Document 10484120
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Internat. J. Math. & Math. Scl.
(1985)75-91
75
Vol 8 No.
FINITE PROPAGATION SPEED AND
KERNELS OF STRICTLY ELLIPTIC OPERATORS
DAVID GURARIE
Department of Mathematics
Case Western Reserve Lniversity
Clee!and, Ohlo 44106
(Received September 20,
ABSTRACT.
and discuss
|984)
We establish estimates of the resolvent and other related kernels
LP-theory for a class of strictly elliptic operators on in
The clas of operators considered in the paper is of the form
A0
leading elliptic part
LP-type
have
and a "singular" perturbation
and are modeled after
Schrdinger
B,
A0 + B wlth he
whose coefficients
operators.
KEY WORDS AND PHRASES. Elliptic operator, resol’env, semigro.p, pert,a-5on
series, finive propagavion speed.
192 MATHEMATICS SUBJECT CLASSIFICATION CODES. 47G05; 35S05
I.
INqRODUCTION.
_
In this paper we shall study the resolven= kernel
and other related "functions of
AO [ aa
(x)D
a
on
n
A"
,
(-A)
-I
for a class of strictly elliptic op=rators
and their perturbations
We are =articu!ar]v_. interesEed in
p
R_(x,y)
LP-theory
A
o
0
+ B.
elliptic operators,
and typical probies :hau arise here are spectral properties ef
A,
c!esedness, essential se!fadcintness, accre:ivity, semigroup generation, etc.
Underlying ali those
kernel
;s
ehe qu=ion of existence and estimates of the reeivent
R_.
Typically the estimate ha3 :he form of a convolution-t::pe bound
xth an L
i-r adial
decreasin functlcn H
In the uniformly elliptic case kernels of the resolvent R_
semiroup
K
e
-tA
(-A)
were studied extensively ([!], [2], [3] eta!.).
particular, Eidelman [i] derived :he following radial bound of
K
-i
In
and
D. GURARIE
76
where
order
m
A,
m
m
and I. Gelfand and G. Shilov [2] applied this
m-I’
A.
estimate to study generalized eigenfunction expansions of
In [4], [5] we obtained similar estimates in a different way starting with
The latter approach allows us to treat perturbations of
the resolvent kernel.
and consequently operators with "nonregular" coefficients.
LP-spectral
results to
A,
We applied these
theory of elliptic operators.
In the present paper we extend the results of [4] to a class of strictly
elliptic operators with possibly unbounded coefficients, obtained by linear
deformations of uniformly elliptic symbols in the -variable,
a
(X,x())
a(x,)
depending smoothly on
Here
x E
x
n.
x
means a matrix function
(ij(x))
Natural examples of such deformations are second order elliptic operators
aij(x)D2...
A
Indeed any quadratic form
2
as a deformation of the simplest one:
Il
a(x,)
6
()I 2
x
aij(x)i
with
can be viewed
"
(aij (x))=.
x
Other examples appear as right (left) invariant operators on nilpotent Lie groups
(see [6]).
For the sake of presentation we shall restrict ourselves to the simplest
case of deformations when
x
(x)l
is scalar
("conformal dilations").
transformations correspond to a multiplication of the leading symbol of
(x)
a positive function
m,
m
order of
A.
The dilating factor
subject to a certain constraint, called "finite propagation
Its precise definition is given in
growth of
at
coefficients of
,
A
2.
(x)
Such
A with
is
speed" condition.
This condition limits the rate of
which can not exceed
can not grow faster than
O(Ixl)
xl
m
In other words leading
for m-th order operators.
Let us observe that growth restrictions on the leading coefficients are
well known in both ordinary (Sturm-Liouville) theory,
A--
d
2
+
-a(x).--dx
(see for instance [7], Ch. 9) and also for partial
differential operators
A
A
I
aij
(x)D’2"lj +
([8]). A sufficient condition for
Weyl’s "limit circle case") is
to be well defined (essentially selfadjoint or
the divergence of the integral
+
dx
+
on
(x)
dr
(r)
(a(r)
(x)!l)
Ixl<r aij
sup
on
n
The latter condition was called in [8] "finite propagation speed."
a(x)ll
Physically,
can be interpreted as a "local propagation speed," then integrals (1.2)
measure the
"amount
of time required to get from a finite point to
".
