ−1;δ−1
q ×
for any fixed excision function χ, where b1 (y, η) = a(y, η) − a0 (y, η) ∈ Scl
η;y
q ; E, e
(b1 )(y, η) is completely determined by (19), and we can
E . Then σ∂ν−1 (b1 )(y, η), σeδ−1
0
form
o
n
µ−1;δ−1
µ−1
(b1 )(y, η) − χ(η)σ∂,e0
(b1 )(y, η) .
a1 (y, η) = χ(η)σ∂ (b1 )(y, η) + χ(y) σeδ−1
0
µ−2;δ−2 q
µ−2;δ−2 q
Then b2 := b1 − a1 ∈ Scl
× q ; E, e
E or b2 = a − a0 − a1 ∈ Scl
×
η;y
η;y
q ; E, e
E . Continuing this procedure successively we get a sequence of symbols ak (y, η) ∈
314
D. Kapanadze – B.-W. Schulze
µ−k;δ−k
Scl
η;y
q
P∞ ×
q ; E, e
E , k ∈
q ×
, with a(y, η) −
PN
µ−(N+1);δ−(N+1)
k=0 ak (y, η) ∈ Sclη;y
q ; E, e
E for all N. Thus we can recover a(y, η) as an asymptotic sum a(y, η) ∼
k=0 ak (y, η).
According to the generalities about symbols with exit behaviour,
cf.
[5], Proposition 1.5,
P∞
y,η
we can produce a(y, η) as a convergent sum a(y, η) =
k=0 χ ck ak (y, η), where χ is
an excision function in q × q and ck a sequence of non-negative reals, tending to infinity
sufficiently fast, as k → ∞.
µ;δ
η;y
µ+ν;δ+κ
(ab)(y, η) ∈ Scl
η;y
ν;κ
q;E ,e
q ×
0 E , b(y, η) ∈ Sclη;y (
q ×
R EMARK 7. a(y, η) ∈ Scl
q × q ; E, e
E , and we have
q ; E, E ) implies
0
σ (ab) = σ (a)σ (b)
(20)
with componentwise multiplication.
µ;δ
q × q ; E, e
E can also be
R EMARK 8. Operator-valued symbols of the classes Scl
η;y
ν;κ
multiplied by scalar ones, namely b(y, η) ∈ Scl
( q × q ). We then obtain (ab)(y, η) ∈
η;y
µ+ν;δ+κ q
Scl
× q ; E, e
E including the symbol relation (20). In particular, we have
η;y
µ;δ
Scl
η;y
q ×
Set
(21)
µ;δ
L (cl)
0;0
q ; E, e
E = hηiµ hyiδ Scl
η;y
q ×
n
µ
q ; E, e
E = Op (a) : a(y, η) ∈ S(cl )
η;y
q ; E, e
E .
q ×
o
q ; E, e
E .
Notice that the subscript “cl” on the left hand side means “classical” both in η and y (in contrast
to the corresponding notation in (17)). Similarly to (8) we have isomorphisms
(22)
µ;δ
Op : S(cl
η;y )
q ×
µ;δ
q ; E, e
E −→ L (cl)
q ; E, e
E
for all µ, δ ∈ . This is a consequence of the same kind of oscillatory integral arguments as in
the scalar case, cf. [5], Proposition 1.11.
Set
s;%
q , E = hyi−% s
q , E ,
E
endowed with the norm kuk s;% ( q ,E) = khyi% uk s ( q ,E) . Then every A ∈ L µ;δ q ; E, e
induces continuous operators
(23)
A:
s;%
q , E −→
s−µ;%−δ
q,e
E
for all s, % ∈ , cf. [5], Proposition 1.21. In addition, A is continuous in the sense
(24)
cf. [5], Proposition 1.8.
A:
q , E −→
q,e
E ,
315
Boundary value problems
µ;δ
R EMARK 9. If we have an operator Op (a) for a(y, η) ∈ S(cl )
η;y
any α ∈ and form the operator hyiα Op (a)hyi−α : ( q , E) → µ;δ
q × q ; E, e
E such that
unique symbol aα (y, η) ∈ S
(clη;y )
q ×
q ; E, e
E , choose
q,e
E . Then, there is a
hyiα Op (a)hyi−α = Op (aα ) .
(25)
Thus, (25) has an extension by continuity to a continuous operator (23) for all s, % ∈ .
µ;δ
This is a direct consequence of hyiα Op (a)hyi−α ∈ L (cl)
phism (22).
q ; E, e
E and of the isomor-
3. Boundary value problems in the half-space
3.1. Operators on the half-axis
The operator-valued symbols in the present set-up will take their values in a certain algebra of
operators on the half-axis. The essential features of this algebra may be described in terms of
block-matrices
(26)
a=
op+ (a)
0
0
0
H s−µ ( + )
Hs ( +)
⊕
⊕
−→
+g:
N−
N+
for µ ∈ and N− , N+ ∈ (s ∈ will be specified below). The operator op+ (a) = r+ op (a)e+
µ
is defined for symbols a(τ ) ∈ Scl ( )tr , i.e., symbols of order µ with the transmission property,
where r+ and e+ are restriction and extension operators as in (4), while op ( . ) denotes the
pseudo-differential action on , i.e.,
ZZ
0
op (a)u(t) =
ei(t −t )τ a(τ )u(t 0 ) dt 0 d̄τ .
(Here, for the moment, we consider symbols with constant coefficients). Moreover, g is a Green
operator of type d ∈ on the half-axis, defined as a sum
!
d
j
X
∂
0
t
g = g0 +
(27)
gj
0 0
j =1
for continuous operators
L 2( + )
⊕
−→
gj :
N−
+
⊕
N+
with
L 2( + )
∗
gj :
⊕
−→
N+
+
⊕
N−
here, ∗ denotes the adjoint with respect to the corresponding scalar products in L 2 ( + )⊕ N± ,
and we set + = ( )| . Let 0 d + ; N− , N+ denote the space of all such operators
+
(27), and let D µ,d + ; N− , N+ denote the space of all operators (26), s > d − 21 , for arbitrary
µ
a ∈ Scl ( )tr and g ∈ 0 d + ; N− , N+ for N− = N+ = 0 we write 0 d + and D µ,d + ,
respectively . The following properties are part of the boundary symbol calculus for boundary
value problems, cf. Boutet de Monvel [3], or Rempel and Schulze [16].
316
D. Kapanadze – B.-W. Schulze
T HEOREM 2. a ∈ D µ,d + ; N0 , N+ and b ∈ D ν,e
D µ+ν,h + ; N− , N+ for h = max(ν + d, e).
+ ; N− , N0
implies ab ∈
T HEOREM 3. Let a ∈ D µ,d + ; N− , N+ where a(τ ) 6= 0 for all τ ∈ , and assume
that a defines an invertible operator (26) for some s0 ∈ , s0 > max(µ, d) − 12 . Then (26) is
invertible for all s ∈ , s > max(µ, d) − 21 . In addition
+
⊕
a:
N−
is invertible, and we have a−1 ∈ D −µ,(d−µ)
−→
+
⊕
N+
+
+
+ ; N+ , N− ; here ν = max(ν, 0).
3.2. Boundary symbols associated with interior symbols
n that gives rise to operator-valued
In this section we introduce a special symbol class on
symbols in the sense of Section 2.4.
µ
µ;δ
Let Scl ( n × n ) defined to be the subspace of all a(x, ξ ) ∈ Scl ( n × n ) vanishing
ξ;x
ξ;x
on the set
TR := x = (y, t) ∈ n : |x| ≥ R, |t| > R|y|
(28)
for some constant R = R(a). In an analogous manner we define the more general space
µ;δ
µ;δ
µ;δ
S µ;δ ( n × n ) . Set Scl ( n × n )tr, = Scl ( n × n ) ∩ Scl ( n × n )tr and
ξ;x
ξ;x
o
n
n
n
µ;δ
µ;δ
n
n × n)
n−1 ×
n
Scl
+.
tr, , + =
+×
tr, = a = ã| + × n : ã(x, ξ ) ∈ Sclξ;x (
ξ;x
n
µ;δ
µ;δ
µ;δ
n
Similarly, we can define the spaces S(cl ) ( n × n ) , Scl ( n × n )tr, , Scl
+×
tr, ,
ξ
ξ
ξ
µ;δ
where clξ means symbols that are only classical in ξ . For a ∈ Scl ( n × n ) we form
ξ
RR i(t −t 0 )τ
op(a)(y, η)u(t) =
e
a(y, t, η, τ )u(t 0 )dt 0 d̄τ and set op+ (a)(y, η) = r+ op(a)(y, η)e+ ,
where r+ and e+ are of analogous meaning on as the corresponding operators r+ and e+ in
Section 2.2.
µ;δ n
n
We also form op+ (a)(y, η) for symbols a(y, t, η, τ ) ∈ Scl
+×
tr, ; the extension
ξ
e+ includes an extension of a to a corresponding ã, though op+ (a)(y, η) does not depend on the
choice of ã.
P ROPOSITION 1. a(x, ξ ) ∈ S µ;δ ( n × n ) implies
n−1 ; H s ( ), H s−µ ( )
n−1 ×
op (a)(y, η) ∈ S µ;δ
for every s ∈ .
The simple proof is left to the reader.
µ;δ
P ROPOSITION 2. a(x, ξ ) ∈ Scl
ξ
op+ (a)(y, η) ∈ S µ;δ
n
+×
n
n−1 ×
tr, implies
n−1 ; H s (
+ ), H
s−µ (
+)
317
Boundary value problems
for every s > − 21 and
op+ (a)(y, η) ∈ S µ;δ
n−1 ; n−1 ×
+ ,
+
.
The proof of this result can be given similarly to Theorem 2.2.11 in [20].
µ;δ n
n
Given a symbol a(x, ξ ) ∈ Scl
+×
tr we call the operator family
ξ
op+ (a|t =0 )(y, η) : H s ( + ) −→ H s−µ ( + ) ,
s > − 21 or op+ (a|t =0 )(y, η) : + −→ + the boundary symbol associated with
a(x, ξ ).
µ;δ
R EMARK 10. For a(x, ξ ) ∈ Scl
ξ;x
n
+×
µ;δ
op+ (a|t =0 )(y, η) ∈ Scl
n−1 ×
η;y
µ;δ
s > − 21 , and op+ (a|t =0 )(y, η) ∈ Scl
η;y
µ;δ
For a(x, ξ ) ∈ Scl
ξ;x
n
n−1 ×
tr we have
n−1 ; H s (
s−µ (
+ ), H
+) ,
n−1 ; + ,
n
+×
+
.
n we form
σ (a) = σψ (a), σe (a), σψ,e (a); σ∂ (a), σe0 (a), σ∂,e0 (a) ,
for σψ (a) = σψ (ã)|
n
n \0) ,
+ ×(
σe (a) = σe (ã)|
µ;δ
with an ã ∈ Scl ( n × n )tr where a = ã|
ξ
n
n,
+ \0 ×
n
n,
+×
σψ,e (a) = σψ,e (ã)|
and
σ∂ (a)(y, η)
:=
σ∂ (op+ (a|t =0 ))(y, η) ,
(y, η) ∈ n−1 ×
σe0 (a)(y, η)
:=
σe0 (op+ (a|t =0 ))(y, η) ,
(y, η) ∈
σ∂,e0 (a)(y, η)
:=
σ∂,e0 (op+ (a|t =0 ))(y, η) ,
(y, η) ∈
n
n \0)
+ \0 ×(
n−1 \ 0 ,
n−1 \ 0 ×
n−1 \ 0 ×
n−1 ,
n−1 \ 0 ,
where the right hand sides are understood in the sense of (18). (Here, e0 is used for the exit
symbol components along y ∈ n−1 , while e indicates exit symbol components of interior
symbols with respect to x ∈ n ).
n
µ;δ
n into a -part and an interior part by
It is useful to decompose symbols in Scl
+×
ξ;x
a suitable partition of unity.
D EFINITION 3. A function χ ∈ C ∞
n
+ if
(i) 0 ≤ χ (x) ≤ 1 for all x ∈
n
+ is called a global admissible cut-off function in
n
+,
(ii) there is an R > 0 such that χ (λx) = χ (x) for all λ ≥ 1, |x| > R,
e and
(iii)g χ (x) = 1 for 0 ≤ t < ε for some ε > 0, χ (x) = 0 for |x| ≥ R, t > R|y|
e
χ (x) = 0 for |x| ≤ R, t > ε̃ for some ε̃ > ε and non-negative reals R and R.
n
n
A function χ ∈ C ∞ + is called a local admissible cut-off function in + if it has the
properties (i), (ii) and
318
D. Kapanadze – B.-W. Schulze
n for some ω̃ ∈ C ∞ ( n ), 0 ≤ ω̃(x) ≤ 1
0
+
n and ω̃(x) = 1 in a neighbourhood of x = 0 and ν = ~| n for some
(iii)l χ (x) = ν(x)(1 − ω(x)) for ω = ω̃|
for all x ∈
+
~ ∈ C ∞ ( n \ 0) with ~(λx) = ~(x) for all λ ∈ + , x ∈ n \ 0, such that for some
y ∈ n−1 with |y| = 1, and certain 0 < ε < ε̃ < 21 we have ~(x) = 1 for all
n
n
x ∈ S n−1 ∩ + with |x − y| < ε and ~(x) = 0 for all x ∈ S n−1 ∩ + with |x − y| > ε̃.
n
µ;δ
n and any (local or global) admissible cut-off function χ we
For a(x, ξ ) ∈ Scl
+×
ξ;x
then get a decomposition
a(x, ξ ) = χ (x)a(x, ξ ) + (1 − χ (x))a(x, ξ )
µ;δ
where a (x, ξ ) := χ (x)a(x, ξ ) ∈ Scl
ξ;x
n .
n
+×
n
µ;δ
and (1 − χ (x))a(x, ξ ) ∈ Sclξ;x
n
+×
R EMARK 11. The operator of multiplication Mχ by any χ ∈ C ∞ ( n ) with χ (λx) =
n
χ (x) for all |x| > R for some R > 0 and λ ≥ 1, can be regarded as an element in L 0;0
cl ( ).
µ;δ
µ;δ
In other words, we have Mχ A, AMχ ∈ L (cl) ( n ) for every A ∈ L (cl) ( n ). If χ and χ̃
are two such functions with supp χ ∩ supp χ̃ = ∅ we have χ A χ̃ ∈ L −∞;−∞ ( n ) for
µ;δ
arbitrary A ∈ L (cl) ( n ). A similar observation is true in the operator-valued case.
3.3. Green symbols
Pseudo-differential boundary value problems are described by a symbol structure that reflects
an analogue of Green’s function and generates boundary (and potential) conditions of elliptic
boundary value problems. This is summarized by the following definition.
µ,0;δ n−1
× n−1 ; N− , N+ of Green symbols of order
D EFINITION 4. The space G
µ ∈ , type 0 and weight δ ∈ is defined to be space of all operator-valued symbols
µ;δ
g(y, η) ∈ Scl
η
µ;δ
η
N+ N− .
n−1 ; L 2 (
+) ⊕
N− , + ⊕
n−1 ×
n−1 ; L 2 (
+) ⊕
N+ , + ⊕
such that
g ∗ (y, η) ∈ Scl
n−1 ×
µ,d;δ n−1
Moreover, the space G
× n−1 ; N− , N+ of Green symbols of order µ ∈
type d ∈ and weight δ ∈ is defined to be the space of all operator families of the form
!
d
j
X
∂
0
t
(29)
g j (y, η)
g(y, η) = g0 (y, η) +
0 0
,
j =1
for arbitrary g j ∈
µ− j,0;δ
G
n−1 ×
n−1 ; N , N , j = 0, . . . , d.
−
+
µ;δ
µ,d;δ n−1
P ROPOSITION 3. Every g(y, η) ∈ G
× n−1 ; N− , N+ belongs to Scl
η
× n−1 ; H s ( + ) ⊕ N− , + ⊕ N+ for every real s > d − 12 .
n−1
319
Boundary value problems
The specific aspect in our symbol calculus near exits to infinity consists of classical el
µ,d;δ n−1
× n−1 ; N− , N+ denote the
ements, here with respect to y ∈ n−1 . Let G,cl
µ,d;δ n−1
subspace of all g(y, η) ∈ G
× n−1 ; N− , N+ of the form (29) for g j (y, η) ∈
µ− j,0;δ n−1
µ,0;δ n−1
× n−1 ; N− , N+ , where G,cl
× n−1 ; N− , N+ is defined to be
G,cl
the space of all
µ;δ
g(y, η) ∈ Scl
η;y
with
µ;δ
g ∗ (y, η) ∈ Scl
η;y
n−1 ×
n−1 ; L 2 (
+) ⊕
N− , + ⊕
n−1 ×
n−1 ; L 2 (
+) ⊕
N+ , + ⊕
(30)
n−1 ×
µ;δ
Scl
η;y
n−1 ; N , N ⊂
−
+
n−1 ×
n−1 ; H s (
+) ⊕
N− , N− .
Similarly to Proposition 3 we have
µ,d;δ
G,cl
N+ + ⊕
for all s > d − 21 . Applying (18) we then get the triple of principal symbols
(31)
σ (g) = σ∂ (g), σe0 (g), σ∂,e0 (g) .
N+ R EMARK 12. There is a direct analogue of Remark 5 in the framework of Green symbols
that we do not repeat in this version in detail. Let us only observe that we can recover g(y, η)
µ−1,d;δ−1 n−1
× n−1 ; N− , N+ by χ(η)σ∂ (g)(y, η)+χ(y){σe0 (g)(y, η)−
from (31) mod G,cl
µ,d;δ n−1
× n−1 ; N− , N+ .
χ(η)σ∂,e0 (g)(y, η)} ∈ G,cl
L EMMA 1. Let φ(y, t) ∈ C ∞ q × + and assume that for some δ ∈ the following
estimates hold: supt ∈ |D αy φ(y, t)| ≤ chyiδ−|α| for all y ∈ q and all α ∈ q , with constants
+
c = c(a) > 0. Then the operator Mφ(y,t ) of multiplication by φ(y, t) fulfils the relation
Mφ(y,t ) ∈ S 0;δ q × q ; L 2 ( + ), L 2 ( + ) .
(32)
Proof. We have to check the symbol estimates
o
n
−1
β
κ (η) D αy Dη Mφ(y,t ) κ(η)
L(
2
+)
≤ chyiδ−|α|
for all α, β ∈ q and all (y, η) ∈ q × q with suitable c = c(α, β) > 0. Because Mφ(y,t ) is independent of η it suffices to consider β = 0. Using κ −1 (η)D αy Mφ(y,t ) κ(η) = D αy Mφ(y,t hyi−1)
we get for u ∈ L 2 ( + )
n
o
−1
β
κ (η) D αy Dη Mφ(y,t ) κ(η)u(t)
L 2(
+)
=
≤
α
D y φ y, thyi−1 u(t) 2
L ( +)
α
−1
sup D y φ y, thyi
kuk L 2 (
t∈
≤
+
chyiδ−|α| kuk L 2 (
+)
.
+)
320
D. Kapanadze – B.-W. Schulze
L EMMA 2. Let φ(y, t) ∈ C ∞ q × + be a function such that there are constants m, δ ∈
α M
such that supt ∈ D y Dt φ(y, t) ≤ chyiδ−|α| htim−M for all y ∈ q , t ∈ + and all
+
α ∈ q , M ∈ , with constants c = c(α, M) > 0. Then we have
(33)
Mφ(y,t ) ∈ S 0,δ q × q ; + , + .
Proof. Let us express the Schwartz space as a projective limit
+ = proj lim hti−k H k ( + ) : k ∈ .
An operator b is continuous in ( + ) if for every k ∈ there is an l = l(k) ∈ such that
kbk
ht i
−l H l (
−k k
+ ),ht i H ( + )
≤c
for certain c = c(k, l) > 0. The symbol estimates for (33) require for every k ∈ an l = l(k) ∈
such that
o
n
−1
(34)
≤ chyiδ−|α| kukht i−l H l ( + ) ,
κ (η) D αy Mφ(y,t ) κ(η)u −k k
ht i
H (
+)
for constants c > 0 depending on k, l, α, for all α and k. Similarly to the proof of Lemma 1 the
η-derivatives may be ignored. Estimate (34) is equivalent to
k α
≤ chyiδ−|α| kvk H l ( + )
(35)
hti D y φ y, thηi−1 hti−l v k
H (
+)
for all v ∈ H l ( + ). Setting l = k + m + for m + = max(m, 0) we get (35) from the system of
simpler estimates
o
n
+
j
Dt hti−m D αy φ y, thηi−1 v(t) L 2(
+)
≤ chyiδ−|α| kvk H l (
+)
+
j
for all 0 ≤ j ≤ k. The function Dt hti−m D αy φ(y, thηi−1 )v(t) is a sum of expressions of the
form
+
j
j
v j1 j2 j3 (t) = chti−m − j1 hηi− j2 Dt 2 D αy φ y, thηi−1 Dt 3 v(t)
for j1 + j2 + j3 = j and constants c = c( j1 , j2 , j3 ). We now employ the assumption on
j φ, namely supt ∈ D αy Dt 2 φ y, thηi−1 ≤ hyiδ−|α| hthηi−1 im− j2 . Using hthηi−1 im− j2 ≤
+
htim− j2 for m − j2 ≥ 0 and hthηi−1 im− j2 ≤ 1 for m − j2 < 0 we immediately get
j kv j1 j2 j3 (t)k L 2 ( + ) ≤ chηiδ−|α| Dt 3 v 2
L (
for all y ∈
+)
q , with different constants c > 0. This gives us finally the estimates (35).
3.4. The algebra of boundary value problems
µ,d;δ n−1
D EFINITION 5. cl
× n−1 ; N− , N+ for (µ, d) ∈ × , δ ∈ , is defined to
be the set of all operator families
op+ (a)(y, η) 0
a(y, η) =
+ g(y, η)
0
0
µ;δ
for arbitrary a(x, ξ ) ∈ Scl
ξ;x
n
+×
n
tr, and g(y, η) ∈
µ,d;δ
G,cl
n−1 ×
n−1 ; N , N .
−
+
321
Boundary value problems
Observe that the components of
(36)
for a(y, η) ∈
σ (a) := σψ (a), σe (a), σψ,e (a); σ∂ (a), σe0 (a), σ∂,e0 (a)
µ,d;δ n−1
× n−1 ; N− , N+ , given by
cl
σψ (a) := σψ (a) ,
and
σ∂ (a)
σe0 (a)
σ∂,e0 (a)
=
=
=
σe (a) := σe (a) ,
σψ,e (a) := σψ,e (a) ,
σ∂ (op+ (a|t =0 ))
0
0
0
σe0 (op+ (a|t =0 ))
0
0
0
σ∂,e0 (op+ (a|t =0 ))
0
+ σ∂ (g) ,
+ σe0 (g) ,
0
+ σ∂,e0 (g)
0
µ−1,d;δ−1
n−1 ×
are uniquely determined by a(y, η), and that σ (a) = 0 implies a(y, η) ∈ cl
n−1 ; N , N .
−
+
ν,e;% n−1
µ,d;δ n−1
× n−1 ; N0 , N+ and b(y, η) ∈ cl
×
Moreover, a(y, η) ∈ cl
