Abstract
In these lectures, which are based on recent joint work with A. Seeger [23], I shall
present sharp analogues of classical estimates
by Peral and Miyachi for solutions of the
standard wave equation on Euclidean space
in the context of the wave equation associated to the sub-Laplacian on a Heisenberg
type group. Some related questions, such
as spectral multipliers for the sub-Laplacian
or Strichartz-estimates, will be briefly addressed. Our results improve on earlier joint
work of mine with E.M. Stein. The new approach that we use has the additional advantage of bringing out more clearly the connections of the problem with the underlying
sub-Riemannian geometry.
0-0
p
Sharp L -estimates for the wave
equation on Heisenberg type
groups
Orleans, April 2008
http://analysis.math.unikiel.de/mueller/
Detlef Müller
April 9, 2008
0-1
Contents
1 Introduction
1.1
0-4
Connections with spectral multipliers
and further facts about the wave equation . . . . . . . . . . . . . . . . . . . 0-13
2 The sub-Riemannian geometry of a Heisenberg type group
0-19
3 The Schrödinger and the wave propagators on a Heisenberg type group
0-27
3.1
Twisted convolution and the metaplectic group . . . . . . . . . . . . . . . . . 0-31
3.2
A subordination formula . . . . . . . . 0-36
4 Estimation of Aλ,k,l,± if k ≥ 1
0-2
0-43
5 Estimation of (1 + L)
5.1
√
−(d−1)/4 i L
e
δ
0-49
Anisotropic re-scaling for k fixed . . . 0-52
6 L2 -estimates for components of the wave
propagator
0-54
7 Estimation for p = 1
0-55
8 Proof of Theorem 1.1
0-62
9 Appendix: The Fourier transform on a
group of Heisenberg type
0-65
0-3
1
Introduction
Let
g = g1 ⊕ g2 ,
with dim g1 = 2m and dim g2 = n, be a Lie algebra
of Heisenberg type, where
[g, g] ⊂ g2 ⊂ z(g) ,
z(g) being the center of g. This means that g is
endowed with an inner product h , i such that g1
and g2 or orthogonal subspaces and the following
holds true:
If we define for µ ∈ g∗2 \ {0} the skew form ωµ on
g1 by
ωµ (V, W ) := µ [V, W ] ,
then there is a unique skew-symmetric linear endomorphism Jµ of g1 such that
ωµ (V, W ) = hµ, [V, W ]i = hJµ (V ), W i
(here, we also used the natural identification of g∗2
with g2 via the inner product). Then
Jµ2 = −|µ|2 I
0-4
(1.1)
Note that this implies in particular that
[g1 , g1 ] = g2 .
As the corresponding connected, simply connected
Heisenberg type Lie group G we shall then choose the
linear manifold g, endowed with the Baker-CampbellHausdorff product
(V1 , U1 ) (V2 , U2 ) := (V1 + V2 , U1 + U2 + 12 [V1 , V2 ])
and identity element e = 0.
Note that the nilpotent part in the Iwasawa decomposition of a simple Lie group of real rank one
is always of Heisenberg type or Euclidean.
As usual, we shall identify X ∈ g with the corresponding left-invariant vector field on G given by
the Lie-derivative
d
Xf (g) := f (g exp(tX))|t=0 ,
dt
where exp : g → G denotes the exponential mapping,
which agrees with the identity mapping in our case.
0-5
Let us next fix an orthonormal basis X1 , . . . , X2m
of g1 , and let us define the non-elliptic sub-Laplacian
L := −
2m
X
Xj2
j=1
on G. Since the vector fields Xj together with their
commutators span the tangent space to G at every
point, L is still hypoelliptic and provides an example
of a non-elliptic “sum of squares operator ” in the
sense of Hörmander ([13]). Moreover, L takes over
in many respects of analysis on G the role which the
Laplacian plays on Euclidean space.
To simplify the notation, we shall also fix an orthonormal basis U1 , . . . , Un of g2 , and shall in the
sequel identify g = g1 + g2 and G with R2m × Rn
by means of the basis X1 , . . . , X2m , U1 , . . . , Un of g.
Then our inner product on g will agree
with the
P2m+n
canonical Euclidean product z · w =
j=1 zj wj
on R2m+n , and Jµ will be identified with a skewsymmetric 2m×2m matrix. Moreover, the Lebesgue
measure dx du on R2m+n is a bi-invariant Haar measure on G. By
d := 2m + n
we shall denote the topological dimension of G. We
0-6
also introduce the automorphic dilations
δr (x, u) := (rx, r 2 u),
r > 0,
on G, and the Koranyi norm
k(x, u)kK := (|x|4 + |4u|2 )1/4 .