It is not clear to what extent(l.2)is necessary to get a well defined
operator A.
Results of [9], [I0], [ii] indicate that it might be "too
restrictive," yet in special cases ([12], [13]) it proved to be necessary in a
certain sense.
FINITE PROPAGATION SPEED AND KERNELS OF ELLIPTIC OPERATORS
77
The form of "finite propagation speed" used in our paper is close but
It is expressed in terms of derivatives of
somewhat stronger than (1.2).
Also the way we apply it in our setting differs from the
rather than integrals.
propagators" (cf. [8], [14]).
standard method of "hyperbolic wave
Under assumptions of
2 we show that m
LP-spaces
defined in different
<
(I
-th
order elliptic operators are well
and their kernels (resolvent,
)
p
"constant coefficient" and "uniformly
semigroup, etc.) share many properties of
elliptic" kernels (cf. [15], [16], [5], [4]).
There is one notable exception
the radial bound (I) central for the argument of [5], [4] is no longer
however:
valid.
(x)
The Hardv-Littlewood
Instead we utilize another Fourier analytic tool
maximal function:
f
f*(x)
n
sup
f(x+y)
ldy
(1.3)
Maximal functions appear in this context when "deformations" of symbols (of
resolvent and related operators) are translated into "deformations" of kernels.
As a result one gets kernels of the type
-nxIKrx-Yl
,(x)J
K(x,y)
K is
where
Ll-radial,
>
(x)
is estimated by the usual
K(x,y) <
bound (I.A).
<
c
is bounded
"Ll-dilation
(1.4)
(x))
In our case (unbounded
,(x)
If
the dilating factor.
0
K
K()
(uniformly elliptic case)
has no longer a single convolution type
K0
But the maximal function (1.3) applies to give
(Kf)(x) < KII f*(x)
f*
f
The maximal operator
is well known to be
LP-operator
[17]), which yields a bound for the
Lp
norm of
<
(I
p
K
<
)
(see for
nst.
Such bounds are
used throughout the paper as a substitute for the radial bound (I.I).
Otherwise our argument is similar to
[4].
First we construct the "free" resolvent
(homogeneous) part of
R
Here
K
0
A,
t
k
(-Ao)-I
A
0
is the leading
K(l-e )-I
is a pseudodifferential operator
is computed explicitly
0
via tle "parametrix series"
K
being the leading svmbol of
R
A,
and
L
(DO)
a
@DO
with symbol
of order
a(x,$)
-a(x,)
-I
whose symbol
( 4).
The "perturbed" resolvent
R
(-A) -I
is constructed via the perturbation
series
(1.6)
78
D. GURARIE
The coefficients of perturbation
b (x)D
B
Such classes are well know’n in the theory of
a
LP-ciasses.
are taken in
Schrdinger
+ V(x)
operators -4
(cf. [18]) and were studied for higher order operators in [15], [16], [5], [4].
b (x)
In our case
{}
can be allowed certain growth at
weights
A
w,
that are fractional powers of
Also the
LP-Sobolev
ca be characterized as a weighted
LP-spaces,
L-domin
B
LP-classes
L
p
with
A0
of
and
space (Theorem 2).
of [5], gives
The central result of the paper, Lemma i, analogous to Le.ma
conditions (in terms of
(x)
depending on
Such conditions are naturally formulated in terms of weighted
of coefficients) for relative boundedness of
BK
A
and estimates the norm of
This result applies in
0
4 to sum series(l.5)and(l.6),and to derive different corollaries. Among them
with respect to
we
ge
LP-spectrum;
a)
LP-closedness
b)
a priori estimates and essential selfadjointness of
c)
resolvent summability, i.e., convergence
d)
existence of a strongly continuous holomorphic semigroup
of
A + B
0
A
and bounds on its
Rf(x)
A
L
in
f(x)
in
2
L
p
and a.e.
-tA}
{e
Re t > 0"
Let us remark that due to the maximal function techniques adopted in the
Ll-space
paper the
kernels
Kd(x,y)
is excluded from consideration.
(x)-nK((x)-l(x-y))
harmonic analysis question.
bearing on
with
At the end of
The
K E L
Ll-theory
of integral
poses an interesting
4 we give two examples which have
Ll-theory.
In conclusion let us mention that the basic
L2-theory
of elliptic operators
([19], [0], [3]) was extended later in a fairly general setting of
pseudodifferential calculus ([21], [9], [22], [6]).