n−1 ; N , N implies (ab)(y, η) ∈ µ+ν,h;δ+% n−1 × n−1 ; N , N
−
−
+ for h = max(ν +
0
cl
d, e) where σ (ab) = σ (a)σ (b) (with componentwise multiplication).
n
Next we define spaces of smoothing operators in the half-space. The space −∞,0;−∞ + ;
N− , N+ is defined to be the set of all block matrix operators
+
+
A K
⊕
⊕
=
:
−→
,
T C
n−1 , N−
n−1 , N+
where
RR
n
n
0 0
0 0
0 0
0 0
(i) Au(y, t) =
n a(y, t, y , t )u(y , t ) dy dt for certain a(y, t, y , t ) ∈ +× +
+
n
= ( n × n )| n × n , u ∈ + ,
+
+
R
P N−
0
0
0
(ii) K v(y, t) =
K
v
n−1 kl (y, t, y )vl (y ) dy for certain
l=1 l l (y, t) for K l vl (y, t) =
n
0
n−1
n
n−1
n
kl (y, t, y ) ∈ + ×
=
×
n−1 , for v = (vl )l=1,...,N− ∈
+×
n−1
N
, − ,
RR
n bm (y, y 0 , t 0 )u(y 0 , t 0 ) dy 0 dt 0 for cer(iii) T u(y) = (Tm u(y))m=1,...,N+ for Tm u(y) =
+
n
= n−1 × n n−1 × n , m = 1, . . . , N+ for
tain bm (y, y 0 , t 0 ) ∈ n−1 × +
+
n
u∈ + ,
P
N− R
0 )v (y 0 ) dy 0
for certain clm (y, y 0 ) ∈ n−1 ×
c
(y,
y
(iv) Cv(y) =
lm
l
l=1
m=1,...,N+
n−1 , l = 1, . . . , N , m = 1, . . . , N .
−
+
n
−∞,d;−∞
+ ; N− , N+ for d ∈ is the space of all operators
!
d
j
X
∂t 0
= 0+
j
0 0
j =1
322
D. Kapanadze – B.-W. Schulze
for arbitrary j ∈ −∞,0;−∞
n
+ ; N− , N+ , j = 0, . . . , d.
µ;δ
µ;δ
Let L cl ( n )^ denote the subspace of all P ∈ L cl ( n ) such that there is an R > 0 with
φ Pψ = 0 for all φ, ψ ∈ C0∞ ( n ) with supp φ, supp ψ ⊆ n \ TR , cf. (28). Moreover, we set
µ;δ
e ∈ L µ;δ ( n )^ . For P = P|
e ∈ L µ;δ ( n )^ we define
e n :P
e n,P
L cl ( n+ )^ = P = P|
cl
cl
+
+
e n
n \0) , σe P
×(
+
e σ (P) = σψ P
(37)
for arbitrary a(y, η) ∈
and
∈ −∞,d;−∞
(39)
µ,d;δ
cl
= Op (a) +
+
n−1 × n−1 ; N , N ,
−
+
n
+ ; N− , N+ . Moreover, we set
n
+ = u.l.c
µ,d;δ
Bcl
e , σψ,e P
n
n
+ ; N− , N+ for (µ, d) ∈
µ,d;δ
D EFINITION 6. The space cl
be the set of all operators
(38)
n
+ \0 ×
µ,d;δ
cl
n
+ \0 ×(
n \0)
!
× , δ ∈ , is defined to
∈
P
0
0 µ;δ
with P ∈ L cl ( n )^
0
n
+ ; N− , N+ .
Similarly, we get the subspaces of so-called Green operators (of order µ, type d, and weight
µ,d;δ n
µ,d;δ n δ) G,cl
+ ; N− , N+ and BG,cl
+ when we require amplitude functions to belong to
µ,d;δ n−1
µ,d;δ n−1
µ,d;δ n
n−1
×
; N− , N+ and RG,cl
× n−1 , respectively. For ∈ cl
+;
G,cl N− , N+ we write ord = (µ; δ).
µ,0;δ n Note that particularly simple elements in Bcl
+ are differential operators
X
(40)
A=
aα (x)Dxα
|α|≤µ
with coefficients aα = ãα |
n
+
δ ( n ).
where ãα (x) ∈ Scl
x
µ,0;0
T HEOREM 4. For every µ ∈ the space Bcl
element R µ that induces isomorphisms
(41)
for all s, % ∈
R µ : H s;%
as well as isomorphisms
Rµ : (42)
−µ,0;0
where R −µ := (R µ )−1 ∈ Bcl
n
+ .
n
+ (cf. the notation (39)) contains an
n −→ H s−µ;%
+
n
+ −→
n
+
n
+ ,
This is a well-known result for % = 0, proved in this form for all s ∈
in Grubb [7];
note that for s ≤ − 12 we have to compose the pseudo-differential operators from the right by an
extension operator l : H s ( n+ ) → H s ( n ), while for s > − 21 we can take e+ . Let us mention
323
Boundary value problems
for completeness that order reductions for s > µ+ − 21 have been constructed before by Boutet
de Monvel [3]. The symbols from [7] have the form
µ
τ
µ
(43)
r− (ξ ) = χ
hηi − iτ
ahηi
ξ = (η, τ ) ∈ n for a sufficiently large constant a > 0 and a function χ ∈ ( ) with F −1 χ
supported in − and χ(0) = 1. It was proved in [20], Section 5.3, that (43) is a classical symbol
n
µ
µ;0
n , and we can set R µ u = r+ Op r µ l
in ξ . In other words, we have r− (ξ ) ∈ Scl
+×
−
tr
ξ;x
where l = e+ for s > − 12 . It is now trivial that R µ induces isomorphisms for all s, %, because
µ;0
the operators with symbols (43) in n belong to L cl ( n ), cf. (9).
Note that the operator R µ is elliptic of order (µ; 0) in the sense of Definition 7 below.
µ,d;δ n
Writing ∈ cl
+ ; N− , N+ in the form (38) we set
σψ ( ), σe ( ), σψ,e ( ) = σψ (a) + σψ (P), σe (a) + σe (P), σψ,e (a) + σψ,e (P) ,
where we use notation from (36) and (37). Moreover, we define
(44)
σ∂ ( ), σe0 ( ), σ∂,e0 ( ) = σ∂ (a), σe0 (a), σ∂,e0 (a) ,
cf. the notation in (36). Finally, we set
(45)
σ ( ) = σψ ( ), σe ( ), σψ,e ( ); σ∂ ( ), σe0 ( ), σ∂,e0 ( ) ,
called the principal symbol of the operator .
Let us set
n o
n
n
µ,d;δ
µ,d;δ
.
symb cl
∈ cl
+ ; N− , N+ = σ ( ) :
+ ; N− , N+
µ,d;δ n
R EMARK 13. The space symb cl
+ ; N− , N+ can easily be defined intrinsically,
i.e., as a space of symbol tuples ( pψ , pe , pψ,e ; p∂ , pe0 , p∂,e0 ) with natural compatibility condi
tions between the components. Then σ : → σ ( ) is a surjective map
n
n
µ,d;δ
µ,d;δ
(46)
σ : cl
+ ; N− , N+ −→ symb cl
+ ; N− , N+ ,
and there is a linear right inverse
(47)
µ,d;δ
op : symb cl
n
+ ; N− , N+
µ−1,d;δ−1
of σ . Moreover, we have ker σ = cl
(46) is called an operator convention.
R EMARK 14. An operator
(48)
:
⊕
N−
n
+ ; N− , N+
n
+ ; N− , N+ . Any choice of a map (47) with
+
n−1 ,
µ,d;δ
−→ cl
n
+ ; N− , N+ induces continuous operators
µ,d;δ
∈ cl
n
−→
n
+
⊕
n−1 ,
N+
.
324
D. Kapanadze – B.-W. Schulze
This is an immediate consequence of the fact that can be written in the form
+ +
e
r Ae
0
+ Op (g) +
(49)
=
0
0
e ∈ L µ;δ ( n )tr := A
e ∈ L µ;δ ( n ) : A
e has the transmission property with respect to
for an A
cl
cl
n
−∞,d;−∞
t = 0 , and ∈
+ ; N− , N+ , where (48) is clear for , while the continuity for
the other ingredients, immediately follow from (10) and (24).
T HEOREM 5.
µ+ν,h;δ+%
cl
∈
n
+ ; N0 , N+ and
µ,d;δ
cl
∈
ν,e;%
cl
n
+ ; N− , N+ for h = max(ν + d, e), and we have
σ(
(with componentwise multiplication). If
n
+ ; N− , N0 implies
) = σ ( )σ ( )
or
is a Green operator then so is
∈
.
The proof of this theorem is very close to the corresponding proof in Boutet de Monvel’s
calculus in local terms. Therefore, we only sketch the typical
novelty in the framework with
e + e for smoothing operators , e and
weights. Compositions of the form ( + ) or
Green operators
= Op (g), e = Op (g̃) in the corresponding spaces are
again of the type
or like (38)). It
Green plus smoothing operator (this can easily
be
verified,
if
we
represent
e + r+ e
ee
e − 2+ ) e
Be+ ,
remains to consider compositions r+ Ae
Be+ that equal r+ A
Be+ + r+ A(1
where 2+ denotes the characteristic function of n+ . The first summand is as desired, while
e − 2+ ) e
r+ A(1
Be+ has to be recognized as an element Op (g) (modulo a smoothing remainder)
e = χ A
e+
for some Green symbol g(y, η) of weight δ + % for |y| → ∞. Here, we can write A
e and e
(1−χ ) A
B = χ e
B +(1−χ ) e
B for a certain global admissible cut-off function χ in n ,
e − 2+ )χ e
cf. Definition 3. Then, r+ χ A(1
Be+ is obviously of the asserted form because the
e − 2+ )χ e
e −
weight contributions for t → ∞ are cut out, while r+ (1 − χ ) A(1
Be+ , r+ χ A(1
e − 2+ )(1 − χ ) e
2+ )(1 − χ ) e
Be+ and r+ (1 − χ ) A(1
Be+ are smoothing, cf. Remark 11.
R EMARK 15. The operator % of multiplication by diag hxi% , hyi% ⊗ id N
0,0;δ n
+ ; N, N and induces isomorphisms
cl
H s n+
H s;% n+
%
⊕
⊕
−→
:
H s n−1 , N
H s;% n−1 , N
belongs to
for all s ∈ . Moreover, we have
%−δ
µ,d;δ
cl
−%
n
=
+ ; N− , N+ µ,d;0
cl
n
+ ; N− , N+
(clearly, the dimensions in the factors of the latter relation are assumed to be N+ and N− ,
respectively).
T HEOREM 6.
(50)
µ,d;δ
∈ cl
n
+ ; N− , N+ induces continuous operators
H s−µ;%−δ n+
H s;% n+
⊕
⊕
:
−→
H s;% n−1 , N−
H s−µ;%−δ n−1 , N+
for all s, % ∈ , s > d − 21 . If ord
the operator (50) is compact.
< (µ; δ) (i.e., the relation < holds for both components)
325
Boundary value problems
e + it suffices
Proof. Write in the form (49). The assertion for is obvious. Concerning r+ Ae
to apply (9) and (10), combined with the properties (6) and (5). It remains Op (g). For simplicity
µ,d;δ n−1
we assume g to be of the form of an upper left corner, i.e., g(y, η) ∈ RG,cl
× n−1 , cf.
the notation in Definition 4. The other entries in the general block matrix case can be treated in
a similar manner which is left to the reader. So we show the continuity
Op (g) : H s;% n+ −→ H s−µ;%−δ n+ ,
s > d − 21 . Applying Theorem 4 it suffices to prove the continuity of
R s−µ Op (g)R −s : H 0;%
n −→ H 0;%−δ
+
n .
+
µ,d;0
The symbol g(y, η) may be written in the form g(y, η) = hyiδ a(y, η) for a(y, η) ∈ RG,cl
n−1 × n−1 . Now, using Theorem 5 we get op (a)R −s = Op (b) + C for some b(y, η) ∈
n
µ−s,0;0 n−1
× n−1 and C ∈ B −∞,0;−∞ + . In the following C may be ignored; we
RG,cl
then have to show the continuity of
R s−µ hyiδ Op (b) : H 0;% n+ −→ H 0;%−δ n+ .
(51)
From the general pseudo-differential calculus with operator-valued symbols we know that
hyiδ Op (b)hyi−δ = Op (c)
s−µ,0;0
with some symbol c(y, η) that in this case again belongs to RG,cl
operator in (51) gets the form
n−1 ×
n−1 . Thus, the
R s−µ Op (c)hyiδ = Op (d)hyiδ + C
0,0;0 n−1
for another C ∈ B −∞,0;−∞ and d(y, η) ∈ RG,cl
× n−1 . We may concentrate on the
proof of the continuity
Op (d)hyiδ : H 0;% n+ −→ H 0;%−δ n+
for arbitrary % ∈ . This is equivalent to the continuity of
hxi%−δ Op (d)hyiδ hxi−% : L 2 n+ −→ L 2
n ,
+
0,0;0
x = (y, t). By the argument used above in connection with (51) we find an f (y, η) ∈ RG,cl
n−1 × n−1 such that
hxi%−δ Op (d)hyiδ hxi−% = hxi%−δ hyiδ−% Op ( f )hyi% hxi−% .
Now φ(y, t) := hyi% hxi−% for % ≥ 0 satisfies the assumptions of Lemma 1 for δ = 0. For
% − δ ≤ 0 we can apply Lemma 1 once again for ψ(y, t) = hxi%−δ hyiδ−% and get
hxi%−δ Op (d)hyiδ hxi−% = Op (Mψ ) Op ( f ) Op (Mφ )
as a continuous operator L 2 n+ → L 2 n+ . Let us now examine the case % ≥ 0 but % −
δ ≥ 0. We then write hxi%−δ hyiδ−% = ψ(y, t)hti%−δ for ψ(y, t) = hxi%−δ hyiδ−% htiδ−% .
326
D. Kapanadze – B.-W. Schulze
The function ψ satisfies the assumptions of Lemma 1, while Mht i%−δ belongs to S 0,0
n−1 ; + ,
+ by Lemma 2. This gives us
Mht i%−δ f (y, η) ∈ S 0,0 n−1 × n−1 ; L 2 ( + ), L 2 ( + )
n−1 ×
Hence, setting again φ(y, t) = hyi% hxi−% , we see that
hxi%−δ hyiδ−% Op ( f )hyi% hxi−% = Op Mψ Op Mht i%−δ f Op Mφ
is a continuous operator L 2 n+ → L 2 n+ . In an analogous manner we can proceed
in
the remaining cases concerning the sign of % and % − δ. The compactness of for ord <
(µ; δ) then follows from the continuity of to spaces of better smoothness and weight and from
corresponding compact embeddings of Sobolev spaces.
3.5. Ellipticity
The principal symbol structure of the preceding section gives rise to an adequate notion of ellipticity of pseudo-differential boundary value problems globally on the half-space.
D EFINITION 7. An operator
if
µ,d;δ
∈ cl
n
n
+ ; N− , N+ is called elliptic (of order (µ; δ))
(i) σψ ( )(x, ξ ) for all (x, ξ ) ∈ + × ( n \ 0),
n
σe ( )(x, ξ ) for all (x, ξ ) ∈ + \ 0 × n ,
n
σψ,e ( )(x, ξ ) for all (x, ξ ) ∈ + \ 0 × ( n \ 0)
are non-zero,
(ii) σ∂ ( )(y, η) for all (y, η) ∈ n−1 × n−1 \ 0 ,
σe0 ( )(y, η) for all (y, η) ∈ n−1 \ 0 × n−1 ,
σ∂,e0 ( )(y, η) for all (y, η) ∈ n−1 \ 0 × n−1 \ 0
are isomorphisms
+
⊕
N−
−→
+
⊕
.
N+
R EMARK 16. Condition (ii) in the latter definition can equivalently be replaced by bijectivities in the sense
Hs( +)
H s−µ +
⊕
⊕
−→
N−
N+
for any s > max(µ, d) − 21 .
µ,d;δ n
−µ,e;−δ n
D EFINITION 8. Given ∈ cl
∈ cl
+ ; N− , N+ , an operator
+ ; N+ ,
n
N− for some e ∈ is called a parametrix of if
− ∈ −∞,dl ;−∞ + ; N− , N− and
n
− ∈ −∞,dr ;−∞ + ; N+ , N+ for certain dl , dr ∈ .
µ,d;δ n
We shall see below that the ellipticity of an operator ∈ cl
+ ; N− , N+ entails the
existence of a parametrix. First we want to construct further examples of elliptic boundary value
problems.
327
Boundary value problems
The Dirichlet problem for c − 1, with the Laplace operator 1 =
c > 0 is represented by the operator
(52)
1 =
c−1
r0
n −→
+
: Hs
For convenience we pass to
(53)
2 =
c−1
Qr0
: Hs
Pn
∂2
j =1 ∂ x 2 , and a constant
j
H s−2 n+
⊕
1
H s− 2 n−1 .
H s−2 n+
⊕
H s−2 n−1
n −→
+
3
where Q is an order reduction on the boundary that we take of the form Q = Op y hηi 2 ,
3
such that Q : H s n−1 −→ H s− 2 n−1 is an isomorphism for all s ∈ . Then we have
2,1;0 n
3
2 ∈ cl
+ ; 0, 1 . We want to show that (53) is an isomorphism for all s > 2 and construct
the inverse. We have for x = (y, t), ξ = (η, τ )
σ∂ ( 2 ) =
Hence
σψ ( 2 ) = |ξ |2 , σe ( 2 ) = c + |ξ |2 , σψ,e ( 2 ) = |ξ |2 ,
!
!
!
|η|2 − ∂t2
c + |η|2 − ∂t2
|η|2 − ∂t2
, σe0 ( 2 ) =
, σ∂,e0 ( 2 ) =
.
3
3
3
|η| 2 r0
hηi 2 r0
|η| 2 r0
2 is elliptic in the sense of Definition 7. First we invert the operator family
(54)
α − ∂t2
βr0
:
+ −→
+
⊕
,
1
3
where α := c + |η|2 2 , β := hηi 2 . Let us write l± (τ ) = α ± iτ ; then l− (τ )l+ (τ ) = α 2 + τ 2
and α 2 − ∂t2 = op+ (l−l+ ) = op+ (l− )op+ (l+ ) the latter identity is true because l− is a minus
function; op+ (l− ) : + → + is an isomorphism . Thus, to invert (54), it suffices to
consider
+
op+ (l+ )
: + −→
⊕
βr0
which is an isomorphism, because
op+ (l+ ) : + → + is surjective and βr0 induces an
isomorphism of ker op+ (l+ ) = γ e−αt : γ ∈
to . Let us form the potential k = k(α) :
→ + , defined by kγ = γ β −1 e−αt , γ ∈ . Then
op+ (l+ )
βr0
−1
op+ l+
1
k =
0
0
1
−1
because r0 op+ (l+
) = 0, and hence
op+ (l+ )
βr0
−1
−1
= op+ l+
k .
,
328
D. Kapanadze – B.-W. Schulze
Consider now a(τ ) = α 2 + τ 2 . The operator in (54) can be written
op+ (a)
βr0
=
op+ (l− )
0
0
1
op+ (l+ )
βr0
and hence
op+ (a)
βr0
−1
=
=
−1
op+ l+
k
−1
−1
op+ l+
op+ l−
−1
op+ l−
0
op+ (a)
βr0
−1
1
!
k .
−1 + −1 Here op+ l+
op l− = op+ a −1 + g for a certain g ∈ 0 0
0
= op+ a −1 + g
+ . It follows altogether
k ,
i.e., we calculated the inverse of (54). Inserting now the expression for α = α(η), β = β(η), we
easily see that the ingredients of
σe0 ( 2 )−1 = op+ a −1 (η) + g(η)
(55)
k(η)
−2,0;0
cl
n−1 × n−1 ; 1, 0 (they are, of course, independent of y), and it is clear
−1 −2,0;0 n
that 2 = Op y σe0 2
which belongs to cl
+ ; 1, 0 . The method of calculating
(55) gives us analogously σ∂ ( 2 )−1 and σ∂,e0 ( 2 )−1 , and
belong to
−1
σ
−1
2
=
−1
|ξ |−2 , c + |ξ |2
, |ξ |−2 ; σ∂ ( 2 )−1 , σe0 ( 2 )−1 , σ∂,e0 ( 2 )−1
It is then obvious how to express
.
−1
1 , namely
−1
1 =
−1
2
1
0
0
Q −1
.
R EMARK 17. Similar arguments apply to the Neumann problem for c − 1 in the halfspace, with r0 ∂t in place of r0
. To get
an element with unified orders we can pass to the boundary
1
operator Rr0 ∂t for R = Op y hηi 2 . We see that
c−1
Rr0 ∂t
2,2;0
∈ cl
n
+ ; 0, 1
is also elliptic in the sense of Definition 7 and even invertible as an operator H s n+ →
−2,0;0 n
H s−2 n+ ⊕ H s−2 n−1 for s > 23 . The inverse belongs to cl
+ ; 1, 0 . We shall
construct in Section 5.3 below a general class of further examples of this kind.
329
Boundary value problems
T HEOREM 7. For every N ∈ there exist elliptic elements
−
0,0;0 n
∈
+ ; N, 0 that induce isomorphisms
N
cl
+
−
⊕
N :
for all s > − 21 , where
n
+
Hs
−
N =
n−1 ,
+ −1
N
.
Hs
N
Proof. Let us start from the above operator
(56)
0=
⊕
−→
+
Hs
N ∈
Hs
n
Hs
N :
+
−→
n
+
n−1 ,
n
+
Hs
N
0,0;0
cl
n
+ ; 0, N and
,
2 and form
s0 −2 e−s0 ∈
2
0,0;0
cl
n
+ ; 0, 1
1,0;0 n for any fixed s0 > 2, where e := R 1 ∈ Bcl
+ is the order reducing element from
Theorem 4 and
:= diag R 1 , R 0 for R 0 = Op y hηi ⊗ id N . Then, setting +
1 = 0 , we
+
can form N inductively by