Notice that this is a homogeneous norm with respect
to the dilations δr , and that L is homogeneous of
degree 2 with respect to these dilations. Moreover,
if we denote the corresponding balls by
Qr (x, u) := {(y, v) ∈ G : k(y, v)−1 (x, u)kK < r},
then the volume |Qr (x, u)| is given by
|Qr (x, u)| = |Q1 (0, 0)| r D ,
where
D := 2m + 2n
is the homogeneous dimension of G. We shall also
have to work with the Euclidean balls
Br (x, u) := {(y, v) ∈ G : |(y − x, v − u)| < r},
with respect to the Euclidean norm
|(x, u)| := (|x|2 + |u|2 )1/2 .
0-7
In the special case n = 1 we may assume that
Jµ = µJ, µ ∈ R, where
0
Im
J :=
−Im 0
This is the case of the Heisenberg group Hm , which
is R2m × R, endowed with the product
(z, u) · (z ′ , u′ ) = (z + z ′ , u + u′ + 12 hJz, z ′ i).
A basis of the Lie algebra hm of Hm is then given by
∂
X1 , . . . , X2m , U , where U := ∂u
is central.
Consider the following Cauchy problem for the
wave equation on G × R associated to L:
∂2u
∂u − (−L)u = 0, u
= f,
= g, (1.2)
2
∂t
∂t t=0
t=0
where t ∈ R denotes time.
The solution to this problem is formally given by
!
√
√
sin(t L)
√
g (x) + (cos(t L)f )(x),
u(x, t) =
L
(x, t) ∈ G × R.
0-8
In fact, the above expression for u makes perfect
sense at least for f, g ∈ L2 (G), if one defines the
functions of L involved by the spectral theorem (no∞
tice that L is essentially
selfadjoint
on
C
(G) [33])
0
R∞
as follows: If L = 0 s dEs denotes the spectral resolution of L on L2 (G), then every Borel function m
on R+ gives rise to a (possibly unbounded) operator
Z ∞
m(L) :=
m(s) dEs
0
on L2 (G). Let us also note that if m is a bounded
spectral multiplier, then, by left–invariance and the
Schwartz kernel theorem, it is easy to see that m(L)
is a convolution operator
Z
m(L)f = f ∗ Km = f (y)Km (y −1 ·) dy,
G
where a priori the convolution kernel Km is a tempered distribution. We also write Km = m(L)δ.
If one decides to measure smoothness properties
of the solution u(x, t) for fixed time t in terms of
Sobolev norms of the form
kf kLpα := k(1 + L)α/2 f kLp ,
0-9
one is naturally led to study√the mapping√propersin(t L)
eit L
as
ties of operators such as (1+L)α/2 or √L(1+L)
α/2
operators on Lp (G) into itself.
For the classical wave equation on Euclidian space,
sharp estimates for the corresponding operators have
been established by Peral [35] and Miyachi [21].
d
In particular, if ∆
√ denotes the Laplacian on R ,
p
d
−α/2 it −∆
is
bounded
on
L
(R
), if
then (1 − ∆)
e
α ≥ (d − 1) p1 − 12 , for 1 < p < ∞. Moreover, (1 −
−
(d−1)/2
2
√
it −∆
is bounded from the real Hardy
∆)
e
space H 1 (Rd ) into L1 (Rd ).
Local analogues of these results hold true for solutions to strictly hyperbolic differential equations
(see e.g. [4], [2],[37]).
Indeed, as has been shown in [2] and [37], the
estimates in [35]and [21] locally hold true more generally for large classes of Fourier integral operators,
and solutions to strictly hyperbolic equations can be
expressed in terms of such operators.
0-10
The problem in studying the wave equation associated to the sub-Laplacian on the Heisenberg group
is the lack of strict hyperbolicity, since L is degenerateelliptic, and classical Fourier integral operator technics do not seem to be available any more. However,
as we shall see, it is still possible to represent solutions to (1.2) as infinite sums of Fourier integral
operators with degenerate phases and amplitudes.