L
p,
p
# 2.
But much less i known about
Two recent works that discuss specifically
Nagel and Stein [6] and Beals [23].
LP-theory
are those by
Both methods allow to treat a variety of
"nonelliptic problems" (subelliptic Lap!acians, degenerate elliptic operators,
etc.), but remain entirely within the pseudodifferential framework in the sense
that symbols (coefficients) are assumed sufficiently smooth.
Another trend ([15], [16], [5]) was to explore what "amount of
nonregularity"
of coefficients yields "well defined"
The present paper continues
the latter trend, but our approach seems to be
extendable to various
"nonelliptic" situations as well. This extension will be discussed
elsewhere.
2. THE CLASS OF OPERATORS.
LP-operators.
Throughout this paper we shall consider operators of the form A
a
A
a (x)D
and a "singular"
0
with the leading elliptic part
perturbation
B
A
0
+ B
b (x)D
The following assumptions will be made on A
and B.
0
I. The leading part A
is assumed to be homogeneous of degree m and have
0
a real symbol a(x,) obtained by the "deformation"
of a uniformly elliptic
symbol a(x,$)
a(x,)
(x,(x))
m(x)(x,)
79
FINITE PROPAGATION SPEED AND KERNELS OF ELLIPTIC OPERATORS
Uniform ellipticity as usual means that
(x)
The dilating factor
(x,)
and the coefficients of
(x).
are subject to the following constraints
Here and elsewhere
’(g)(x) <
const
61-igl;
l(fl)(x)l
const
-131; I
u
()
<
Igi <
<
(2.1)
m
(2.2)
m
Du.
denote a partial derivative
Condition (2.1) is
close but somewhat stronger than the "finite propagation speed" condition [8].
-th
order operators of the above type requires
The latter adjusted for m
dr
+’
(r)
[+
dr )
g(r)
(r)
(2.3)
6(x)
sup
whereas (2.1) implies
’
C
I-I
(x)
Both conditions do not allow
II.
singularities
b
[
B
Perturbation
Pa
+"
to grow faster than
b (x)D
a
O(Ix I)
as
has coefficients with local
For each term
Lloc
a
ngl+
b (x)D
of
a
B
x
.
LP-type
we introduce its
"fractional order"
n__+ la
p
d
and require
d
m.
a
The latter is needed in order to have
B
bounded relative to
We also need some control of
at
{},
b
way to express it is in terms of weighted
Precisely, we take
L
b
Pa
depending on
LP-spaces, p {f:
LW
A
0
A convenient
6
IfwlPdx < }
with the weight
W
-d
w
(2.5)
w
Let us notice that the above class of operators is closed under the
adjunction
(b
of
A*
A
P lal, w
a
a
(x) D
the
provided the coefficients of
LP-weighted
g
Sobolev space of order
are sufficiently smooth
I’I)
Indeed the adjoint
is
(a)a(a-)D
Da(aa ")
Remembering that
a
m
.
with
and
g
subject to (2.1)
estimate the g-order coefficient of the adjoint
a
(a-g)
y ’+y=a-
(m) () () ()
(2.2), we
80
D. GURARIE
as
b
a
(-)
-II)
(w
i.e. belong to the above class of
3
Namely, interpolating in
(fractional) order
and the exponent
s
LP-class
(for
p
parameters:
Sobole
b
gt
w
of the weight
t
of
of
b
Similar argument applies to derivatives of the coefficients
perturbations.
B.
L
are
coming from the top order coefficients:
A*
So the lower order terms of
one can show
that
b 6
LPs;t-
(Y)
b
[qs_iyl; t-IYl
6
eq ep + IYI
where
A + B includes
0
The latter is important both
This implies in particular that the class of perturbations
A0
all symmetric operators with the leading part
for the argument below and in various applications.
3.
THE MAIN LEM.