 +


+ + A N−1 A +
A+
0
1
+
A
AN



 N−1
+
+
1
=  TN−1
:=  TN−1
A+
0 
N =
1 .
T1+
TN+
0
1
T1+
Here,
+
1 =
A+
1
T1+
. Moreover, from the above construction of
+ −1
.
that we may set −
N :=
N
−1
2
and Theorem 4 it follows
3.6. Parametrices and Fredholm property
T HEOREM 8. Let
(57)
µ,d;δ
∈ cl
n
+ ; N− , N+ be elliptic. Then
H s;% n+
H s−µ;%−δ n+
⊕
⊕
:
−→
H s;% n−1 , N−
H s−µ;%−δ n−1 , N+
is a Fredholm operator for every s > max(µ, d) − 21 and every % ∈ , and has a parametrix
−µ,(d−µ)+ ;−δ n
+
∈ cl
+ ; N+ , N− where dl = max(µ, d) and dr = (d −µ) (cf. the notation
in Definition 8).
The proof of this theorem will be given below after some preparations.
R EMARK 18. Applying Remark 15 we can reduce the proof of Theorem 8 to the case δ =
µ,d;0 n
0. In other words, it suffices to consider the operator %−δ −% ∈ cl
+ ; N− , N+ .
Furthermore, we can reduce orders and pass to
s0 −µ %−δ −% −s0
0,0;0 n
0 := (+) + ; N− , N+
(−) ∈ cl
330
D. Kapanadze – B.-W. Schulze
for any choice of s0 > max(µ, d), where (±) = diag R 1 , R 0N± for R 0N± := Op hηi ⊗
id N± , cf. similarly (56). Clearly,
of 0 , and the
the ellipticity of is equivalent to that construction of a parametrix 0 for 0 gives us immediately a parametrix of , namely
0
= −% (−)0 0 (+)
−s
(58)
So we mainly concentrate on the case
0;0
Let p(x, ξ ) ∈ Scl
(59)
n
+×
n
σψ ( p) 6= 0
0,0;0
∈ cl
s −µ %−δ
.
n
+ ; N− , N+ .
tr, be a symbol with
for all
(x, ξ ) ∈
(60)
σe ( p) 6= 0
for all
(x, ξ ) ∈
(61)
σψ,e ( p) 6= 0
for all
(x, ξ ) ∈
Set
(62)
n
n \ 0 ,
+×
n
n,
+ \0 ×
n
n \ 0 .
+ \0 ×
0 (y, η) := op+ ( p|
b11
t =0 )(y, η) ,
and consider the operator families
2
0 (y, η) , σ 0 b0 (y, η) , σ 0 b0 (y, η) : L 2 (
(63)
σ∂ b11
+ ) −→ L ( + ) ,
e
∂,e
11
11
0
0
0
σ∂ b11
for (y, η) ∈ n−1 × n−1 \ 0 , σe0 b11
for (y, η) ∈ n−1 \ 0 × n−1 , σ∂,e0 b11
for (y, η) ∈ n−1 \ 0 × n−1 \ 0 .
These are families of Fredholm operators parametrized by the corresponding sets of (y, η)variables.
P ROPOSITION 4. For every ε > 0 there exists an R = Rε > 0 such that
σ∂ b0 (y, η) − σ∂,e0 b0 (y, η)
(64)
11
11
L2( +) < ε
for all |y| > R and η ∈ n−1 \ 0,
σ∂ b0 (y, η) − σe0 b0 (y, η)
(65)
11
11
L(
for all |y| > R and |η| > R,
σe0 b0 (y, η) − σ∂,e0 b0 (y, η)
(66)
11
11
for all |y| ∈ n−1 \ 0 and |η| > R.
2
L(
2
+)
<ε
+)
<ε
Proof. Let us first verify (64). Both op (σψ ( p)|t =0 )(y, η) and op (σψ,e ( p)|t =0 )(y, η) can be
regarded as parameter-dependent families of pseudo-differential operators L 2 ( + ) → L 2 ( + )
with parameter y ∈ n−1 , smoothly dependent on η with |η| = 1.
But
(67)
op σψ ( p|t =0 ) − σψ,e ( p|t =0 ) (y, η)
is of order −1 in the parameter. A well-known result on operator norms of parameter-dependent
pseudo-differential operators, cf., e.g., [30], Section 1.2.2, tells us that the L 2 ( + ) -norm of
331
Boundary value problems
(67) tends to zero for |y| → ∞, in this case uniformly for |y| = 1. Thus, composing (67) from
the right with e+ and from the left with r+ we get relation (64) for all |y| ≥ R, R = Rε , and
|η| = 1.
In a similar way we can argue for (66), now with η ∈ n−1 \ 0 as parameter and smooth
dependence on y with |y| = 1. This gives us relation (66). Estimate (65) is then an obvious
consequence of (64) and (66).
C OROLLARY 1. Under the conditions of Proposition 4 there exists an R = Rε > 0 such
that the Fredholm families
0 (y, η) : L 2 (
2
σ∂ b11
for
0 ≤ |y| ≤ R , |η| = R
+ ) −→ L ( + )
and
2
0 (y, η) : L 2 (
for
|y| = R , 0 ≤ |η| ≤ R ,
σe0 b11
+ ) −→ L ( + )
0 (y, η) − σ 0 b0 (y, η)
<
ε
for
all
|y| = |η| = R.
satisfy σ∂ b11
e 11
(L 2 ( ))
+
Let ε > 0, and set
o
n
Tε = (y, η) ∈ 2(n−1) : |y| = |η| = Rε ,
Dε = Tε × [0, 1] ,
and form
Zε
j
=
j
=
Hε
n
o
(y, η) ∈ 2(n−1) : |y| ≤ Rε + j , |η| = Rε ,
n
o
(y, η) ∈ 2(n−1) : |y| = Rε , |η| ≤ Rε + j
j
j
j
for j = 0, 1, ∞. Define the spaces ε = Z ε ∪d Hε ∪b Dε / ∼, where ∪d is the disjoint
union, while ∪b is the disjoint union combined with the projection to the quotient space, given
j
j
by natural identifications Tε ∩ Z ε ∼
= Tε × {1}.
= Tε × {0}, Tε ∩ Hε ∼
Write for abbreviation Z ε = Z ε0 , Hε = Hε0 , ε = 0ε . Moreover, let Dε,τ := Tε × [0, τ ]
and ε,τ := Z ε ∪d Hε ∪b Dε,τ , 0 ≤ τ ≤ 1, where ∪b is defined by means the identifications
Tε ∩ Z ε ∼
= Tε × {0}, Tε ∩ Hε ∼
= Tε × {τ }. Thus ε = ε,1 , and we set ε = ε,0 .
Define an operator function F(m), m ∈ ε , by the following relations:
0 (y, η) for m = (y, η) ∈ Z ,
F(y, η) = σ∂ b11
ε
0 (y, η) for m = (y, η) ∈ H ,
F(y, η) = σe0 b11
ε
0
0
F(y, η, δ) = δσ∂ b11 (y, η) + (1 − δ)σe0 b11 (y, η) for m = (y, η, δ) ∈ Dε .
From Corollary 1 we get
F(y, η, δ) − F y, η, δ 0 (68)
L(
2
+)
≤ |δ − δ 0 |ε
for all (y, η, δ), (y, η, δ 0 ) ∈ Dε , 0 ≤ δ, δ 0 ≤ 1. We have
(69)
L 2( + ) ,
F ∈ C ε,
and F| Z ε , F| Hε are continuous families of Fredholm operators. Relation (68) shows that (69) is
a family of Fredholm operators for all m ∈ ε , provided ε > 0 is sufficiently small. We then get
332
D. Kapanadze – B.-W. Schulze
an index element ind ε F ∈ K ( ε ). Because of K ( ε,τ ) ∼
= K ( ε ) for all 0 ≤ τ ≤ 1, ind
represents, in fact, an element in K ( ε ) that we denote by
0 (y, η), σ 0 b0 (y, η) ∈ K ( ) .
(70)
ind ε σ∂ b11
ε
e
11
ε
F
0 (y, η) for an elliptic symbol
Our next objective is to check, whether the operator family b11
n
0;0
n
p(x, ξ ) ∈ Scl
+×
tr can be completed to a block matrix valued symbol
ξ;x
b0 (y, η) =
(71)
0
b11
b21
b12
b22
0;0
(y, η) ∈ Scl
η;y
L 2( + )
E=
⊕
,
(72)
e
E=
N−
n−1 ×
n−1 ; E, e
E ,
L 2( + )
⊕
N+
with suitable N− , N+ , such that the homogeneous symbols
(0)
(73)
σ∂ b0 (y, η) ∈ Sη ,
(y, η) ∈ n−1 × n−1 \ 0 ,
(0)
σe0 b0 (y, η) ∈ S y ,
(y, η) ∈ n−1 \ 0 × n−1 ,
(74)
(0;0)
(75)
σ∂,e0 b0 (y, η) ∈ Sη;y ,
(y, η) ∈ n−1 \ 0 × n−1 \ 0 ,
are isomorphisms.
n
0;0
n
T HEOREM 9. Let p(x, ξ ) ∈ Scl
+ ×
tr be (σψ , σe , σψ,e )-elliptic, i.e., relations
ξ;x
(59), (60) and (61) are fulfilled. Then the following conditions are equivalent:
(i) The families of Fredholm operators L 2 ( + ) → L 2 ( + )
0 (y, η) for (y, η) ∈ n−1 ×
n−1 \ 0 ,
σ∂ b11
(76)
n−1 \ 0 × n−1 ,
0 (y, η) for (y, η) ∈
(77)
σe0 b11
0 (y, η) for (y, η) ∈
n−1 \ 0 ×
n−1 \ 0
(78)
σ∂,e0 b11
can be completed to D 0,0
(75), respectively.
+ ; N− , N+ -valued families of isomorphisms (73), (74) and
(ii)
(79)
0 , σ 0 b0 ∈ π ∗ K ({+}) ,
ind ε σ∂ b11
e
+
11
where π+ : ε → {+} is the projection of ε to a single point {+}, (K ({+}) =
).
Proof. (i) ⇒ (ii): In the construction of the proof we choose ε > 0 sufficiently small. Assume
that we have isomorphism-valued symbols (73), (74) and (75), associated with the given upper
left
on ε , associated with
N F(m)
corners
(76), (77)
N family
and (78). Then the above Fredholm
0 , σ 0 b0
+ −
− , i.e., ind
σ∂ b11
which
F ∈
has the property ind ε F =
e 11
ε
0 , σ 0 b0
∼ ∗
implies ind ε σ∂ b11
e 11 ∈ = π+ K ({+}).
0 , σ 0 b0
∗ ({+}) implies the existence of
(ii) ⇒ (i): Condition ind ε σ∂ b11
e 11 ∈ π+ K
0
0
N
+
− N− . Replacing N± by N± +
numbers N± ∈ with ind ε σ∂ b11 , σe0 b11 =
333
Boundary value problems
M for sufficiently large M and denoting the enlarged numbers again by N± we find operator
families
k(m) : N− −→ L 2 ( + ) ,
t (m) : L 2 ( + ) −→ N+ ,
q(m) : N− −→ N+ ,
such that
(80)
f (m) :=
F(m)
t (m)
k(m)
q(m)
:
L 2( + )
L 2( + )
⊕
⊕
−→
N−
N+
is a family of isomorphisms,
continuously parametrized by ε . It is evident that they can be chosen as D 0;0 + ; N− , N+ -valued functions, similarly to the construction of bijective boundary
symbols in the local algebra of boundary value problems with the transmission property. In
addition it is clear that the functions k(m), t (m) and q(m) can be chosen to be smooth in (y, η).
Let us now define a Fredholm family F 1 (m) for m ∈ 1ε by
F 1 (m) = F(m)
for m ∈ ε ,
0 (y, η) + λσ 0 b0 (y, η)
F 1 (m) = (1 − λ)σ∂ b11
∂,e
11
for Rε ≤ |y| ≤ Rε + 1, |η| = Rε , where λ = Rε − |y|,
0 (y, η) + λσ 0 b0 (y, η)
F 1 (m) = (1 − λ)σe0 b11
∂,e
11
for |y| = Rε , Rε ≤ |η| ≤ Rε + 1, where λ = Rε − |η|.
Estimates (64) and (66) show that F 1 is a family of Fredholm operators on 1ε , provided
ε > 0 is sufficiently small. We can construct a family of isomorphisms
(81)
f 1 (m) :=
F 1 (m)
t 1 (m)
k 1 (m)
q 1 (m)
L 2( + )
L 2( + )
−→
,
⊕
⊕
:
N−
N+
m ∈ 1ε , similarly as f (m) (if necessary, we take N− , N+ larger than before), where f 1 |
Since F(m) is a-priori given on ∞
ε , we can also form
f˜1 (m) :=
F(m)
t 1 (m)
k 1 (m)
q 1 (m)
ε
= f.
L 2( + )
L 2( + )
:
⊕
−→
⊕
,
N−
N+
m ∈ 1ε . Due to (64) and (66) this is a family of Fredholm operators. Clearly, we may choose
f˜1 (m) in such a way that f˜1 | 1 and f˜1 | 1 are smooth in (y, η). Let us finally look at ∞
ε .
Zε
Hε
The operator function f 1 , first given on 1ε , canonically extends to ∞
ε by homogeneity of order
1
∞ denote this extension,
zero to ∞
ε \ ε in y and η. Let f
∞
F (m) k ∞ (m)
∞
(82)
f (m) :=
t ∞ (m) q ∞ (m)
i.e., f ∞ | 1 = f 1 . Since f ∞ is obtained by homogeneous extension of a family of isomorε
phisms, it is again isomorphism-valued. Moreover, we can also form
F(m) k ∞ (m)
f˜∞ (m) :=
∞
∞
t (m) q (m)
334
D. Kapanadze – B.-W. Schulze
which is a family of isomorphisms because of the corresponding property of (82) and relations
(64) and (66).
0
Then, to get (73), (74) and (75), it suffices to
define σ∂ (b )(y, η) as the extension by homogeneity 0 in η of f˜∞ | Z ε∞ to n−1 × n−1 \ 0 , σe0 (b0 )(y, η) as the extension by homogeneity
0 in y of f˜∞ | Hε∞ to n−1 \ 0 × n−1 and σ∂,e0 (b0 )(y, η) as the extension by homogeneity 0
in y and η of f˜∞ |{|y|=Rε +1,|η|=Rε +1} to n−1 \ 0 × n−1 \ 0 . To justify the notation in
(73), (74) and (75) (i.e., to generate the latter homogeneous
functions in terms of a symbol (71))
we can first form b00 (y, η) = χ(η)σ∂ (b0 )(y, η) + χ(y) σe0 (b0 )(y, η) − χ(η)σ∂,e0 (b0 )(y, η) ∈
n−1 × n−1 ; E, e
E , cf. the second part of Remark 5, and then define b0 (y, η) by replacS 0;0
clη;y
0 (y, η).
ing the upper left entry of b00 (y, η) by b11
R EMARK 19. Notice that Theorem 9 is an analogue of the Atiyah-Bott condition for the
existence of elliptic boundary conditions to an elliptic operator A, cf. also Section 4.4 below.
∗ n−1 gives
The canonical projection T ∗ n−1 → n−1 restricted
to the subset ε ⊂ T
n−1