Interesting information about solutions to (1.2)
have been obtained by Nachman [32]. Among other
things, Nachman showed that the wave operator on
Hm admits a fundamental solution supported in a
“forward light cone”, whose singularities lie along
the cone Γ formed by the characteristics through
the origin. Moreover, he computed the asymptotic
behaviour of the fundamental solution as one approaches a generic singular point on Γ. His method
does, however, not provide uniform estimates on these
singularities, so that it cannot be used to prove Lp estimates for solutions to (1.2). What his results do
reveal, however, is that Γ is by far more complex
for Hm than the corresponding cone in the Euclidian case. This is related to the underlying, more
complex sub-Riemannian geometry, which we shall
briefly discuss.
0-11
Nevertheless, the following theorem is true, which
improves on earlier work in [31] on the Heisenberg
group, by also including the endpoint regularity, and
which is a direct analogue of the results by Peral and
Miyachi in the Euclidean setting.
Theorem 1.1.
√
e L
(1+L)α/2
i
extends to a bounded oper-
ator on Lp (G), for 1 < p < ∞, when α ≥ (d −1)| p1 −
d−1
1
|,
and
for
p
=
1,
if
α
>
2
2 .
The proof will be based on some “Hardy space
type result” for the endpoint p = 1.
Main difference compared to the Euclidean setting: there is not just one single Hardy space, but
a whole sequence of local Hardy spaces associated
with the problem, whose definitions are based on
some interesting interplay between two different homogeneities, namely those coming from the isotropic
dilations of the underlying Euclidean structure of G
and the one coming from the non-isotropic automorphic dilations on G.
0-12
Remark 1.2. The same result holds for
sin
√
L
√
α−1
L(1+L) 2
,
or with the factors (1 + L)−α/2
(respectively (1 +
√
−α
L)−(α−1)/2
)
replaced
by
(1
+
L)
, (respectively
√ −(α−1)
(1 + L)
).
Notice that the restriction to time t = 1 in our
theorem is inessential, since L is homogeneous of
degree 2 with respect to the automorphic dilations
(z, t) 7→ (rz, r 2 t), r > 0.
1.1
Connections with spectral multipliers and further facts about the
wave equation
If m is a bounded spectral multiplier, then clearly the
operator m(L) is bounded on L2 (G). An important
question is then under which additional conditions
on the spectral multiplier m the operator m(L) extends from L2 ∩ Lp (M ) to an Lp -bounded operator,
for a given p 6= 2. If so, m is called an Lp -multiplier
for L.
0-13
Fix a non-trivial cut-off function χ ∈ C0∞ (R) supported in the interval [1, 2], and define for α > 0
kmksloc,α := sup kχ m(r·)kH α ,
r>0
where H α = H α (R) denotes the classical Sobolevspace of order α. Thus, kmksloc,α < ∞, if m is
locally in H α , uniformly on every scale. Notice also
that, up to equivalence, k · ksloc,α is independent of
the choice of the cut-off function χ. Then the following analogue of the classical Mihlin-Hörmander
multiplier theorem holds true (see M. Christ [3], and
also Mauceri-Meda [19]):
Theorem 1.3. If G is a stratified Lie group of homogeneous dimension D, and if kmksloc,α < ∞ for
some α > D/2, then m(L) is bounded on Lp (G) for
1 < p < ∞, and of weak type (1,1).
Observe that, in comparizon to the classical case
G = Rd , the homogeneous dimension D takes over
the role of the Euclidian dimension d. Because of
this fact, which is an outgrowth of the homogeneity
of L with respect to the automorphic dilations, the
condition α > D/2 in Theorem 1.3 appeared natural
and was expected to be sharp for a while.
0-14
The following result, which was found in joint
work with E.M. Stein [24], and independently also
by W. Hebisch [12] a bit later, came therefore as a
surprise:
Theorem 1.4. For the sub-Laplacian L on a Heisenberg type group G, the statement in Theorem 1.3 remains valid under the weaker condition α > d/2 instead of α > D/2.
Theorem 1.1 can be used to give a new, short
proof of this (sharp) multiplier theorem, based on
the following simple Subordination Principle, respectively variants of it:
√
ei L
(1+L)α/2
extends to
Proposition 1.5. Assume that
a bounded operator on L1 (G), and let β > α + 1/2.
Then there is a constant C > 0, such that for any
multiplier ϕ ∈ H β (R) supported in [1, 2], the corresponding convolution kernel Kϕ = ϕ(L)δ is in L1 (G),
and
kKϕ kL1 (G) ≤ CkϕkH β (R) .