In this section we shall prove the main Lemma, which gives condition for
b(x)D
B
relative boundedness of the operator
a
A
with respect to
B
provides an estimate of the operator norm of the product
0
I, and
of
"parametrix
o
(’-A
O)
This result will enable us to prove the convergence of series (1.5) and (1.6)
4.
and to derive all consequences in
Let
A
a(x,D)
0
2 with symbol
be an elliptic operator of
(2)
K(x,z)
K (x,z),z
K
Define a DO
a(x,d(x))
a(x,)
I
a parametrix of (-Ao)
-a(x,)
-n/2
|
(2.1)
is
(2.5) of
LP-bounded,
BKII
C
where
(x
a(x )
Let
ei’z
(3.1)
2, then for all
I
<
p
-<.
p,
C
<
sinS/2
depends on
n+l
p
e
b(x) 6 Lw’
6 range a(x,)
(x))
with symbol
-a(x, ) d
P
LEMMA I.
y,
x
and
satisfy assumptions
+,
the operator
BK
and its norm is estimated as follows
r
-l+d/m
the
L
re
iO
-norm of
W
,d
n___ +
d
b(x)
P
<
m
(3.2)
and the leading symbol
a(x,)
PROOF.
The operator
multiplication with
DO’s
BK
b(x)
consists of a
We start with
DO
DK.
DK
followed by a
By the usual product formula of
its symbol is equal
(x,)
[
()*-88--a)
"x
We apply the "iterated chain rule" to partial derivatives of
(3.3)
1
(_-)
k
8xS(a)
C81- ...8
k
(-a)-k-l’[’[(83Ja)
x
’i
(3.4)
FINITE PROPAGATION SPEED AND KERNELS OF ELLIPTIC OPERATORS
81
k
8J
B
summation taken over all partitions
{B J}
into the sum of multi-indices
i
CBI
with certain combinatorial coefficients
represented as
o
As a result
k.
"8
is
(3.5)
i(x)i(x’)
where
k
product of
partial derivatives of the
k
(3.6)
a(x, $)
coefficients of
and
vk+l’
.(x,E)
k
where the multiindex
(-a)
II
km
+
LP-type
In order to treat a possible
I
I-i
singularity of the coefficient
E(x
we
shall use two interpolation inequalities for convolution and multiplication.
i
K*fll p < llgll q f!l P
< llbil
i
i
f
(I-
i
(Young
)
+ I
i
(HIder)
(3.7)
Young’s inequality the Hardy-Littlewood-Sobolev
Lq-weak
K 6
(see for instance [17], Ch. 3). The relation
including a stronger version of
inequality, where
between
p,
p’
and
is the same but
q
p
is excluded in the latter case.
i
Rephrasing (3.7) in the reciprocal scale
convolution with
and
(or
Lq-weak)
P
one can say that a
6 [0,i]
shifts the
LP-class
to the left by the
r
)
shifts it to the right by
while a multiplication with b E L
(i
q
bK*f is bounded if the LP-classes of b
The product of two operations f
i
i
I
K are "Young-dual":
r
q
in Lemma I,
This relation explains the limitation on the scale of
amount
!.
r
K 6 L
q
LP-spaces
P
p
(we do not want a
convolution-type
K
term
to
"push" the
LP-class
of
f out of the scale [0,i]). It also explains the definition of the "fractional
p
is "equivalent" by (3.7) to
order" (2.3). Indeed, a multiplication with b 6 L
s
;s
a convolution with the "fractional Laplacian" A
(_)s/2
b(x)D=K
Now we return to the operator
argument accurate.
fractional Laplacian
and want to make the above heuristic
It is convenient to multiply and divide
A
s,
--n
s
P
BKSA -s
O,
(s
[
of
o.
(_ a)
A
-s
by the
(bi)MiA-S
a combination of multiplication operators
and a fractional Laplacian
pa
BK
Then we get from (3.5)
=)
(b i
DO’s
Mi
with symbols
(3.8)
k+l
The order of
o.1
is
II
+
s
m(k+l)
which
D. GURARIE
82
by (3.6) is equal to
d
m
p
-m-
[8] <
0
i8
We "pull out" the absolute value of complex
re
in the denominator of
(3.8), and by the homogenuity of a write o. in terms of a "uniformly elliptic"
symbol
oi
--i
e -a(x,)
r
-i/m
i(x’)
7 i(x;
r
I
-I + d/m
Slim;
d
t
I’vl
6 (x)E)
(3.1o)
where
(bi)MiA-s
(x)
S
j
[ol
Isl.
mk + d
s
will joint the left (multiplication) factor of
Remembering that
k
]aaj(J),
-t
6
The weight
+
+
d
is a product of derivatives
#i
we get
i
k
i
(b/d)(a(.BJ)/m-lSJl)
d+km-] B
By hypothese (2.1), (2.2), (2.5) of
whereas all terms of the product
(b i)
factor
i
product
iMi A
(3.7).
t
-s
6
LP
are
r
possible.
o
1
pa,
A
-s
of the
M.l
(x,pK)
o.
in terms of the "uniformly elliptic" kernel
of a
is nonpositive by (3.9).