: |y| ≤ Rε . Condition (79) can equivalently be
us a projection πε : ε → Bε := y ∈
written
0 , σ 0 b0 ∈ π ∗ K (B ) ,
ind ε σ∂ b11
ε
e
ε
11
since Bε is contractible to a point {+}.
0;0
C OROLLARY 2. Given a symbol p(x, ξ ) ∈ Scl
ξ;x
n
+ ×
n
tr that is σψ , σe , σψ,e -
0 (y, η) = op+ ( p|
elliptic, under the condition (79) for b11
t =0 )(y, η) we find a
0,0;0 n−1
× n−1 ; N− , N+
b(y, η) ∈ cl
for suitable N− , N+ ∈ , such that (76), (77) and (78) are isomorphims L 2 ( + ) ⊕ N− →
L 2 ( + )⊕ N+ , cf. Definition 7. To construct b(y, η) it suffices to define b(y, η) by replacing the
upper left entry of b00 (y, η) by op+ ( p )(y, η) for p (x, ξ ) = χ (x) p(x, ξ ) with some global
admissible cut-off function χ , cf. Definition 3.
0,0;0 n P ROPOSITION 5. Let G ∈ BG,cl
+ be an operator such that A := 1 + G is elliptic in
0,0;0 n e
e
e
the sense of Definition 7. Then there is a G ∈ BG,cl
+ such that A := 1 + G is a parametrix
n
e − 1, A
eA − 1 ∈ B −∞,0;−∞ + .
of A, i.e., A A
Proof. Let us first observe that for every g ∈ 0 0 + i.e., g ∈
L 2 ( + ) with g, g ∗ :
L 2 ( + ) → ( + ) being continuous, cf. Section 3.1, we have ag, ga ∈ 0 0 + for every
a ∈
L 2 ( + ) . Then, if 1 + g : L 2 ( + ) → L 2 ( + ) for a g ∈
L 2 ( + ) is invertible,
we have a = (1 + g)−1 ∈
L 2 ( + ) and a(1 + g) = 1 = a + ag, i.e., a = 1 + g̃ for
0
g̃ = −ag ∈ 0
+ . Analogous conclusions are valid for the symbols σ∂ (1 + G), σe0 (1 + G)
and σ∂,e0 (1 + G). Then, setting
g̃∂ (y, η)
:=
σ∂ (1 + G)−1 (y, η) − 1 ,
g̃e0 (y, η)
:=
σe0 (1 + G)−1 (y, η) − 1 ,
g̃∂,e0 (y, η)
:=
σ∂,e0 (1 + G)−1 (y, η) − 1 ,
0,0;0 n−1
we can form g̃(y, η) := χ(η)g̃∂ (y, η) + χ(y) g̃e0 (y, η) − χ(η)g̃∂,e0 (y, η) ∈ RG,cl
×
n−1 cf. Remark 5. For G
e1 = Op y (g̃) we then have (1 + G) 1 + G
e1 = 1 + G
e2 where
335
Boundary value problems
e2 ∈ B −1,0;−1 n+ . Then G
e j ∈ B − j,0;− j n+ for all j , and we can carry out the asymptotic
G
G,cl
2
G,cl
P
−1,0;−1 n j ej
sum ∞
+ (which is just a version of the
j =0 (−1) G 2 in the class of operators 1 + BG,cl
e3 ∈
formal Neumann series argument in our operator class). In other words, we can find a G
n
−1,0;−1 n −∞,0;−∞
e
e
BG,cl
+ such that 1 + G 2 1 + G 3 = 1 + C for C ∈ B
+ . Because
0,0;0 n e
e
e
e
e
1 + G 1 1 + G 3 = 1 + G for some G ∈ BG,cl
+ we get(1 + G) 1 + G = 1 + C. Similar
0,0;0 n e
e
−∞,0;−∞ n .
e
e
arguments from the left yield a G ∈ BG,cl
+ with 1 + G (1 + G) − 1 ∈ B
+
e
e= G
e mod B −∞,0;−∞ n+ . In other words
Then a standard algebraic argument gives us G
e= 1 + G
e is as desired.
A
Proof of Theorem 8. As noted
the case µ = d =
in Remark 18 we maycontent ourselves with
δ = 0. The ellipticity of
with respect to σψ ( ), σe ( ), σψ,e ( ) allows us to form a
symbol p(x, ξ ) = χ(ξ )σψ ( )−1 (x, ξ ) + χ(x) σe ( )−1 (x, ξ ) − χ(ξ )σψ,e ( )−1 (x, ξ ) ∈
−1
−1
−1
n
0;0
n , where σ ( p), σ ( p), σ
. We
Scl
e
ψ
ψ,e ( p) = σψ ( ) , σe ( ) , σψ,e ( )
+×
tr
ξ;x
now observe that p(x, ξ ) meets the assumption of Theorem 9. In fact, the original symbol a(x, ξ )
belonging to satisfies these conditions
because
the assumed
bijectivities just correspond to the
ellipticity of with respect to σ∂ ( ), σe0 ( ), σ∂,e0 ( ) . Hence relation (80) with respect to
a(x, ξ ) is fulfilled. This implies the corresponding relation with respect to p(x, ξ ) because the
∗ K ({+}) is just the inverse of that for a(x, ξ ). By construction we have
index element in π+
−1;−1 n
n . This yields
p(x, ξ )a(x, ξ ) = 1 + r (x, ξ ) for an r (x, ξ ) ∈ Scl
+×
tr
ξ;x
p(x, ξ )#a(x, ξ ) = 1 + r̃ (x, ξ )
(83)
n
n . A formal Neumann series argument, applied to 1+ r̃ (x, ξ )
+×
tr
−1;−1 n
n
in terms of the Leibniz multiplication # gives us a symbol q̃(x, ξ ) ∈ Scl
+×
tr such
ξ;x
n
n
−∞;−∞
. Setting p̃(x, ξ ) = (1 +
that (1 + q̃(x, ξ ))#(1 + r̃ (x, ξ )) = 1 mod S
+ ×
n
q̃(x, ξ ))# p(x, ξ ) from relation (83) we get p̃(x, ξ )#a(x, ξ ) = 1 mod S −∞;−∞ + × n .
−1;−1
for an r̃ (x, ξ ) ∈ Scl
ξ;x
Applying Corollary 2 to p̃(x, ξ ) we can generate a b(y, η) of the asserted kind, more precisely
0,0;0 n−1
b(y, η) ∈ cl
× n−1 . Then the operator
R 0
e := Op (b) +
0 0
for R := (1 − χ (x))Opx ( p̃), cf. Definition 3, has the property e = + for some ∈
0,0;0 n
e
+ ; N+ , N+ . Since and are both elliptic also + is elliptic. Applying Theorem 7
cl
to N = N+ we can pass to the elliptic operator N+ ( + ) −1
N+ that has the form 1 + G for a
n
0,0;0 n 0,0;0
e
e
G ∈ BG,cl
+ . Proposition 5 gives us a G ∈ BG,cl
+ such that (1 + G) 1 + G = 1 + C
n
e
for a C ∈ B −∞,0;−∞ + . It follows that N+ e −1
N+ 1 + G = 1 + C for an element
n
C ∈ B −∞,0;−∞ + . This yields
e −1
e N = −1 (1 + C) N = 1 +
+
+
N 1+G
N
+
for a remainder
∈ −∞,0;−∞
+
n
+ ; N+ , N+ . Hence,
e −1 1 + G
e N ∈
0 :=
+
N+
0,0;0
cl
n
+ ; N+ , N−
336
D. Kapanadze – B.-W. Schulze
is a right parametrix of . In an analogous manner we can construct a parametrix from the left;
then a standard argument shows that 0 is also a left parametrix. In other words, when we go
back to the original orders of Theorem 8, we get a parametrix by formula (58), where its type
is (d − µ)+ and the types dl and dr of remainders are an immediate consequence of Theorem 5.
The Fredholm property of (57) follows from the fact that the remainders are compact operators
in the respective spaces, since they improve smoothness and weight. This completes the proof
of Theorem 8.
4. The global theory
4.1. Boundary value problems on smooth manifolds
The calculus of boundary value problems that we intend to develop in Section 4.2 below on a
manifold with exits to infinity will be a substructure of a corresponding calculus on a general
(not necessarily compact) smooth manifold with smooth boundary. This is, in fact, Boutet de
Monvel’s algebra [3] that we employ as the corresponding background. Concerning details, cf.
the monograph of Rempel and Schulze [16] or Schulze [30], Chapter 4. For future references we
want to give a brief description.
Let M be a smooth manifold with smooth boundary ∂ M, choose vector bundles E, F ∈
Vect(M), J − , J + ∈ Vect(∂ M) and set v = E, F; J − , J + . We then have the space −∞,0 (M;
v) of all smoothing operators C0∞ (M, E) ⊕ C0∞ (∂ M, J − ) → C ∞ (M, F) ⊕ C ∞ ∂ M, J + of
∞
type 0 that are given by corresponding C kernels, smooth up to boundary (in the corresponding
variables on M). Integrations refer to Riemannian metrics on M and ∂ M that we keep fixed in
the sequel, further to Hermitian metrics in the occurring vector bundles. Assume that the Riemannian metric on M induces the product metric of (∂ M) × [0, 1) in a collar neighbourhood
of ∂ M. Incidentally we employ 2M, the double of M, obtained by gluing together two copies
of M along ∂ M by an identification diffeomorphism. On M we have the space Diff j (M; E, F)
of all differential operators of order j acting between sections in the bundles E and F. Then
−∞,d (M; v), the space of all smoothing operators on M of type d ∈ , is defined to be the set
of all
d
X
Dj 0
= 0+
j
0
0
j =1
for arbitrary 0 , . . . , d ∈ −∞,0 (M; v) and D j ∈ Diff j (M; E, F). To introduce the space
of Green operators on M we first consider an open set  ⊆ n−1 , n = dim M, and define
µ,d
n−1 ; k, m; N , N , the space of all Green symbols of order µ and type d, to be
−
+
G ×
the set of all
!
d
j
X
0
∂
t
g(y, η) = g0 (y, η) +
g j (y, η)
0 0
j =1
ν,0
n−1 ; k, m; N , N
for g j (y, η) ∈
−
+ 3
G ×
g(y, η) is given by the conditions
ν  × n−1 ; L 2
k ⊕ N− , m ⊕ N+ ,
g(y, η) ∈ Scl
+,
+,
ν  × n−1 ; L 2
m ⊕ N+ , k ⊕ N− ,
g ∗ (y, η) ∈ Scl
+,
+,
µ− j,0
×
G
n−1 ; k, m; N , N . Here
−
+
x = (y, t) is the splitting of variables in local coordinates near ∂ M (cf. analogously Definition 4), and k, m, N− and N+ are the fibre dimensions of E, F, J − and J + , respectively.
337
Boundary value problems
µ,d
Now G (M; v) is defined to be the set of all operators of the form 0 + for arbitrary
∈ −∞,d (M; v) and operators 0 that are concentrated in a collar neighbourhood of ∂ M
µ,d
and are locally finite sums of operators of the form Op (g) for certain g(y, η) ∈ G  ×
n−1 ; k, m; N , N . The pull-backs refer to charts U →  ×
−
+
+ for coordinate patches U
near ∂ M and trivializations of the involved bundles; “ 0 concentrated near ∂ M” means that for
certain functions φ, ψ ∈ C ∞ (M) that equal 1 in a collar neighbourhood of ∂ M and 0 outside
another collar neighbourhood of ∂ M we have 0 =
φ 0
ψ , cf. similar notation in (92)
below.
µ
e F
e for E,
e F
e ∈ Vect (2M) denote the subspace of all A
e ∈ L µ 2M;
Finally, let L cl 2M; E,
tr
cl
e F
e (classical “in ξ -variables”) pseudo-differential operators on 2M of order µ acting beE,
e F
e that have the transmission property with respect to ∂ M.
tween sections of the bundles E,
s (M, E) of smoothness s ∈
s
We employ the standard Sobolev spaces Hcomp
(M, E), Hloc
s
for bundles E ∈ Vect (M). “comp” and “loc” are understood in the sense Hcomp
(M, E) =
s
s
s
e
e ∈ Vect (2M) with E = E|
e M.
e
2M,
E
for
any
E
Hcomp 2M, E
,
H
(M,
E)
=
H
M
M
loc
loc
e F
e and E = E|
e ∈ L µ 2M; E,
e M , F = F|
e M we can form r+ Ae
e + , where e+ is
For every A
cl
tr
the extension by zero from intM to 2M and r+ the restriction from 2M to intM; this gives us
continuous operators
s−µ
s
e + : Hcomp
r+ Ae
(M, E) −→ Hloc (M, F)
for all s > − 21 .
D EFINITION 9. The space µ,d (M; v) for µ ∈ , d ∈ , v = E, F; J − , J + , is defined
to be the set of all operators
+ +
e
r Ae
0
(84)
=
+
0
0
e ∈ L µ 2M; E,
e F
e and
for arbitrary A
cl
tr
An operator
µ,d
∈ G (M; v).
∈ µ,d (M; v) induces continuous operators
s−µ
s
Hcomp
(M, E)
Hloc (M, F)
⊕
⊕
:
−→
s−µ
s
Hcomp
∂ M, J −
Hloc ∂ M, J +
for all s > d − 21 (which entails continuity between C ∞ sections). In particular, if M is compact,
“comp” and “loc” are superfluous, and we get continuous operators
(85)
:
H s (M, E)
H s−µ (M, F)
⊕
⊕
−→
H s ∂ M, J −
H s−µ ∂ M, J + .
The principal symbol structure of
∈ µ,d (M; v) consists of a pair
σ ( ) = σψ ( ), σ∂ ( ) ,
where σψ ( ), the homogeneous principal interior symbol of order µ, is a bundle homomorphism
∗
∗
e ∗
σψ ( ) := σψ A
(86)
T M\0 : πψ E −→ πψ F ,
338
D. Kapanadze – B.-W. Schulze
for πψ : T ∗ M \ 0 −→ M, and σ∂ ( ), the homogeneous principal boundary symbol of order µ,
a bundle homomorphism