0-15
Proof. Observe first that (1 + L)−ε δ ∈ L1 (G) for
any ε > 0. This follows from the formula
R ∞ ε−1 −t(1+L)
−ε
1
δ dt
(1 + L) δ = Γ(ε) 0 t e
R ∞ ε−1 −t
1
e pt dt
=
Γ(ε) 0 t
and the fact that the heat kernel pt := e−tL δ is
a probability measure on G. Write
√
ϕ(s) = ψ( s)(1 + s)−γ , with γ > α/2,
and put g := (1 + L)−γ δ ∈ L1 (G). Then ||ψ||H β ≃
||ϕ||H β , and
√
√
−γ
Kϕ = ψ( L)((1 + L) δ) = ψ( L)g
√
R∞
it L
=
c −∞ ψ̂(t)e
g dt.
And, our assumption in combination with some easy
multiplier estimates (cf. [31]) and the homogeneity
of L yields
√
it L
ke
gk1 . k(1+t2 L)α/2 gk1 . k(1+t2 )α/2 (1+L)α/2 gk1 ,
hence
kKϕ k1 .
Z
∞
−∞
|ψ̂(t)|(1 + |t|)α k(1 + L)α/2 gk1 dt.
0-16
But, (1 + L)α/2 g = (1 + L)α/2−γ δ ∈ L1 , hence, by
Hölder’s inequality and Plancherel’s theorem,
1/2
|ψ̂(t)(1 + |t|)β |2 dt
kKϕ ||1 .
1/2
R
∞
2(α−β)
(1
+
|t|)
dt
. kψkH β .
·
−∞
R
∞
−∞
Q.E.D.
1
d
+
=
Indeed, we may choose β > d−1
2
2
2 in this
subordination principle. This is just the required
regularity of the multiplier in Theorem 1.4, and one
can in fact deduce this theorem from Theorem 1.1
by means of a refinement of the above subordination
principle and standard arguments from CalderónZygmund theory.
In contrast to the close analogy between the wave
equation on G and on Euclidean space expressed in
Theorem 1.1, there are other aspects where the solutions behave quite differently compared to the classical setting.
0-17
One instance of this is the “almost” failure of
dispersive estimates for solutions to (1.2) on Heisenberg groups. Indeed, in [1], Bahouri, Gérard and Xu
prove that solutions to (1.2) on Hm with sufficiently
smooth initial data in L1 satisfy
ku(t)k∞ ≤ C|t|−1/2 ,
and that the exponent −1/2 is optimal, whereas for
the Laplacian on Rd a dispersive estimate holds with
exponent −(d−1)/2. Correspondingly, the Strichartz
estimates proven in [1] are much weaker than the
Euclidean analogues would suggest.
A related result is the “almost” failure of (spectral) restriction theorems for Hm proven in [25].
0-18
2
The sub-Riemannian geometry of a Heisenberg type group
Let M be a smooth manifold, and let D ⊂ T M be
a smooth distribution, i.e., a smooth subbundel of
T M. A sub-Riemannian structure (D, g) on M is
given by a smooth distribution D and a Riemannian metric g on D. For p ∈ M and X, Y ∈ D(p)
1/2
we write hX, Y ig := gp (X, Y ) and kXk = hX, Xig .
M, endowed with such a structure, is called a subRiemannian manifold.
An absolutely continuous curve γ : [0, T ] → M
in M is then called horizontal, if
γ̇(t) =
d
γ(t) ∈ D(γ(t))
dt
for almost every t ∈ [0, T ]. As in Riemannian geometry, the length of a horizontal curve γ is then defined
by
Z T
L(γ) :=
kγ̇(t)kg dt.
0
0-19
The Carnot–Carathéodory distance on (M, D, g)
is then defined by
dCC (p, q) :=
inf{L(γ) : γ is a horinzontal curve joining p with q},
where we use the convention that inf ∅ := ∞.
We shall here be interested in the special situation where D = span {X1 , . . . , Xk }, where X1 , . . . , Xk
are smooth vector fields on M which form an orthonormal system with respect to g at every point.
Then a curve γ is horizontal
Pk iff there are functions
aj (t) such that γ̇(t) =
j=1 aj (t)Xj (γ(t)) for a.e.
1/2
RT P
2
dt.
t ∈ [0, T ], and L(γ) =
j aj (t)
0
Moreover, if these vector fields are bracket generating, i.e., if they satisfy Hörmander’s condition,
then by the Chow-Rashevskii theorem dCC (p, q) <
∞ for every p, q ∈ M, and dCC is a metric on M
that induces the topology of M.