DO o.l
Strictly negative order,
A negative order
l
DO
<
Z
SI, 0
DO M.
is
with
of kernels, i.e.
p-n i(x; 0-1z)
M.l(x,z)
The order
Ll_dilation
.
We shall show that a
and estimate its norm.
results in the dual
which represents
L
"complete" each other in the sense of interpolation formulas
(l<p<)
-i/m6
is
and the "convolution" (right) factor
We first observe that the dilation of symbols
O
d
Thus the "multiplication: (left)
L
It remains to study the middle term.
LP-bounded
b/
2 the first factor
..
(3.11)
So two cases are
O.
is well known to have an
Ll-radial
convolution-type bound
[Bi(x,z)[ <
cH(lzl)
(3.12)
83
FINITE PROPAGATION SPEED AND KERNELS OF ELLIPTIC OPERATORS
where
c depends on a finite number (N-- n + I) symbol class semlnorms of
(see for inst. [5]).
A straightforward evaluation of seminorms yields
sup
x,(
llNl(aoi)( )I
<
C
in e/2]N,
IBI
N
whence the constant in estimate (3.2) of Lemma i.
From (3.11), (3.12) it follows that the kernel
Ll-function
so-called p-dilation of the radial
"’
.IMi(x,z) < co-n[x)H[io"
If
Mo
1
Ll-convolution
for all
<
i
p
<
<
o(x)
were bounded
0
the usual
H
H
(x)z)
$
M4(x,z)
z
p(x)
p
with
x
is bounded by the
r
-i/m
6 (x)
Y.
H < H
LP-boundedness
(uniformly elliptic case), then
e
would immediately imply the result:
(cf. [4], [5]).
6(x)
In case of unbounded
and
of
the maximal
function estimate applies,
Hlllf*(x)
(Hof)(x) <
LP-boundedness
This yields
-
M
of
in all spaces
1
I
<
p
<
and proves
Lemma I in the first case.
2
o’l
Order
c(x)l +
0
+
0
$i
We split
E
a multiple of the
0
SI,0
Oo(X,E)
a homogeneous of order zero (in
)
over the unit sphere
and a negative order
i
sbol
Let us illustrate this splitting for
(
_
c(x),
H
{0}
of the kernel
Correspondingly kernel
c(x)l + M
+ M_,
0
O,
6 S
sbol
1,0
i.e.
/(x,g)d
c(x)
while the second is obviously in
0
sufficiently regular at
radial bound
k
with the zero mean-value
(-a)a
a
/a
terms:
en
m)
the first te splits into the sum of
o
3
into the sum of
"identity" (in the -variable) function"
-m
SI, 0
and
Notice tha
which is important for the existence of "global"
$ (x,D)
(cf. [5]).
splits into the sum of three kernels:
c(x)
a multiplication with "nice" (bounded) function
Calderon-Zygmund kernel
!
o(X’Z)
(homogeneous of degree
-n
z)
in
and an
L -adially bounded kernel
The first two of them are invariant under the 0-dilation
(z)
-(-z)
while the third is dominated by the maximal function as in case i
completes the proof.
is
o
This
a
D. GURARIE
84
&.
AND A AND THEIR APPLICATIONS.
A
0
After Lemma I we can study the convergence of series (1.5) and (1.6), that
0
0
-1
give resolvent kernels R
and R
We start with R
(-A)
RESOLVENT KERNELS OF
(-A0)-1
THEOREM i.
Series (1.5) converges absolutely in the complement of a
parabolic region
1
<
p
<
,
.DO’s
of
its symbol
o
k
i
<m
k
By the product formula
I.
D(a)8(_)x
(a 6
e
8;
i
e
(-A0)K
L
LP-spaces
in all
A.
of
of the type that appear in Lemma i.
ioi
a(Sjj
a
<181
e(x,)
expands into the sum
i(x)
-
0
R
We recall that the operator
PROOF.