E0 ⊗ +
∗

σ∂ ( ) : π ∂
⊕
J−
(87)


F0 ⊗ +
 −→ π ∗ 
⊕
∂
J+


for π∂ : T ∗ (∂ M) \ 0 → ∂ M. Alternatively, σ∂ ( ) may be regarded as a homomorphism

σ∂ ( ) : π∂∗ 
(88)

 0

F ⊗ H s−µ ( + )
E0 ⊗ Hs( +)
∗
 −→ π 

⊕
⊕
∂
J−
J+
for all s > d − 12 , cf. Remark 16. Setting symb µ,d (M; v) = σ ( ) : ∈ µ,d (M; v) there
is a map
op : symb µ,d (M; v) −→ µ,d (M; v)
with σ ◦ op = id on the symbol space. We have σ ( ) = 0 ⇒
compact, the operator (85) is compact when its symbol vanishes.
∈
µ−1,d (M; v); if M is
T HEOREM 10. Let M be compact; then ∈ µ,d (M; v), v = E 0 , F; J0 , J + , and ∈
ν,e (M; w), w = (E, E ; J − , J ), implies
∈ µ+ν,h (M; v◦w), for h = max(ν +d, e), v◦
0
0
−
+
w = E, F; J , J , and we have σ (
) = σ ( )σ ( ) (with componentwise multiplication).
An analogous result holds for general M when we replace the composition by
φ for a
∞
compactly supported φ ∈ C (M) where σ (
)
=
σ
(
).
)σ
(
φ
φ
An operator ∈ µ,d (M; v) is called elliptic, if both (86), and (87) are isomorphisms (the
second condition is equivalent to the bijectivity of (88) for all s > max(µ, d) − 21 ).
T HEOREM 11. Let M be compact. Then the following conditions are equivalent:
(i)
∈ µ,d (M; v) is elliptic,
(ii) the operator (85) is Fredholm for some s = s0 > max(µ, d) − 21 .
is elliptic, then (85) is a Fredholm operator for all s > max(µ, d) − 21 , and there is a
+
parametrix ∈ −µ,(d−µ) M; v−1 of in the sense
If
(89)
−
∈ −∞,dl (M; vl ) ,
−
∈ −∞,dr (M; vr )
for dl = max(µ, d), vl = (E, E; J − , J − ), dr = (d − µ)+ , vr = F, F; J + , J + .
R EMARK 20. Ellipticity of ∈ µ,d (M; v) for non-compact M entails the existence of a
+
parametrix ∈ −µ,(d−µ) M; v−1 , where (89) is to be replaced by
ψ
φ
−
φ ∈
−∞,dl (M; v ) ,
l
φ
ψ
−
φ ∈
−∞,dr (M; v )
r
for arbitrary φ, ψ ∈ C0∞ (M) with φψ = φ (and
φ,
ψ being the multiplication operators,
containing evident tensor products with the identity maps in the respective vector bundles).
339
Boundary value problems
4.2. Calculus on manifolds with exits to infinity
In this paper a manifold M with boundary and conical exits to infinity is defined to be a smooth
manifold with smooth boundary containing a submanifold C that is diffeomorphic (in the sense
of manifolds with boundary) to (1 − ε, ∞) × X with a smooth compact manifold X with smooth
boundary Y , where M \ C is compact. Concerning the local descriptions we proceed similarly
to Section 2.3 above. To simplify the considerations we assume (without loss of generality) that
there is a smooth manifold 2M without boundary (the double of M) where 2M has conical exits
to infinity, cf. Section 2.3, with 2X being the base of the infinite part of 2M that is diffeomorphic
to (1−ε, ∞)×(2X). Here 2X, the double of X, is obtained from two copies of X, glued together
along the common boundary Y by an identification diffeomorphism to a smooth closed compact
manifold.
To describe the pseudo-differential calculus of boundary value problems on M we mainly
concentrate
on C; the calculus on the “bounded” part of M has been explained in Section 4.1.
ej
If U
denotes an open covering of 2M of analogous meaning as (11), we have the
j =1,...,N
ej
subsystem U
of “infinite” neighbourhoods. Without loss of generality we can
j =L+1,...,N
ej ∩ ∂ M = ∅ for j = L + 1, . . . , B, U
ej ∩ ∂ M 6 = ∅
choose the numeration in such a way that U
for j = B + 1, . . . , N, for a certain L + 1 ≤ B ≤ N. Similarly to Section 2.3 we have charts
ej −→ V
ej ,
χ̃ j : U
j = B + 1, . . . , N
ej = x ∈ n : |x| > 1 − ε, x ∈ V 1 for certain open sets V
e1 ⊂ S n−1 , n = dim(2M).
where V
|x|
j
j
e
We may (and will) assume that U j has the form 2U j for an infinite neighbourhood U j on M,
ej = 2U j along U
ej ∩ ∂ M, where
U j ∩ ∂ M 6= ∅, that is glued together with its counterpart to U
n−1
ej ∩ ∂ M → V
ej ∩
and
χ̃ j : U
(90)
ej ∩ n+ =: V j ,
χ j := χ̃ j U : U j −→ V
j
j = B + 1, . . . , N .
Let U ⊂ M be a neighbourhood of M that equals U j for some B + 1 ≤ j ≤ N, and
n
let χ : U → V ⊂ + be the chart corresponding to (90). We call U a local admissible
neighbourhood and any φ ∈ C ∞ (U ) a local admissible cut-off function on M if φ = χ ∗ ~
n
for some local admissible cut-off function ~ in + (that is supported in V ), cf. Definition 3.
∼
Moreover, the above-mentioned infinite part C = (1 − ε, ∞) × X of M allows us to define
global admissible neighbourhoods on M, namely sets of the form (1 − ε, ∞) × Y × [0, β) for
some (small) β > 0, where Y × [0, β) denotes a corresponding collar neighbourhood of Y in
X. Then a φ ∈ C ∞ (M) is called a global admissible cut-off function
on
M if 0 ≤ φ ≤ 1,
h
supp φ ⊂ 1 − 2ε , ∞ × Y × [0, β), φ = 1 for m ∈ (1, ∞) × Y × 0, β2 , and φ(λm) = φ(m)
h
for all λ ≥ 1, m ∈ (R, ∞) × Y × 0, β2 for some R > 1.
e ∈ Vect (2M) such that E = E|
e M . In SecGiven a vector bundle E ∈ Vect (M) we fix an E
s;%
s;%
e
tion 2.3 we have defined weighted Sobolev spaces H
2M, E for s, % ∈ . Let H0 (M, E)
s;%
e with supp u ⊆ M. Similarly, denoting by M−
denote the subspace of all u ∈ H
2M, E
s;%
e M . Let r+ be the
the negative counterpart of M in 2M, we have H0 (M− , E − ) for E − = E|
−
operator of restriction to int M = M \ ∂ M, and set
n
o
e .
(91)
H s;% (M, E) = r+ u : u ∈ H s;% 2M, E
340
D. Kapanadze – B.-W. Schulze
e /H s;% (M− , E − ) which gives
There is then an isomorphism of (91) to the space H s;% 2M, E
0
us a Banach space structure on (91) (in fact, a Hilbert space structure) via the quotient topology.
Similarly to (12) we introduce the Schwartz space (M, E) of sections in E.
µ;δ
e F
e for E,
e
e F
e ∈ Vect (2M) denote the subspace of all A
e ∈ L µ;δ 2M; E,
Let L cl 2M; E,
tr
cl
+
e
F that have the transmission property with respect to ∂ M. Then, if e is the operator of extene + for arbitrary A
e ∈ L µ;δ 2M; E,
e
sion by zero from M to 2M, analogously to (4) we form r+ Ae
cl
e
F tr and get continuous operators
e + : H s;% (M, E) −→ H s−µ;%−δ (M, F)
r+ Ae
for all s > − 21 and % ∈ .
In order to introduce the global space of pseudo-differential boundary value problems on
M we first introduce the smoothing elements of type 0. Let E, F ∈ Vect (M), J − , J + ∈
Vect (∂ M). Recall that all bundles are equipped with Hermitian metrics (homogeneous of order
zero in the axial variable of the conical exits). Moreover, on M and ∂ M we have fixed Riemannian metrics such that the metric on ∂ M is induced by that on M. There are thenassociated
measures dm on M and dn on ∂ M. Now −∞,0;−∞ (M; v) for v = E, F; J − , J + is defined
to be the space of all operators
=
C11
C21
C12
C22
:
H s;% (M, E)
−→
⊕
H s;% ∂ M, J −
(M, F)
⊕
∂ M, J +
ˆπ
s, % ∈ such that Ci j are integral operators with kernels ci j , where c11 (m, m 0 ) ∈ (M, F)⊗
(M, E ∗ ), c12 (m, n 0 ) ∈ (M, F)⊗ˆ π (∂ M, (J −)∗ ), c21 (n, m 0 ) ∈ ∂ M, J + ⊗ˆ π (M, E ∗ ),
ˆ π (∂ M, (J − )∗ ) and
c22 (n, n 0 ) ∈ ∂ M, J + ⊗
(C11 u)(m) =
Z
M
c11 m, m 0 , u m 0 E dm 0
j ;δ
with (·, ·) E denoting the pointwise pairing in the fibers of E, etc. Let Diffcl (M; E, E) be the
space of all differential operators of order j on M (acting on sections of the bundles E) that
j ;δ
belong to L cl (M; E, E) (cf., in particular, formula (40)). Then the space −∞,d;−∞ (M; v) of
all smoothing operators of type d ∈ is defined to be the set of all
= 0+
d
X
Dj
j
0
j =1
j ;0
0
0
for arbitrary j ∈ −∞,0;−∞ (M; v) and D j ∈ Diffcl (M; E, E).
Next we introduce the space of classical Green operators on M, that is an analogue of
µ,d;δ n
+ ; N− , N+ , cf. Definition 6. First, for arbitrary k, m ∈ there is an evident blockG,cl
µ,d;δ n
matrix version G,cl
in this space is continuous in the
+ ; k, m; N− , N+ . Every operator
sense
H s−µ;%−δ n+ , m
H s;% n+ , k
−→
⊕
⊕
:
H s;% n−1 , N−
H s−µ;%−δ n−1 , N+
341
Boundary value problems
n
for s > d − 21 . If ~ and ϑ are local admissible cut-off functions in + , we have
µ,d;δ n
(92)
~
ϑ ∈ G,cl
+ ; k, m; N− , N+ ,
µ,d;δ n
for every
∈ G,cl
~ is the operator of multiplication by
+ ; k,m; N− , N+ , where
diag ~⊗id m , ~| n−1 ⊗id N+ and similarly ϑ . Given bundles E, F ∈ Vect (M), J − , J + ∈
Vect (∂ M), an operator
H s;% (M, E)
H s−µ;%−δ (M, F)
:
⊕
−→
⊕
H s;% ∂ M, J −
H s−µ;%−δ ∂ M, J +
(93)
is said to be supported in a global admissible neighbourhood of ∂ M if there are global admissible
=
∈
cut-off functions φ , ψ on M such that
φ
ψ . Similarly, we say that a
n
µ,d;δ n
satisfies a relation
+ ; k, m; N− , N+ is supported in a local admissible set in + if
G,cl
=
~
ϑ for certain local admissible cut-off functions ~ and ϑ. If χ : U → V is
one of the charts (90), we have an associated chart χ 0 : U ∩ ∂ M → V ∩ n−1 , and there are
corresponding trivializations of the bundles E, F and J − , J + , respectively. χ gives rise to a
push-forward of operators
H s;% n+ , k
H s−µ;%−δ n+ , m
−→
,
χ∗ φ
⊕
⊕
ψ :
H s;% n−1 , N−
H s−µ;%−δ n−1 , N+
where k, m and N− , N+ are the fibre dimensions of the bundles E, F and J − , J + , respectively,
and φ, ψ local admissible cut-off functions supported by U .
µ,d;δ
Now G,cl (M; v) for v = E, F; J − , J + is defined to be the set of all operators =
0 + 1 + , where
(i)
∈ −∞,d;−∞ (M; v) and
0 is supported in ∪ B+1≤ j ≤N U j , cf. (90), where
χ j∗
φj 0
ψj ∈
µ,d;δ
G,cl
n
+ ; k, m; N− , N+
for arbitrary local admissible cut-off functions φ j and ψ j on M supported in U j , B + 1 ≤
j ≤ N.
(ii)
1 is an operator (93) that is supported in a collar neighbourhood of the boundary of the
finite part M, i.e., ∂ M \ C , and it is a Green operator of order µ and type d in Boutet de
Monvel’s algebra on M \ C.
It can be easily proved that this is a correct definition; in fact, the operators in the space
n
n
+ ; k, m; N− , N+ , supported in an admissible set in + , are invariant under the transition maps generated by the charts and corresponding trivializations of the involved bundles.
µ,d;δ
G,cl
µ,d;δ
D EFINITION 10. The space cl
(M; v) for µ ∈ , d ∈ , δ ∈ and v = E, F; J − ,
+
−
+
J , E, F ∈ Vect (M), J , J ∈ Vect (∂ M), is defined to be the set of all operators
(94)
=
+ +
e
r Ae
0
0
0
+
e ∈ L µ;δ 2M; E,
e F
e (with E|
e M = E, F|
e M = F) and
for arbitrary A
cl
tr
µ,d;δ
∈ G,cl (M; v).
342
D. Kapanadze – B.-W. Schulze
µ,d;δ
(M; v), v = E, F; J − , J + , induces continuT HEOREM 12. Every operator ∈ cl
ous operators
H s;% (M, E)
H s−µ;%−δ (M, F)
:
⊕
−→
⊕
H s;% ∂ M, J −
H s−µ;%−δ ∂ M, J +
for all real s > d − 21 and all % ∈ . In particular,
(M, E)
:
is also continuous in the sense
(M, F)
⊕
−→
∂ M, J −
⊕
∂ M, J +
.
This result is an easy consequence of Theorem 6 and Remark 14.
Similarly to the global principal symbol structure of operators on a closed manifold with
exit to infinity, cf. Section 2.3, we now introduce global principal symbols for an operator
µ,d;δ
∈ cl
(M; v), v = E, F; J − , J + for E, F ∈ Vect (M), J − , J + ∈ Vect (∂ M). The
e in (94). According to formulas (13), (15), (16), we
principal interior symbols only depend on A
e , σe A
e , σψ,e A
e for any A
e ∈ L µ;δ 2M; E,
e F
e , where
have σψ A
cl
∗e
∗
e : π∗ E
e
σψ A
ψ −→ πψ F , πψ : T (2M) \ 0 −→ 2M ,
e : πe∗ E
e −→ πe∗ F
e , πe : T ∗ (2M)|(2X )∧ −→ (2X)∧
σe A
∞,
∞
∧
∗
∗
∗
e :π E
e
e
σψ,e A
ψ,e −→ πψ,e F , πψ,e : T (2M) \ 0 (2X )∧ −→ (2X)∞ .
∞
Restricting this to M (and taking for the projections the same notation) we get
∗
∗
∗
e ∗
σψ ( ) := σψ A
(95)
T M\0 : πψ E −→ πψ F , πψ : T M \ 0 −→ M ,
∗
∗
∗
∧
e ∗
∧ −→ X ∞ ,
(96)
σe ( ) := σe A
T M| ∧ : πe E −→ πe F , πe : T M| X ∞
X∞
(97)
∗
∗
e ∗
σψ,e ( ) := σψ,e A
(T M\0)| X ∧ : πψ,e E −→ πψ,e F ,
∞
∧ .
πψ,e : T ∗ M \ 0 X ∧ −→ X ∞
∞
Concerning the principal boundary symbol components we first have
 0