0-20
Let us now consider the sub-Riemannian structure on a group G of Heisenberg type, where D =
span {X1 , . . . , X2m }. Here, the Carnot–Carathéodory
distance is left-invariant, i.e.,
dCC (x, y) = kx−1 ykCC ,
where kxkCC := dCC (0, x) is again a homogeneous
norm on G. Observe that there exists a constant C ≥
1 such that
C −1 kxkCC ≤ kxkK ≤ CkxkCC ,
(2.1)
since any two homogeneous norms are equivalent
[7]. In analogy with wave equations on Riemannian
manifolds, one expects that singularities of solutions
propagate along “geodesics” of the underlying geometry. However, it turns out that the notion of
geodesic of a sub-Riemannian manifold is not as clear
as one might expect. The perhaps best approach to
the problem of determining length minimizing curves
in this context is by means of control theory, as it
is outlined in the article [18] by Liu and Sussman,
which I warmly recommend.
One type of geodesic can be constructed following
classical geometrical optics ideas:
0-21
If x ∈ M, we endow the dual space D(x)∗ with
the dual metric gx∗ to gx on D(x) ⊂ T ∗ M. This
allows to define the length of a cotangent vector
(x, ξ) ∈ T ∗ M as kξkg := kξ|D(x) kgx∗ . The function
H : T ∗ M → R given by
H(x, ξ) := 12 kξk2g
is called the Hamiltonian of the sub-Riemannian manifold M. Observe that this is, apart from the factor
1
2 , just the principal symbol of the associated subLaplacian. Since T ∗ M is a symplectic manifold in
a canonical
P way with respect to the symplectic form
ωM := j dξj ∧ dxj (in canonical coordinates), we
may associate to H the Hamiltonian vector field H,
i.e., ωM (X, H) = dH(X) for every X ∈ T M. The integral curves (x(t), ξ(t)) of H are called bicharacteristics. They satisfy the Hamilton-Jacobi equations
∂H
ẋ(t) =
(x(t), ξ(t)),
∂ξ
˙ξ = − ∂H (x(t), ξ(t)),
∂x
and H is constant along every bicharacteristic. A
normal geodesic of the sub-Riemannian manifold M
will then be the space projection x(t) of a bicharacteristic (x(t), ξ(t)) along which the Hamiltonian H
does not vanish. This notion is in accordance with
the idea that the wave front set of a solution to our
0-22
wave equation should evolve along the bicharacteristics of the wave equation.
It is known that every length minimizing curve
in a sub-Riemannian manifold M is either such a
normal geodesic, or an “abnormal geodesic” [18]. I
won’t give a definition of the latter ones here, since
it is known [11] that groups of Heisenberg type (and
more generally, Metivier groups) do not admit nontrivial abnormal geodesics.
However, it should be warned here that in general
abnormal geodesics do exist, even in nilpotent Lie
groups (see [11] for an example of a group of step
4, and [34] for an example of step 2). This fact has
been ignored in some cases in the literature, which
had led to some confusion.
On a Heisenberg type group G, one easily finds
that the Hamiltonian is given by
H = 21 |ξ − 12 Jµ x|2 ,
if we write coordinates for T ∗ G as ((x, u), (ξ, µ)) ∈
(R2m × Rn ) × (R2m × Rn ).
0-23
The Hamilton-Jacobi equations are then given by
∂H
∂µ
∂H
µ̇ = −
.
∂u
∂H
,
∂ξ
∂H
ξ˙ = −
,
∂x
u̇ =
ẋ =
By left-invariance, it suffices to condsider bicharacteristics starting at the origin, i.e., x(0) = 0, u(0) =
0. We also put η := ξ − 21 Jµ x. Then H = 12 hη, ηi, so
that |η| is constant along the bicharacteristic. Also
µ = const., and
ẋ
η̇
= η, ξ˙ = − 12 Jµ η
= ξ˙ − 12 Jµ ẋ = −Jµ η,
hence
η(t) = e−tJµ α,
if we put ξ(0) = η(0) := α. This implies x(t) =
1−e−tJµ
α, and since Jµ2 = −|µ|2 I, we have
Jµ
sin( 2t |µ|) − t Jµ
x(t) = 2
e 2 α.