,
about positive real axis in
and defines the resolvent
L_
m
Namely,
b(x))
plays the role of
and
k
i(x’)
(-a)k+l
e;
8 +
e
1
II
181
(k+l)m
So the whole argument of Lemma I can be repeated for
DO’s
o.(x,D)
have strictly negative order,
1
Hence by Lemma i each term
M
is estimated as
]ILl1 <
with
C
depending on
C
<
[liMi[l
[sin 8/21
L-norms
of
and
i
i Lk
expansion (1.5), we see that
n+l
-i.
r
-
X
(I
p
[8[/m
i
<
<
p
i
) --,
m
)
(4.1)
Returning to series
converges absolutely if the right hand side
0
of (4.1)
C
r
Isin 8/21 n+l
This condition gives a parabolic region
<
(4.2)
i.
(-A0)ROf
RO(-A0)f
for all
f
f
,
about positive real axis in
whose complement we have a well defined bounded operator
0
We want to show that R
is the resolvent of A
0
p
(I)
f
f 6 e
(II)
L
and consequently the whole operator
L
of
Notice that all
L"
in the
LP-domain
R
0
K(I-L)
in
-I
that is
of
A
0
(-Ao)K
L.
It
The first formula immediately follows from the relation
I
0
shows that R
is the right resolvent of A
To show the existence of the left
0
resolvent (II) we shall use duality between
Namely: the adjoint of
the right resolvent of
operator
A*
0
in
L
p
A
0
(+
-
in
i
LP-spaces.
L
p
I)
becomes the left resolvent of the adjoint
and vice versa.
We can not apply the above result directly to
the right resolvent, since
whose lower order terms
A
0
e
A
0
to show the existence of
is not of homogeneous order.
{beD }lel<m
But
A
A
+ B,
0
0
have coefficients bounded "relative to
A
0
FINITE PROPAGATION SPEED AND KERNELS OF ELLIPTIC OPERATORS
85
Therefore the "perturbation" Theorem (Theorem 4
below) applies to prove the
existence of the right resolvent for A
outside of another parabolic region
0
Once both resolvents, right and left, are
shown to exist they must be equal. This
completes the proof of Theorem i.
’
LP-domain
Next we apply Lemma i to characterize the
A0,
of
Pp(Ao),
<
p
i
-th
<
.
For constant coefficient elliptic operators P (A
is known to be the m
P
i v’m"
and the same is true for uniformly elliptic
O)
(I-A)m/2LP
Sobolev space
[4]).
operators (see
THEOREM 2.
For strictly elliptic operators of the above type we have
The
with the weight
LP-domain
6
w
IP’m
w
is equal to the m
0
I lwf(a) IP
{f:
(-Ao)fll
and the norms
A
of
m
in
Pp (AO)
dx
<
-th
weighted Sobolev spce
}
!lwDfll.p
and
in
Ip, m are equivalent.
w
We have to show that the operator w(l-) m/2-0 is bounded and
0
-I
invertible in L v
But R
K(I-L)
It suffices to show boundedness and
PROOF
invertibility of the operator
Lemma I.
w(l-&)m/2K.
T
Boundedness follows directly from
Indeed
w(l-
&)m/2K [
wDK.
c
is the sum of terms that appear in Lemma i.
To prove invertibility it is convenient to use
>
large
0
(l-&)
instead of
m/2
reverse two terms in the product
(-A)
in the definition of
w(%-A)
m/2
ip,m
w
m/2
for sufficiently
norm and also to
We observe that
g)m/2
w(k
where order
bounded.
B
<
m
Then for sufficiently large
Hence the inverse
[(l
Now we can invert
T
But
(l
-I
A0w-i 0 +
+ B]w,
and the coefficients
i
B(;k
as
A)m/2
[(
A)
T
X
A)-m/2[[
m/2 + B] -I
w(%
[
c w
v’/w
are all
the operator norm
<
1/2.
exists and is bounded.
A)m/2R0
A0)w-I
(
b
(4.3)
[(
using (4.3)
A)
with uniformly elliptic
m/2
+ B]
-I
0 w-iA0
(4.4)
and
of the same type
B.
Finally "pulling
out"
-m/2
A)
A)-m/2B
[I + (
(\-A)
-I
m/2
from the right factor in (4.4),
the problem reduces to
LP-boundedness
of the
D. GURARIE
86
operator
small"
,
(0 + )(k &)-m/2
S
and "relatively
Indeed the operator S
,DO’s { (x)D( g)m/2:
combination of zero- and negative order
former are given by Calderon-Zygmund kernels, hence
.