 0
E ⊗ +
F ⊗ +
 −→ π ∗ 

σ∂ ( ) : π∂∗ 
(98)
⊕
⊕
∂
−
+
J
J
µ,d;δ
(M; v) ⊂ µ,d (M; v), E 0 =
for π∂ : T ∗ (∂ M) \ 0 → ∂ M, according to the inclusion cl
E|∂ M , F 0 = F|∂ M , cf. Section 4.1. Moreover, the e0 - and (∂, e0 )-components of (44) (in the
corresponding (m×k) block matrix-valued version) have
a simple invariant meaning with respect
to the transition maps from the local representations of on the infinite part of M. The system of
the local boundary (e0 - and (∂, e0 )-) symbols in the sense of (44) gives us bundle homomorphisms
 0
 0


F ⊗ +
E ⊗ +
∗
∗
 −→ π 0 

(99)
σe0 ( ) : πe0 
⊕
⊕
e
J−
J+
343
Boundary value problems
∧
∧ → Y∞ and
for πe0 : T ∗ (∂ M)|Y∞
 0
 0


E ⊗ +
F ⊗ +
∗ 
 −→ π ∗ 0 

σ∂,e0 ( ) : π∂,e
(100)
⊕
⊕
0
∂,e
−
J
J+
∧
∧ → Y∞ . Note that for π∂,e0 : (T ∗ (∂ M) \ 0)|Y∞
+ may be replaced by Sobolev spaces on
the half-axis for s > d − 12 , cf. analogously Section 4.1. Let
σ ( ) = σψ ( ), σe ( ), σψ,e ( ); σ∂ ( ), σe0 ( ), σ∂,e0 ( )
µ,d;δ
µ,d;δ
µ,d;δ
(M; v), and set symb cl
(M; v) = σ ( ) :
(M; v) . We then
∈ cl
∈ cl
for
have a direct generalization of Remark 13; the obvious details are left to the reader.
Note that there are natural compatibility properties between the components of σ ( ).
(101)
ν,e;%
µ,d;δ
(M; v), v = E 0 , F; J0 , J + , and ∈ cl (M; w), w =
cl
µ+ν,h;δ+%
(E, E 0 ; J − , J0 ), implies
∈ cl
(M; v ◦ w) for h = max(ν + d, e) and v ◦ w =
) = σ ( ) σ ( ) (with componentwise multiplication).
E, F; J − , J + , and we have σ (
T HEOREM 13.
∈
Theorem 13 is the global version of Theorem 5 and, in fact, a direct consequence of this
local composition result.
4.3. Ellipticity, parametrices and Fredholm property
µ,d;δ
D EFINITION 11. An operator ∈ cl
(M; v) for v = E, F; J − , J + is called elliptic
of order (µ, δ) if all bundle homomorphisms (95), (96), (97), (98), (99), (100) are isomorphisms.
Similarly to Remark 16, in the conditions for (98), (99), (100) we may replace
H s ( + ) and H s−µ ( + ), respectively, for s > max(µ, d) − 21 .
+ by
µ,d;δ
D EFINITION 12. Given
∈ cl
(M; v) for v = E, F; J − , J + an operator
∈
−µ,e;−δ
−1
−1
+
−
M; v
for v = F, E; J , J and some e ∈ is called a parametrix of if
cl
∈ −∞,dl ;−∞ (M; vl ) ,
∈ −∞,dr ;−∞ (M; vr )
for certain dl , dr ∈ , and vl = (E, E; J − , J − ), vr = F, F; J + , J + .
−
−
Note that the Theorem
13 entails σ ( )−1 = σ ( ) (with componentwise inversion) where
is a parametrix of .
T HEOREM 14. Let
(102)
µ,d;δ
∈ cl
(M; v) be elliptic. Then
H s−µ;%−δ (M, F)
H s;% (M, E)
−→
⊕
:
⊕
H s−µ;%−δ ∂ M, J +
H s;% ∂ M, J −
is a Fredholm operator for every s > max(µ, d) − 21 and every % ∈ , and has a parametrix
−µ,(d−µ)+ ;−δ
∈ cl
M; v−1 , where dl = max(µ, d) and dr = (d − µ)+ (cf. the notation in
Definition 12).
344
D. Kapanadze – B.-W. Schulze
The proof of this result can be given similarly to
Theorem 8. Alternatively, themethods
of Section 3.6 can also be used to first construct σ ( )−1 and to form e := op σ ( )−1 ∈
+
−µ,(d−µ) ;−δ M; v−1 . Then we get e − ∈ −1,e;−1 (M; v ) for some e, and we get
l
itself by a formal Neumann series argument.
µ,d;δ
(M; v) be elliptic. Then we have elliptic regularity of soR EMARK 21. Let
∈ cl
lutions in the following sense. u = f ∈ H s−µ;%−δ (M, F) ⊕ H s−µ;%−δ ∂ M, J + for any
s > max(µ, d)− 21 and % ∈ and u ∈ H r;−∞ (M, E)⊕ H r;−∞ (∂ M, J − ), r > max(µ, d)− 12 ,
implies u ∈ H s;% (M, E) ⊕ H s;% (∂ M, J − ).
in a standard manner. Composing u = f from the left by we get
In fact, we can argue
s;%
s;%
−
−
u = (1 + )u ∈ H
yields the assertion.
(M, E) ⊕ H
(∂ M, J ) and u ∈
(M, E) ⊕ (∂ M, J ) which
R EMARK 22. From Remark 21 we easily obtain that the kernel of is a finite-dimensional
subspace of (M, E)⊕ (∂ M, J − ) (and as such independent of s and %). Moreover, itcan easily
be shown that there is a finite-dimensional subspace − ⊂ (M, F) ⊕ ∂ M, J + such that
im + − = H s−µ;%−δ (M, F) ⊕ H s−µ;%−δ ∂ M, J + for all s, where im means the image
in the sense of (102). Thus ind (the index of (102)) is independent of s > max(µ, d) − 21 and
of % ∈ .
µ,d;δ
− +
R EMARK
23. Let i ∈ cl (M; vi ), vi = E, F; Ji , Ji , i = 1, 2, be elliptic operators where 1 has the same upper left corner as 2 ; then there is an analogue of Agranovich
Dynin formula for the indices ind i , i = 1, 2: There exists an elliptic operator ∈ L 0;0
cl ∂ M;
J2+ ⊕ J1− , J1+ ⊕ J2− such that
ind 1 − ind 2 = ind
.
The idea of the proof is completely analogous to the corresponding result for a compact,
smooth manifold with boundary, cf. Rempel and Schulze [16], Section 3.2.1.3. The operator
can be evaluated explicitely by applying
reductions of orders and weights (cf., also Theorem 22
below) and using a parametrix of 2 .
4.4. Construction of global elliptic boundary conditions
An essential point in the analysis of elliptic boundary value problems is the question whether an
element
µ,d;δ
A ∈ Bcl
(103)
(M; E, F)
that is elliptic with respect to the interior symbol tuple (σψ (A), σe (A), σψ,e (A)) can be regarded
as the upper left corner of an operator
(104)
µ,d;δ
∈ cl
(M; v)
for
v = E, F; J − , J +
for a suitable choice of bundles J − , J + ∈ Vect (∂ M) and additional entries of the block matrix,
is elliptic in the sense of Definition 11. We want to give the general answer and
such that
by this extend the well-known Atiyah-Bott condition from [1]. Atiyah and Bott formulated a
345
Boundary value problems
topological obstruction for the existence of Shapiro-Lopatinskij elliptic boundary conditions for
elliptic differential operators on a compact smooth manifold (concerning the corresponding conditions for pseudo-differential boundary value problems cf. Boutet de Monvel [3]). To formulate
the result in our situation, without loss of generality we consider the case µ = d = δ = 0. The
general case is then a consequence of a simple reduction of orders, types and weights, applying
Theorem 22 and Remark 29 below. The constructions for Theorem 9 above can be generalized
0,0;0
to a given (σψ , σe , σψ,e )-elliptic operator A ∈ Bcl
(M; E, F) as follows. Starting point are
the boundary symbols
as operator families
σ∂ (A)(y, η)
for
(y, η) ∈ T ∗ (∂ M) \ 0 ,
σe0 (A)(y, η)
for
σ∂,e0 (A)(y, η)
for
∧ ,
(y, η) ∈ T ∗ (∂ M)|Y∞
(y, η) ∈ T ∗ (∂ M) \ 0 ∧ ,
Y∞
E 0y ⊗ L 2 ( + ) −→ Fy0 ⊗ L 2 ( + ) ,
in contrast to (98)-(100)we now prefer L 2 ( + ) instead of + , according to the considerations in Section 3.6 . For points y ∈ ∂ M belonging to the infinite exit to infinity ∼
=
(1 − ε, ∞) × Y∞ it makes sense to talk about |y| > R (this simply means that the associated
axial variable is larger than R). First there is an obvious analogue of Proposition 4 that refers to
points (y, η) ∈ T ∗ (∂ M) for y ∈ (1 − ε, ∞) × Y∞ .
P ROPOSITION 6. For every ε > 0 there exists an R = Rε > 0 such that
<ε
σ∂ (A)(y, η) − σ∂,e0 (A)(y, η)
(105)
E 0 ⊗L 2 ( ),F 0 ⊗L 2 ( )
for all |y| > R and η 6= 0,
σ∂ (A)(y, η) − σe0 (A)(y, η)
(106)
for all |y| > R and |η| > R,
σe0 (A)(y, η) − σ∂,e0 (A)(y, η)
(107)
y
E 0y ⊗L 2 (
+
y
+
0
2
+ ),Fy ⊗L ( + )
E 0y ⊗L 2 (
0
2
+ ),Fy ⊗L ( + )
for all |y| ∈ (1 − ε, ∞) × Y∞ and |η| > R.
C OROLLARY 3. There is an R = Rε > 0 such that
σ∂ (A)(y, η) − σe0 (A)(y, η)
E 0 ⊗L 2 (
y
0
2
+ ),Fy ⊗L ( + )
for all |y| = |η| = R.
and
For ε > 0 we set
Tε = (y, η) ∈ T ∗ (∂ M) : |y| = |η| = Rε ,
Zε
j
=
j
=
Hε
<ε
<ε
<ε
Dε = Tε × [0, 1]
(y, η) ∈ T ∗ (∂ M) : y ∈ ∂ M \ {|y| > Rε + j } , |η| = Rε ,
(y, η) ∈ T ∗ (∂ M) : |y| = Rε , |η| ≤ Rε + j
346
D. Kapanadze – B.-W. Schulze
j
j
j
for j = 0, 1, ∞. Moreover, let ε = Z ε ∪d Hε ∪b Dε / ∼, with ∪d being the disjoint union and
∪b the disjoint union combined with the projection to the quotient space that is given by natural
j
j
identifications Tε ∩ Z ε ∼
= Tε × {0}, Tε ∩ Hε ∼
= Tε × {1}. Write Z ε = Z ε0 , Hε = Hε0 , ε = 0ε .
Furthermore, for 0 ≤ τ ≤ 1 we set Dε,τ := Tε × [0, τ ] and form ε,τ := Z ε ∪d Hε ∪b Dε,τ ,
0 ≤ τ ≤ 1, where ∪b is the disjoint union combined with the projection from the identification
Tε ∩ Z ε ∼
= Tε × {0}, Tε ∩ Hε ∼
= Tε × {τ }. We now introduce an operator function F(m), m ∈ ε ,
as follows:
(108)
F(y, η)
=
σ∂ (A)(y, η)
for
m = (y, η) ∈ Z ε ,
(109)
F(y, η)
=
σe0 (A)(y, η)
for
m = (y, η) ∈ Hε ,
(110) F(y, η, δ)
=
δσ∂ (A)(y, η) + (1 − δ)σe0 (A)(y, η)
for
m = (y, η, δ) ∈ Dε .
We then have an operator family
F(m) : E 0y ⊗ L 2 ( + ) −→ Fy0 ⊗ L 2 ( + )
continuously depending on m ∈ ε , and F is Fredholm operator-valued, provided ε > 0 is
sufficiently small. This gives us an index element ind ε F ∈ K ( ε ). For analogous reasons as
above in connection with (70) we form
(111)
ind ε σ∂ (A)(y, η), σe0 (A)(y, η) ∈ K ( ε ) ,
ε := ε,0 ⊂ T ∗ (∂ M). The canonical projection T ∗ (∂ M) → ∂ M induces a projection πε :
ε → Bε where
Bε := ∂ M \ {y ∈ ∂ M : |y| > Rε } .
Given an arbitrary (σψ , σe , σψ,e )-elliptic operator (103) we set
(112)
s −µ −δ
0
A R −s
E
A 0 = R F0
s −µ
s −µ,0;0
−s
−s ,0;0
for any s0 > max(µ, d)− 21 , where R F0
∈ Bcl0
(M; F, F) and R E 0 ∈ Bcl 0
(M; E,
E) are order reducing operators in the sense of Remark 29, and −δ a weight reducing factor
0,0;0
on M of a similar meaning as that in Remark 15. Then we have A 0 ∈ Bcl
(M; E, F), and A 0
is also (σψ , σe , σψ,e )-elliptic. In the sequel the choice of the specific order and weight reducing
factors is unessential.
The following theorem is an analogue of the Atiyah-Bott condition, formulated in [1] for
the case of differential operators on a smooth compact manifold with boundary, and established
by Boutet de Monvel [3] for pseudo-differential boundary value problems with the transmission
property.
T HEOREM 15. Let M be a smooth manifold with boundary and conical exits to infinity,
µ,d;δ
E, F ∈ Vect (M), and let A ∈ Bcl
(M; E, F) be a (σψ , σe , σψ,e )-elliptic operator. Then
there exists an elliptic operator (104) having A as the upper left corner if and only if the operator
(112) satisfies the condition
(113)
ind ε σ∂ (A 0 ), σe0 (A 0 ) ∈ πε∗ K (Bε ) ,
for a (sufficiently small) ε > 0, πε : ε → Bε .
If (113) holds, for any choice of the additional bundles J − , J + ∈ Vect (∂ M) in the sense of
(104) we have
ind ε σ∂ (A 0 ), σe0 (A 0 ) = πε∗ J + | Bε − J − | Bε .
(114)
347
Boundary value problems
Proof. First note that the criterion of Theorem 15 does not depend on the choice of order re0,0;0
ductions. Moreover, such reductions allow us to pass from A 0 ∈ Bcl
(M; E, F) and an
0,0;0
(M; v) with A 0 as upper left corner to the corresponding operators
cl
µ,d;δ
µ,d;δ
∈ cl (M; v). Thus, without loss of generality we assume
A ∈ Bcl
(M; E, F) and
associated
0 ∈
µ = d = δ = 0 and talk about A and , respectively. Clearly, the existence of an elliptic ∈
0,0;0
0,0;0
(M; v), v = E, F; J − , J + , to a given ((σψ , σe , σψ,e )-) elliptic A ∈ Bcl
(M; E, F)
cl
implies
ind ε σ∂ (A), σe0 (A) = πε∗ J + | Bε − J − | Bε ,
(115)
because the role of the bundles J − , J + in the components of (σ∂ ( ), σe0 ( ), σ∂,e0 ( )) is just
that they fill up the Fredholm families (σ∂ (A), σe0 (A), σ∂,e0 (A)) to block matrices of isomorphisms; combining this with Corollary 3 we get the desired index relation. Conversely assume
that (115) holds. Then the construction of an elliptic operator in terms of A takes place on
the level of boundary symbols. In other words, the Fredholm families have to be first completed
to block matrices of isomorphisms. This can be done when we also include (110) into the construction, in order to deal with continuous
Fredholm families, and then drop the “superfluous”
part on Dε . Thus the first step to find is to fill up F(m), m ∈ ε , to a family of isomorphisms
(m) =
F(m)
t (m)
K (m)
Q(m)
E 0y ⊗ L 2 ( + )
Fy0 ⊗ L 2 ( + )
:
−→
,
⊕
⊕
J y−
J y+
m ∈ ε . Here we employ the fact that the additional finite-dimensional vector spaces corresponding to the entries (m)i j for i + j > 1 are fibres in some bundles J − and J + on Bε ,
using the hypothesis on F(m), further local representations with respect to y ∈ Bε and the
invariance under the transition maps.
Similarly to the local theory we find (m) (locally) in
form of D 0,0 + ; k, k; N− , N+ -valued families (here, k is the fibre dimension both of E and
F, and N± are the fibre dimensions J ± , and we employ a corresponding generalization of the
notation of Section 3.1 to (k × k)-matrices in the upper left corners), smoothly dependent on
(y, η) on Z ε or Hε . In this construction ε > 0 is chosen sufficiently small, i.e., R = Rε large
enough. The construction so far gives us
σ∂ ( )| Z ε and σe0 ( )| Hε . Extending σ∂ ( )| Z ε (by
κλ -homogeneity)
for all η 6= 0 and σe0 ( )| Hε (by usual
homogeneity)
for all |y| ≥ Rε we get
σ∂ ( ) and σe0 ( ) everywhere. Next we form σ∂,e0 ( ) = σe0 (σ∂ ( )) = σ∂ (σe0 ( )). Thus
we have an elliptic symbol tuple σ ( ) := (σψ ( ), σe ( ), σψ,e ( ); σ∂ ( ), σe0 ( ), σ∂,e
0 ( )),
where the first three components equal the given ones, namely (σψ ( ), σe ( ), σψ,e ( )). By
0,0;0
virtue of σ ( ) ∈ symb cl
(M; v) we can apply an operator convention
to get
0,0;0
0,0;0
op : symb cl
(M; v) −→ cl
(M; v)
itself.
R EMARK 24. As is well-known for compact smooth manifolds with boundary there are in
general elliptic differential operators that violate the Atiyah-Bott condition. An example is the
Cauchy-Riemann operator ∂ z in a disk in the complex plane. One may ask what happens for ∂ z ,
: Im z ≥ 0}. In this case the Atiyah-Bott condition is, of course,
say, in a half-plane {z ∈