|µ|
(2.2)
For simplicity, let us now assume that n = 1, i.e.,
that we are dealing with a Heisenberg group. Then
0-24
Jµ = µJ, with µ ∈ R, and we have
u̇
t
sin(
− 2t Jµ
1
2 |µ|) −tJµ
he
α, Je
αi
= hη, − 2 Jxi = −
|µ|
sin( 2t |µ|)
t
2
= −
hα, Je Jµ αi.
|µ|
t
Looking at the Taylor series of e 2 Jµ and using the
fact that J is skew and satisfies J 2 = −I, this yields
u̇ =
µ
2 t
2
|α|
sin
( |µ|),
2
|µ|
2
if µ 6= 0. Consequently,
(
sin( 2t |µ|) cos( 2t |µ|)
t
2 µ
)
|α|
(2 −
2,
|µ|
|µ|
u(t) =
0, if µ = 0.
if µ 6= 0
(2.3)
The same formula remains valid on an arbitrary Heisenberg type group [34]. Observe that the curve γ(t) :=
(x(t), u(t)) is a normal geodesic iff α 6= 0, and it
has constant
speed kγ̇kg = |ẋ(t)| = |η| = |α|, since
P
γ̇(t) = j ẋj (t)Xj (γ(t)).
Denote by Σ1 the set of endpoints of all geodesics
of length 1 (see figure 1). These endpoints are given
by (2.2),(2.3), with t = 1, where α can be chosen
0-25
arbitrarily within the unit sphere in R2m and µ in
Rn . Therefore,
s−sin s cos s sin s Σ1 = {(x, u) ∈ G : |x| = s , 4|u| = s2
for some s ∈ R.}
(2.4)
We then expect that Σ1 is exactly the singular
support of any of the wave propagators
√
√
√
sin(t L)
it L
√
δ, cos(t L)δ or e
δ,
L
for time t = 1, and this turns out to be true.
Singular support of a wave emenating from a single point in the Heisenberg group
Figure 1: Σ1
The “outer hull” of the set Σ1 is just the unit
sphere with respect to the Carnot-Carathéodory distance.
0-26
Let us finally mention the well-known fact that
solutions to our wave equation have finite propagation speed (see [20], or [27] for an elementary proof
for Lie groups), √i.e., if Pt denotes any of the wave
√
sin(t L)
propagators √L δ, cos(t L)δ, then, for t ≥ 0,
supp Pt ⊂ {(x, u) ∈ G : k(x, u)kCC ≤ t}
3
(2.5)
The Schrödinger and the wave
propagators on a Heisenberg
type group
One principal problem in proving estimates for solutions to wave equations consists in finding sufficiently explicit expressions for the associated wave
propagators. For strictly hyperbolic equations, such
expressions can be given by means of Fourier integral
operators, at least locally in space and time. These
technics, however, do not apply to wave equations
associated to non-elliptic Laplacians, and no sufficiently explicit expressions for the wave propagators
are known to date in these more general situations,
0-27
except for the case of Heisenberg type groups and a
few related examples. What allows us to give such
expressions on Heisenberg type groups is the existence of explicit formulas for the Schrödinger propagators eitL δ on these groups.
There are several possible approaches to these
formulas. One would consist in appealing to the
Gaveau-Hulanicki formula [9],[16] for the heat kernels e−tL δ, which can be derived by means of representation theory and Mehler’s formula, and then
trying to apply physicists approach and replacing
time t by complex time −it.
We shall here sketch another approach, which is
more far-reaching and has turned out to be useful
also in the study of questions of solvability of general
second order left-invariant differential operators on
2-step nilpotent Lie groups (see [28], also for further
references to this topic).
Let G be a group of Heisenberg type as before.
Given f ∈ L1 (G) and µ ∈ g∗2 = R2m , we define the
partial Fourier transform f µ of f along the center by
Z
f (x, u)e−2πiµ·u du (x ∈ R2m ).
f µ (x) :=
Rn
0-28
Moreover, if we define the µ–twisted convolution
of two suitable functions ϕ and ψ on g1 = R2m by
Z
(ϕ ∗µ ψ)(x) :=
ϕ(x − y)ψ(y)e−iπωµ (x,y) dy,
R2m
then
(f ∗ g)µ = f µ ∗µ g µ ,
where f ∗ g denotes the convolution product of the
two functions f, g ∈ L1 (G). Accordingly, the vector fields Xj are transformed into the µ-twisted first
order differential operators Xjµ such that (Xj f )µ =
Xjµ f µ , and the sub-Laplacian is transformed into the
µ-twisted Laplacian Lµ , i.e.,
(Lf )µ = Lµ f µ = −
2m
X
(Xjµ )2
j=1
(say for f ∈ S(G)). An easy computation shows that
explicitly
Xjµ
∂
=
+ iπωµ (x, Xj ).