[!7], ch. 2).
i-<. p
0
for uniformly elliptic
which is well known (cf. [4])
Ll-radially
The latter have
L
p
bounded kernels, hence
II < m}.
<
1
for all
is a
L
<
p
p
The
(see
for all
In many applications, including the perturbation series (1.6) one needs to
-I and
0
K (I
R
0.
estimate the operator norm of BR
By Theorem I
-i
is easily shown to converge to I as
(I-L)
{iarg
I
>
>
0}
L)
uniformly in any sector
The problem is thus reduced to estimating
is provided by Lemma I.
If
COROLLARY i.
Lp
for all
max {d
d
symbol
<
i
p
A
B
and
0
BROil
<
<
{p
rain
satisfy assumptions of
r
!Ib=il)
(C
The latter
2, then
-l+d/m
(4.5)
Isin /21 k+l
E
and
"fractional order of
is the
BKII
R0
(where
B,
exists).
C
constant
Here
depends on
and the
p
a(x,)
From (4.5) immediately follows
COROLLARY 2 (cf. [5] [4], [16]).
LP-domains
A
of
and
p(A)
A
B
The operator
Ao-bounded
is
and the
are equal
0
6p,m,w
p(A0)
-m,
w
i
<
<
p
rain
Ip
Let us notice that the relative bound in the right hand side of (4.5) can be
made as small as one likes, taking sufficiently large
"fractional" order
<
d
r,
provided the
In this case the perturbation series (1.6 converges
m.
absolutely in the complement of a parabolic region
’
The limit case,
LP-type
d
{: relative bond
m,
<
I}
is important, as it gives the "optimal
In this case one can claim somewhat less.
perturbations.
PROPOSITION I.
Operator
R,
with arbitrarily small
BR?II
: II
>
<
R;
and large
,
i
in any regioD
lard I >
}
L
and
perturbation
d
<
m- i
"very small" b"
B
b"!l
P !ib"’l <
splits into the sum
hence
while the second
L
B’RO!l
B"R?I<
(4.6)
depending on
To prove Proposition i we cut each coefficient
b’
amount" of
singularity of lower order coefficients and also allows "top order"
e.
B’ + B".
b
E L
p
into two parts:
Consequently, te whole
The first term has order
becomes small outside of a parabolic region
1
in some region
fR,O"
and can be made as small as one likes.
From Proposition i easily follows
Here
.’
depends on
This proves the Proposition
87
FINITE PROPAGATION SPEED AND KERNELS OF ELLIPTIC OPERATORS
>0
C=C
>
For any
COROLLARY 3 (a priory estimate).
there exists
0
s.t.
Bf!I p <
!IAofll p + cIIfll p
Lwp’m
f 6
(4.7)
Along with Kato-Rellich Theorem a priory estimate (4.7) implies essential
2
Proposition i
of a formally symmetric operator A + B.
L
0
selfadjointness in
LP-type singularity of
operators in LP-spaces. Namely,
a(x)D of i has leading
also yields a large class ("optimal" in the sense of
coefficients) of "well defined" elliptic
THEOREM 3.
m(x)
coefficients
LP-spaces
I
p
<
<
p
If
p
Lp,m
D p (A)
2)
min
(2.4), then
Al[P,m
w
closure {A
mln
is formally symmetric and
C
selfadjoint on
A
is "well defined" in all
A iC
0}
in the sense that
and
w
A
ll--m
satisfy (2.3)
6 Lw
a
A
If an operator
i)
min
{p=} >
2
A
then
is essentially
O
Theorem 3 extends many of earlier known results [8], [9], [24], [41, [ii],
LP-theory.
[15], [16] on essential selfadjointness and
R
Now we turn to the resolvent
(cf. [5], [4])
THEOREM 4.
Let
of
A
A.
A0 + B
be an operator of Theorem 3.
Then
(I)
I
for all
<
<
p
rain
{pa
R
(1.6).
and is given by an absolutely convergent series
(II)
(resolvent summability) for all
f(x)
(Rf)(x)
in L
(-A)
the resolvent
f
L
p
(i
<
p
<
exists in
{p})
rain
R,
the family
uniformly in
as
p-norm.