violated, too, but the operator ∂ z is worse. In fact, there is no constant c ∈ such that c + ∂z is
(σψ , σe , σψ,e )-elliptic, such that also for that reason there are no global elliptic operators in
the half-plane with σψ ( ) = σψ ∂ z .
348
D. Kapanadze – B.-W. Schulze
5. Parameter-dependent operators and applications
5.1. Basic observations
As noted in the beginning the theory of pseudo-differential boundary value problems on a manifold with exits is motivated by a number of interesting applications. In this connection boundary value problems appear as parameter-dependent operator families, where parameters λ ∈
are involved like additional covariables in the symbols. All essential notions and results have
reasonable analogues in the parameter-dependent case, though there are some specific new aspects. The parameter-dependent ellipticity that we formulate below is also of interest for the
(non-parameter-dependent) algebras themselves, insofar, as we shall see, they provide a tool to
construct order reducing elements within the algebras in a transparent way.
µ
First we have a direct analogue of the symbol classes with the transmission property Scl ×
n+l
+ × ξ,λ tr , cf. Section 2.2, where ξ is to be replaced by (ξ, λ). Concerning symbol estimates,
the parameter-dependent case is not a new situation; in estimate (1) we admitted independent
dimensions of x- and ξ -variables, anyway. Similarly, we can talk about weighted symbol classes
S µ;δ nx × n+l
ξ,λ , where ξ in the estimates (7) is replaced by (ξ, λ). The material of Section 2.3
on weighted symbols that are classical in x and ξ has an evident parameter-dependent analogue,
in other words, we have the symbol classes
µ;δ
n × n+l
(116)
Scl
x
ξ,λ
ξ,λ;x
including the (λ-dependent) principal symbols
σ (a) = σψ (a)(x, ξ, λ), σe (a)(x, ξ, λ), σψ,e (a)(x, ξ, λ)
for all a(x, ξ, λ) belonging to (116), with σψ (a)(x, ξ, λ) being given on n × n+l \ 0 ,
σe (a)(x, ξ, λ) on ( n \ 0) × n+l and σψ,e (a)(x, ξ, λ) on ( n \ 0) × n+l \ 0 . We set
µ,δ n
µ;δ
n × n+l , where Op is a bijection beL cl
; l = Opx (a)(λ) : a(x, ξ, λ) ∈ Scl
x
ξ,λ;x
tween the parameter-dependent symbol and operator spaces for all µ, δ ∈ . Then, in particular,
L −∞;−∞ n ; l = l , L −∞;−∞ n ,
where L −∞;−∞ ( n ) is identified with ( n × n ).
If M is a manifold with exits to infinity in the sense of Section 2.3, we also have the global
µ;δ
spaces of (classical) parameter-dependent operators L cl M; E, F; l for E, F ∈ Vect(M).
(Clearly, there is also the non-classical context, but we want to employ homogeneous principal
symbols; thus we content ourselves with the classical case). The parameter-dependent homoge
µ;δ
neous principal symbols for A ∈ L cl M; E, F; l are bundle homomorphisms
(117)
∗ E −→ π ∗ F ,
σψ (A) : πψ
ψ
(118)
σe (A) : πe∗ E −→ πe∗ F ,
(119)
πψ : T ∗ M × l \ 0 −→ M ,
l −→ X ∧ ,
∧ ×
πe : T ∗ M| X ∞
∞
∗
∗
∗
l
σψ,e (A) : πψ,e E −→ πψ,e F ,
πψ,e : T M ×
\ 0 X −→ X ∞ ,
∞
here, 0 means (ξ, λ) = 0.
µ;δ
µ;δ
Notice that A ∈ L cl M; E, F; l implies A(λ0 ) ∈ L cl (M; E, F) for every fixed λ0 ∈
l . Clearly, the associated principal symbols σ (A(λ )), σ
ψ
ψ,e (A(λ0 )) do not depend on λ0 . In
0
349
Boundary value problems
this connection we also call (117), (118) and (119) the parameter-dependent principal symbols
µ;δ
of A(λ). Every A(λ) ∈ L cl M; E, F; l gives rise to families of continuous operators
(120)
A(λ) : H s;% (M, E) −→ H s−µ;%−δ (M, F)
for all s, % ∈ . Let ν ≥ µ and set
(121)
bµ,ν (λ) =
hλiµ
hλiµ−ν
for ν ≥ 0 ,
for ν ≤ 0 .
We then have the following result:
µ;δ
T HEOREM 16. Let A(λ) ∈ L cl M; E, F; l be regarded as a family of continuous operators
A(λ) : H s;% (M, E) −→ H s−ν;%−δ (M, F)
for every ν ≥ µ. Then there is a constant m > 0 such that the operator norm fulfils the estimate
(122)
kA(λ)k
H
s;% (M,E),H s−ν;%−δ (M,F )
for all λ ∈ l .
≤ mb
µ,ν (λ)
We have no explicit reference for this result, though the proof is not really difficult; so the
details are left to the reader.
µ;δ
An operator A(λ) ∈ L cl M; E, F; l is called parameter-dependent elliptic if (117),
(118) and (119) are isomorphisms.
M; E, F; l be parameter-dependent elliptic. Then
−µ;−δ
there is a parameter-dependent parametrix P(λ) ∈ L cl
M; F, E; l , i.e.,
µ;δ
T HEOREM 17. Let A(λ) ∈ L cl
P(λ)A(λ) − I ∈ L −∞;−∞ M; E, E; l ,
A(λ)P(λ) − I ∈ L −∞;−∞ M; F, F; l .
Moreover, there is a C > 0 such that (120) are isomorphisms for all |λ| ≥ C and all s, % ∈ .
The proof of the first part of the theorem is straightforward, the second assertion is a direct
consequence.
Next let M be a smooth manifold with smooth boundary, not necessarily compact. There
is then a direct parameter-dependent analogue of the class of pseudo-differential boundary value
problems µ,d (M; v), cf. Definition 9, namely
µ,d M; v; l .
(123)
To define (123) we simply have to replace the ingredients of (84) by the corresponding parameter
µ
+ and (λ), respectively. Here, A(λ)
e
e
e F;
e l with
dependent versions r+ A(λ)e
∈ L cl 2M; E,
tr
obvious meaning of notation (recall that “cl” here only means “classical” in the covariables,
µ,d
though M may be non-compact) and (λ) ∈ G M; v; l , also being defined along the
lines of the class without parameters (all symbols simply contain λ as extra covariable, i.e.,
(ξ, λ) instead of ξ in the interior and (η, λ) instead of η near the boundary), and the parameterdependent smoothing operators are given by
−∞,d M; v; l = l , −∞,d (M; v) ,
(124)
350
D. Kapanadze – B.-W. Schulze
where −∞,d (M; v) is equipped with its standard Fréchet topology.
For
∈ µ,d M; v; l we have parameter-dependent homogeneous principal symbols,
namely
∗ E −→ π ∗ F ,
σψ ( ) : π ψ
πψ : T ∗ M × l \ 0 −→ M ,
(125)
ψ
(126)
π∂ : T ∗ (∂ M) × l \ 0 −→ ∂ M .
σ∂ ( ) : π∂∗ E −→ π∂∗ F ,
∈ µ,d M; v; l implies (λ0 ) ∈ µ,d (M; v) for every fixed λ0 ∈ l , and we call (125),
(126) the parameter-dependent
principal symbols of (λ) if we want to distinguish them from
the usual ones of (λ0 ) that are independent of λ0 .
An element ∈ µ,d M; v; l is called parameter-dependent elliptic if (125), (126) are
isomorphisms.
T HEOREM 18. Let
∈
µ,d M; v;
l
be parameter-dependent elliptic. Then there is
+
a parameter-dependent parametrix
∈ −∞,(d−µ) M; v−1 ; l in a similar sense as in
Remark 20; here, the remainders are smoothing in the sense of (124).
The proof is similar to that of Theorem 14 above.
T HEOREM 19. Let M be a compact smooth manifold with boundary, and let
v; l be parameter-dependent elliptic. Then there is a C > 0 such that
(λ) :
∈ µ,d M;
H s (M, E)
H s−µ (M, F)
⊕
−→
⊕
H s ∂ M, J −
H s−µ ∂ M, J +
are isomorphisms for all |λ| ≥ C and all s > max(µ, d) − 12 .
Theorem 19 is a direct corollary of Theorem 18.
R EMARK 25. In the cases that we discussed so far in the parameter-dependent set-up (i.e.,
“closed” manifolds with exits to infinity or smooth compact manifolds with boundary), where
elliptic operators induce isomorphisms between the Sobolev spaces for large |λ|, we can easily conclude that the inverse maps belong λ-wise to the corresponding algebras in the nonparameter-dependent sense (as such they are reductions of orders in the algebras). It suffices
to observe that when 1 + {smoothing operator} in one of our algebras is invertible, the inverse is
of analogous structure and can be composed with the parametrix. This can even be done in the
parameter-dependent framework for large |λ|, such that, in fact, the inverses for large |λ| are also
in the corresponding parameter-dependent class.
5.2. Boundary value problems for the case with exits to infinity
In the preceding section we extended some “standard” pseudo-differential algebras to the parameter-dependent variant, namely the algebra on a “closed” smooth manifold M with (conical) exits to infinity and the algebra of boundary value problems with the transmission property on a smooth manifold M with boundary (compact or non-compact). Now we formulate
the calculus on a smooth manifold M with boundary and (conical) exits to infinity. In other
words, we extend the material of Sections 3.2, 3.3, 3.4, 3.5, 3.6, 4.2 and 4.3 to the parameterdependent case. This is to a large extent straightforward; so we content ourselves with the
351
Boundary value problems
basic definitions and crucial points. Let us consider the parameter-dependent variant of symbol
and operator spaces of Section 2.4 that is an operator-valued generalization of the corresponding scalar symbols and operator spaces, respectively, as they are studied in Section 2.3. Similarly to the remarks in the beginning of the preceding section the essential constructions for the
operator-valued symbols with parameters are practically the same as those without parameters.
µ;δ
In particular, we have the parameter-dependent spaces of symbols S(cl) q × q+l ; E, e
E =:
µ;δ
E; l based on strongly continuous groups of isomorphisms {κτ }τ ∈ +
S(cl) q × q ; E, e
on E, {κ̃τ }τ ∈ + on e
E, and the parameter-dependent spaces of pseudo-differential operators
µ;δ
µ;δ q
l
e
E; l . In the latter operator space M is, of course, a
; E, E;
or L (cl) M; E, e
L (cl)
“closed” manifold with conical exits to infinity. Let us examine the behaviour of the operator norm with respect to the parameter λ. In the present situation the corresponding analogues of
estimates (122) refer to global weighted Sobolev spaces
s;% (M, E)
(127)
s (M,
(on a closed manifold M with conical exits to infinity) that are defined as subspaces of loc
−%
s
q
E) locally modelled by hyi
( , E)|0 for q = dim M and suitable open subsets 0 ⊂ q
that are conical in the large (recall that the global weighted spaces H s;% (M) in Section 2.3 have
been introduced by a similar scheme). Recall that for strongly continuous groups of isomorphisms {κτ }τ ∈ + on E and {κ̃τ }τ ∈ + on e
E there are constants K and K̃ , respectively, such
that
e
K
kκτ k (E) ≤ chτ i K , kκ̃τ k e
E ≤ c̃hτ i
for all τ ∈ + and certain constants c, c̃ > 0; hτ i = 1 + τ 2
1
2.
µ;δ
T HEOREM 20. Let A(λ) ∈ L (cl) M; E, e
E; l be regarded as a family of continuous operators
A(λ) : s;% (M, E) −→ s−ν;%−δ M, e
E
for some ν ≥ µ. Then there is a constant m > 0 such that the operator norm fulfils the estimate
kA(λ)k
for all λ ∈ l , cf. (121).
s;% (M,E),
s−ν;%−δ
e,ν+K + K
e(λ)
M,e
E ≤ mbµ+K + K
For the case of compact M and spaces s (M, E) = s;0 (M, E) a similar theorem is
proved in Behm [2]. This extends to the non-compact case with conical exits and weighted spaces
in a similar manner as in the scalar situation; for the corresponding technique, cf. Dorschfeldt,
Grieme, and Schulze [5] and Seiler [32].
Let us now return to the case of a smooth manifold M with boundary and conical exits to
infinity. First, there is the space
−∞,d;−∞ M; v; l := l ; −∞,d;−∞ (M; v)
(128)
E, F; J − , J + of parameter-dependent smoothing operators on M, using
−∞,d;−∞ (M; v) in its canonical Fréchet topology.
Another simple ingredient of the class
µ,d;δ M; v; l
(129)
for v
=
352
D. Kapanadze – B.-W. Schulze
+ , where
e
that will be defined below is the space of all operator families r+ A(λ)e
e F;
e l .
e ∈ L µ;δ 2M; E,
A(λ)
(130)
cl
tr
Here, “cl” means classical in covariables and variables in the local representations on conical
subsets of M; the transmission property including parameters with respect to ∂ M has been defined in Section 5.1; the interpretation of the weight δ at infinity is the same as in Section 2.3.
µ,d;δ n
Furthermore, we have a direct analogue of cl
+ ; k, m; N− , N+ , cf. the m × k block
matrix-version of Definition 6 in the parameter-dependent case, namely
n
µ,d;δ
l .
+ ; k, m; N− , N+ ;
cl
and in equality (38) that we alAn inspection of all ingredients shows that (except for
ready defined above) the only new point is to replace the amplitude function a(y, η) in (38) by
µ,d;δ n−1
× n−1+l ; k, m; N− , N+ , the parameter-dependent
a(y, η, λ) from the space cl
m × k-block matrix version of the corresponding space in Definition 5. Finally, we get (129) by
a straightforward generalization of the constructions for Definition 10. In fact, only the abovementioned ingredients are involved, except for evident invariance properties (under transition
maps) of corresponding subspaces of parameter-dependent Green operators and localization by
admissible cut-off functions (those are the same as for the case without parameters). Summing
up we have introduced all data of the following definition.
µ,d;δ
D EFINITION 13. The space cl
M; v; l for µ ∈ , d ∈ , δ ∈
E, F; J − , J + is defined to be the set of all operator families
+
+ 0
e
r A(λ)e
(131)
(λ) =
+ (λ) ,
0
0
and v =
µ;δ
l , for arbitrary A(λ)
e
e F;
e l (with E|
e M = E, F|
e M = F) and
∈ L cl 2M; E,
tr
µ,d;δ
l .
G,cl M; v;
λ∈
µ,d;δ
R EMARK 26. By definition we have cl
hand side is understood in the sense of (123).
(λ) ∈
M; v; l ⊂ µ,d M; v; l where the right
e on
Applying the definition of global parameter-dependent symbols (117), (118), (119) to A
2M and restricting them to M (similarly to (95), (96), (97)) we get the parameter-dependent
principal interior symbols
∗ E −→ π ∗ F ,
(132)
σψ (A) : πψ
πψ : T ∗ M × l \ 0 −→ M ,
ψ
(133)
(134)
σe (A) : πe∗ E −→ πe∗ F ,
l −→ X ∧ ,
∧ ×
πe : T ∗ M| X ∞
∞
∧ .
∗
∗
∗
l
σψ,e (A) : πψ,e E −→ πψ,e F ,
πψ,e : T M ×
\ 0 X ∧ −→ X ∞
∞
A direct generalization of (98), (99), (100) to the parameter-dependent case gives us the parameter-dependent principal boundary symbols
 0
 0


F ⊗ +
E ⊗ +
∗
∗
 −→ π 

(135)
σ∂ ( ) : π ∂ 
⊕
⊕
∂
J−
J+
353
Boundary value problems
for π∂ : T ∗ (∂ M) × l \ 0 → ∂ M,
 0


 0
E ⊗ +
F ⊗ +
 −→ π ∗0 

σe0 ( ) : πe∗0 
(136)
⊕
⊕
e
−
+
J
J
l → Y ∧ and
∧ ×
for πe0 : T ∗ (∂ M)|Y∞
∞
 0
 0


F ⊗ +
E ⊗ +
∗
∗
 −→ π 0 

(137)
σ∂,e0 ( ) : π∂,e0 
⊕
⊕
∂,e
J−
J+
∧ . Further explanation to the latter bundle homomorfor π∂,e0 : T ∗ (∂ M) × l \ 0 Y ∧ → Y∞
∞
phisms is unnecessary, because the only novelty are the additional covariables λ ∈ l .
µ,d;δ
µ,d
∈ cl
M; v; l implies (λ0 ) ∈ cl (M; v) for every fixed λ0 ∈
R EMARK 27.
l , and the symbols σ ( (λ )), σ
ψ
ψ,e ( (λ0 )), σ∂ ( (λ0 )), σ∂,e0 ( (λ0 )) do not depend on λ0 . If
0
necessary we point out that (132), (133), (134), (135), (136), (137) are the parameter-dependent
principal symbols of (λ).
R EMARK 28. There is an obvious analogue of the composition result of Theorem
13 for
the parameter-dependent case, including the symbol rule, where in the present case σ ( ) is the
tuple of parameter-dependent principal symbols (132)-(137), similarly to (101).
µ,d;δ
D EFINITION 14. An operator ∈ cl
M; v; l is called parameter-dependent elliptic if all principal symbol homomorphisms (132)-(137) are isomorphisms. An operator (λ) ∈
−µ,e;−δ
M; v−1 ; l for some e ∈ is called a parameter-dependent parametrix if
cl
(λ) (λ) − ∈ −∞,dl ;−∞ M; vl ; l ,
(λ) (λ) − ∈ −∞,dr ;−∞ M; vr ; l
for certain dl , dr ∈ , and vl = (E, E; J − , J − ), vr = F, F; J + , J + .
T HEOREM 21. Let M be a smooth manifold with boundary and (conical) exits to infinity,
µ,d;δ
and (λ) ∈ cl
M; v; l , v = E, F; J − , J + , be parameter-dependent elliptic. Then
−µ,(d−µ)+ ;−δ
there exists a parameter-dependent parametrix (λ) ∈ cl
M; v−1 ; l , where
the types in the remainders are dl = max(µ, d), dr = (d − µ)+ . Moreover,
(138)
H s−µ;%−δ (M, F)
H s;% (M, E)
−→
⊕
(λ) :
⊕
H s−µ;%−δ ∂ M, J +
H s;% ∂ M, J −
is a family of Fredholm operators of index 0 for every s > max(µ, d)− 12 , and there is a constant
C > 0 such that (138) are isomorphisms for all |λ| ≥ C.
The basic idea of proving results of this type has been briefly discussed in Remark 25 above.
Also in the present situation of Theorem 21 we first construct a parameter-dependent
parametrix
(λ) by
inverting
the
parameter-dependent
principal
symbol
of
(λ)
and
get
(λ)
(λ) − =
(λ),
(λ)
(λ)
−
=
(λ),
with
smoothing
operators
in
Boutet
de
Monvel’s
algebra.
By
r
l
virtue of (128) it is fairly obvious that + {smoothing operator} is invertible in the same class,
such that it can be composed with (λ).
354
D. Kapanadze – B.-W. Schulze
T HEOREM 22. Let M be a smooth manifold with boundary and (conical) exits to infinity,
and let E ∈ Vect (M), µ ∈ , δ ∈ . Then there exists a parameter-dependent elliptic element
µ,0;δ
µ;δ
M; E, E; l that induces isomorphisms
R E (λ) ∈ cl
µ;δ
R E (λ) : H s;% (M, E) −→ H s−µ;%−δ (M, E)
(139)
−µ,0;−δ
µ;δ
for all s, % ∈ and all λ ∈ l , and we have R E (λ)−1 ∈ cl
M; E, E; l . Similarly,
there exists a parameter-dependent elliptic element
for every J ∈ Vect (∂ M) and ν, δ ∈
ν;δ
l that induces isomorphisms
R 0 ν;δ
(λ)
∈
L
∂
M;
J,
J
;
J
cl
(140)
for all s, % ∈
ν;δ
R 0 J (λ) : H s;% (∂ M, J ) −→ H s−ν;%−δ (∂ M, J )
−1 ∈ L −ν;−δ ∂ M; J, J ; l .
and all λ ∈ l , and we have R 0 ν;δ
J (λ)
cl
R EMARK 29. Combining the latter theorem with Remark 27, by inserting any fixed λ0 ∈
µ,0;δ
(M; E, E)
cl
l into (139) and (140) we get order reducing operators in the operator spaces
and L ν;δ (∂ M; J, J ), respectively.
R EMARK 30. The concept of algebras of parameter-dependent operators can also be formulated for more general parameter sets 3 ⊆ q that have the property λ ∈ 3 ⇒ cλ ∈ 3 for
all c ≥ 1. Examples are
3 = l \ 0,
3 = l \ {|λ| < C}
or single rays 3 in q . For such 3 all our operator classes have corresponding variants, e.g.,
µ;δ
µ,d
L cl (M; E, F; 3), cf. Theorem 16, cl (M; v; 3), cf. formula (123), etc. The behaviour
of operators in these spaces for small λ ∈ 3 remains unspecified; we assume, for instance,
smoothness in λ. The parameter-dependent symbols now refer to λ ∈ 3, and we have evident
generalizations of the corresponding parameter-dependent ellipticities and parametrices.
Let us explicitly formulate a corresponding extension of Theorem 21 in the version with 3:
µ,d;δ
(M; v; 3), v = E, F; J − , J + , be parameter-depencl
−µ,(d−µ)+ ;−δ
M;
dent elliptic. Then there exists a parameter-dependent parametrix (λ) ∈ cl
v−1 ; 3 , with the above-mentioned types dl and dr of remainders. Furthermore, the operators
(138) are Fredholm and of index 0 for all s > max(µ, d)− 12 , there is a constant C > 0 such that
−µ,(d−µ)+ ;−δ
M; v−1 ; 3C for
(138) are isomorphisms for all |λ| ≥ C, and we have −1 ∈ cl
3C = {λ ∈ 3 : |λ| ≥ C}.
T HEOREM 23. Let
(λ) ∈
µ,d;δ
R EMARK 31. The spaces cl
(M; v; 3), v = E, F; J − , J + , (as well as the other
operator spaces, e.g., from Sections 4.1 or 5.1) can easily be generalized to the case of DouglisNirenberg orders (DN-orders) with a corresponding ellipticity; the results carry over to the variant with DN-orders.
The Douglis-Nirenberg generalization refers to representations of the bundles as direct sums
E = ⊕km=1 E m , F = ⊕ln=1 Fn , J − = ⊕bi=1 Ji− , J + = ⊕cj =1 J +
j . Operators are then represented as block matrices, composed with diagonal matrices of order reductions on M and ∂ M,
355
Boundary value problems
respectively. The constructions are straightforward, so we do not really discuss the details, but in
some cases below we need notation. This concerns DN-orders for the boundary operators, where
(λ) =
(λ) e(λ) −1 (λ)
µ,d;δ
with e(λ) ∈ cl
(M; v; 3), v = E, F; ⊕bi=1 Ji− , ⊕cj =1 J +
j , and
!
!
1
0
1
0
γ
(λ) =
(λ) =
,
β
0 diag R 0 j j (λ)
0 diag Q 0 i i (λ)
with 1 denoting identity operators referring to F and E and order reducing operator families
β
β ;0
∂ M; Ji− ; Ji− ; 3 ,
i = 1, . . . , b ,
Q 0 i i (λ) ∈ L cli
γ
γ ;0
+
R 0 j j (λ) ∈ L clj
∂ M; J +
j = 1, . . . , c ,
j ; Jj ;3 ,
βi , γ j ∈ . Such a situation is customary in elliptic boundary value problems for differential
operators with differential boundary conditions. In this particular case all bundles Ji− are of fibre
dimension 0, while parametrices refer to the case that the bundles J +
j are of fibre dimension 0.
Theparameter-dependent ellipticity of (λ) is defined to be the parameter-dependent ellipticity
of e(λ), and we get a parameter-dependent parametrix of (λ) by (λ) = (λ) e(λ) −1 (λ)
when e(λ) denotes a parameter-dependent parametrix of e(λ).
In any case the involved orders are known and fixed. Therefore, given (β1 , . . . , βb ) and
(γ1 , . . . , γc ), we set
n
o
µ,d;δ
µ,d;δ
(λ) e(λ) −1 (λ) : e(λ) ∈ cl
(141)
(M; v; 3) =
(M; v; 3) .
cl
−µ,(d−µ)+ ;−δ
Parametrices then belong (by notation) to cl
M; v−1 ; 3 .
5.3. Relations to the edge pseudo-differential calculus
In this section we want to discuss relations between our calculus of boundary value problems
on non-compact manifolds with exits and the theory of boundary value problems in domains
with edges. Particularly simple edge configurations occur in models of the crack theory. In local
terms the situation can be described by 2 \ + × , where  ⊆ q plays the role of a
crack boundary (for crack problems in 3 we have q = 1), 2 is the normal plane to the crack
boundary, and + ⊂ 2 is a coordinate half-axis corresponding to the intersection of the crack
with 2 . This situation is studied in detail in Kapanadze and Schulze [11]. A special aspect
of this approach is that the crack boundary is regarded as an edge and 2 \ + as an infinite
model cone with the origin of 2 as the tip of the cone. More precisely, the cone consists of a
configuration, where two copies of + constitute the slit in 2 with separate elliptic boundary
conditions on the ±-sides. The edge symbol calculus for this situation may be regarded as a
parameter-dependent “infinite” cone theory, consisting of the calculus on a bounded part of the
cone near the tip and that in the exit sense elsewhere. The latter calculus treats both sides of the
slit separately, and its contribution can be formulated in terms of parameter-dependent boundary
value problems in the half-space, together with a localization. We now formulate a result that is
typical for this theory. Let
X
β
A=
aαβ (x, y)Dxα D y
|α|+|β|≤m
356
D. Kapanadze – B.-W. Schulze
be a differential operator in U ×  3 (x, y), U ⊆ n open, containing the origin, and  ⊆ q
open (for simplicity we assume A to be scalar; the considerations for systems are analogous),
with coefficients aαβ ∈ C ∞ (U × ). For the edge symbol calculus the coefficients are to be
frozen at x = 0. This gives us an operator family
X
aαβ (0, y)Dxα ηβ : H s n+ −→ H s−m n+ .
σ∧ (A)(y, η) :=
|α|+|β|=m
Set (κλ u)(x) =
n
λ2
u(λx), λ ∈ + . Then we have
σ∧ (A)(y, λη) = λm κλ σ∧ (A)(y, η)κλ−1
(142)
for all λ ∈ + and all y, η.
Let A be elliptic, and let T = (r0 B1 , . . . , r0 B N ) be a vector of trace operators, r0 u = u|
for differential operators
X
j
β
bαβ (x, y)Dxα D y
Bj =
n−1 ,
|α|+|β|≤m j
j
with coefficients bαβ ∈ C ∞ (U × ) and m j < m for all j . Assume that the boundary value
problem
in U ∩ n+ ,
Au = f
(143)
Tu = g
on U ∩ n−1
is elliptic in the sense that the trace operators satisfy the Shapiro-Lopatinskij condition with
respect to A clearly, N is known by the problem, e.g., if m is even and n + q ≥ 3, we have
N = m2 . Let us form
σ∧ (T )(y, η) = (σ∧ (T1 )(y, η), . . . , σ∧ (TN )(y, η))
1
P
j
where σ∧ (T j )(y, η) = r0 |α|+|β|=m j b j α (0, y)Dxα ηβ : H s n+ −→ H s−m j − 2
s > m j + 21 . Then
n−1 ,
1
σ∧ T j (y, λη) = λm j + 2 σ∧ T j (y, η)κλ−1
(144)
for all λ ∈ + and all (y, η) ∈  × ( q \ 0). We have
σ∧ (A)(y, η)
n
m,d;0
σ∧ ( )(y, η) :=
(145)
∈ C ∞ , cl
; v; q \ 0
+
σ∧ (T )(y, η)
with the above-mentioned interpretation of the order superscript and γ j = −m + m j + 21 , j =
1, . . . , N, while the numbers β j disappear in this case , d = max j (m j + 1), v = (1, 1; 0, N).
T HEOREM 24. Let
(146)
=
A
T
σ∧ ( )(y, η) : H s
be elliptic in U × . Then

n −→ 

+
H s−m n+
⊕
1
⊕ Nj=1 H s−m j − 2
n−1



is a family of invertible operators for all (y, η) ∈  × ( q \ 0) and all s > max(m, d) − 21 , and
we have
n
−m,0;0
q \0 ,
−1
σ∧ ( )−1 (y, η) ∈ C ∞ , cl
+; v ;
cf. the notation in Remark 30 and formula (141).
357
Boundary value problems
Proof. By assumption A is elliptic in U × , i.e., σψ (A)(x, y, ξ, η) 6= 0 for all (x, y) ∈ U ×
P
α β
 and (ξ, η) 6= 0. Thus the η-dependent family a(y, η) =
|α|+|β|=m aαβ (0, y)D x η of
differential operators with respect to x (smoothly dependent on y) is parameter-dependent elliptic
with parameters η ∈ n−1 \ 0 with respect to (σψ , σe , σψ,e ), uniformly on compact subsets with
respect to y). For similar reasons, the Shapiro-Lopatinskij condition of the original boundary
value problem (143), i.e., the invertibility of
σ∂
A
T
x 0 , y, ξ 0 , η : + −→
+
⊕
N
for all (x 0 , y) ∈ U ∩ n−1 × , (ξ 0 , η) 6= 0, gives us parameter-dependent ellipticity with
parameter η ∈ n \ 0 with respect to (σ∂ , σe0 , σ∂,e0 ) (uniformly on compact subsets with respect
to y). Applying Theorem 21 we find for every y ∈  a constant C > 0 (which can obviously be
chosen uniformly on compact subsets of ) such that (146) is invertible for |η| > C. Because
of the κλ -homogeneity of σ∧ ( )(y, η) (i.e., relations (142) and (144)) we get the invertibility of
(146) for all η 6= 0. Concerning the asserted nature of the inverse we can apply Theorem 21.
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AMS Subject Classification: 35S15, 58J40.
David KAPANADZE
A. Razmadze Mathematical Institute
Academy of Sciences of Georgia
1, M. Alexidze Str., TBILISI 93, GEORGIA
e-mail: Bert-Wolfgang SCHULZE
Institut für Mathematik
Universität Potsdam
Postfach 601553, 14415 POTSDAM, GERMANY
e-mail: 360
D. Kapanadze – B.-W. Schulze