The first statement is already proved, the second follows from the estimates
of
eemma
i (cf. [4], [5]).
After Theorem 2 we can obtain a variety of other "analytic multipliers"
{e(A)} e
and "summation families"
by Cauchy integration of
Rr..
(A)
Indeed, the
formula
(A)
F
defines a nice
(bounded)
operator
(A)
with respect to the measure
contour
provided
du
sin
is the family of functions
{re+-i;
r
> ro},
0
<
<
{t()
and
chosen arbitrarily small due to
Thus we get
e -t
is integrable over the
cr-l+d/m
/21n+ I
d.
}t" ieHere contour
a circle {roe
Ii > }.
the shape of QR,"
F
One such example
consists of two rays
Angle
can be
88
D. GURARIE
e
A
An operator
THEOREM 5.
-tA
of Theorem 3 generates a holomorphic semigroup
p
min
p
Re t > 0 in all L
I
<
in the right half plane
Moreover, the family of functions
e
-tAf (x)
{p}
f(x)
as
0
t
in
LP-norm
REMARK i.
As in [4] ( ), all the above results extend to strictly elliptic
Rn
operators on certain Riemannian manifolds diffeomorphic to
and strictly
elliptic systems.
REMARK 2.
n/E
E
potentials:
V
V(xl-x j)
[
n)
is a subspace of
(E
on
(x
m,
(see [5]).
I
=
3
3N
6
B
of the perturbation
b (x)
which are
One such class consists of
3.
can be allowed in
spaces
{b
More general classes of coefficients
defined on quotient
p
on E,
like "N-body"
L
In this case condition (2.4)
i<j
of
dim E
2 should read:
lal
+
Pe
classes is discussed in [15], [16].
Izl
L
Another extension of
It consists of all functions
p
b (x)
whose
-s
H (z)
(Izl i, and s n m + a) (a "local
is bounded. This condition is close to so called
singularity" of
"Rollnik condition" in the theory of Schrdinger operators (see [25]).
convolution with
Da(-Ao)-I)
We shall conclude this section with two examples, which have bearing on
Ll-theory
This theory poses the following
of the above class of operators.
given an
interesting harmonic analysis problem:
dilating factor
>
(x)
O,
L
function
K(x)
x-y
is
I
K
when the kernel
I
n(x) K( (x))
Two examples below indicate that the situation in
compared to
LP-theory
>
(p
"finitely propagating"
Ll-radially
are no longer sufficient for
O(Ixl)
be "slower" than
Namely, an
i)
Ll-case
.
bounded.
K
and a
the growth of
and some additional relation between
Ixi
I
becomes more subtle
bounded
LI:
and a
L
K
6
must
appears.
and
is
K(lK(x,y)
It is well known (see, for instance, [17], Appendix)
homogeneous of degree
p
y-PK(l-y) dy < =. The latter is obviously
that K is L
if and only if
I.
but fails for p
true for p > i,
as
2vrx on R and write the operator
XAMPLE 2. Now we take (x)
EXAMPLE I.
(x)
-I.
Take
on
Then
J
K6f
I
(K6f)(x)
(x)y) dy.
K(y)f(x
Then
f K(y)I
IIKflll
We introduce a new variable
whole real line
R
and on the interval
line
and
Rx
13
_y2
0
u
(d)
0
<
u
I
(x)y
x
u
but the function
into three intervals
the Jacobian
f(x- 6(x)y)dx dy.
I
u
y(X)
(x)
The range of
u
has a critical point
x
Y
I
(=d__u_)
0
y
2
"three-fold" (see figure) We divide the
(0;2y2];
(2y2;+) On I
(-;0];
is
12
is bounded from both sides,
c
is the
c2 <
13
I
FINITE PROPAGATION SPEED AND KERNELS OF ELLIPTIC OPERATORS
89
Hence
+
I
d(x)y) dx
f(x
<
13
1
i
fl[ I
yIt is only a neighborhood of the critical point x 0
is computed explicitely.
The Jacobian
to "blow up".
that causes
()
du
y
I +_
(-fx)
y2
[y2,2y2]
[0,
on
/y2+u
"+"
on
Hence
I 12
0
0
(x)y) dx
f(x
f(u)
2
-y
-y
2
K
be
v
-u
K(y)
iff the convolution
II)
K
A similar condition on
Ixl
and the order of integration yields
+ 