∂xj
What turns out to be important for us is that
µ
the one-parameter Schrödinger group eitL , t ∈ R,
generated by Lµ can be computed explicitly:
0-29
Proposition 3.1. For f ∈ S(G),
itLµ
e
f = f ∗µ γtµ ,
t ∈ R,
where γtµ ∈ S ′ (R2m ) is a tempered distribution given
by the imaginary Gaussian
γtµ (x)
−m
=2
m
2
π
|µ|
e−i 2 |µ| cot(2πt|µ|)|x| ,
(sin(2πt|µ|)
whenever sin(2πt|µ|) 6= 0. Moreover, by well-known
formulas for the inverse Fourier transform of a generalized Gaussian [14], its Fourier transform is
µ
γc
t (ξ) =
2π
1
i |µ|
tan(2πt|µ|)|ξ|2
e
m
(cos(2πt|µ|))
whenever cos(2πt|µ|) 6= 0.
0-30
3.1
Twisted convolution and the metaplectic group
It is not difficult to reduce the proof of Proposition
3.1 to the case of the Heisenberg group G = Hm ,
which we shall consider now. For any real, symmetric matrix A := (ajk )j,k=1,...,2m let us put
S := −AJ,
(3.1)
0
Im
where J :=
is as in the definition of
−Im 0
the product in Hm . It is easy to see that the mapping
A 7→ S is a vector space isomorphism of the space of
all real, symmetric 2m × 2m matrices onto the Lie
algebra
sp(m, R) := {S : tSJ + JS = 0}.
Let us define
∆S :=
2m
X
ajk Xj Xk ,
j,k=1
S ∈ sp(m, R),
where A is related to S by (3.1). Then one verifies
easily that for any S1 , S2 ∈ sp(m, R), we have
[∆S1 , ∆S2 ] = −2U ∆[S1 ,S2 ] ,
0-31
(3.2)
where U denotes again the central basis element of
hm . Applying a partial Fourier transform along the
(here one-dimensional) center, this leads to the following commutation relations among the µ-twisted
convolution operators ∆µS :
[∆µS1 , ∆µS2 ] = −4πiµ ∆µ[S1 ,S2 ] .
Since ∆µS is formally self-adjoint, this means that
the mapping
i
S 7→ 4πµ
∆µS
(3.3)
is a representation of sp(m, R) by (formally) skewadjoint operators on L2 (R2m ).
Let us consider the case µ = 1 (it is not difficult
to reduce considerations to this case). In this case,
we just speak of the twisted convolution, and write
ϕ × ψ in place of ϕ ∗1 ψ.
In [15], R. Howe has proved for this case that
the map (3.3) can be exponentiated to a unitary
representation of the metaplectic group Mp (m, R), a
two-fold covering of the symplectic group Sp(m, R).
The group Mp (m, R) can in fact be represented by
twisted convolution operators of the form f 7→ f ×γ,
0-32
where the γ’s are suitable measures which, generically, are multiples of purely imaginary Gaussians
eA (z) := e−iπ
t
z·A·z
,
with real, symmetric 2m × 2m matrices A. In particular, one has
t
∆1S
i 4π
e
f = f × γt,S ,
t ∈ R.
(3.4)
The distributions γt,S have been determined explicitly in [22]. To indicate how this can be accomplished, let us argue on a completely formal basis:
If eA , eB are two Gaussians as before such that
det(A + B) 6= 0, one computes that
eA ×eB = [det(A+B)]−1/2 eA−(A−J/2)(A+B)−1(A+J/2) ,
where a suitable determination of the root has to
be chosen. Choosing A = 12 JS1 , B = 21 JS2 , with
S1 , S2 ∈ sp(m, R), and assuming that S1 and S2
commute, one finds that
0-33
e 12 JS1 × e 21 JS2
= 2n (det(S1 + S2 ))−1/2 e 12 J[S1 S2 +I)(S1 +S2 )−1 ] .
This reminds of the addition law for the hyperbolic cotangent, namely
coth(x + y) =
coth x coth y + 1
.
coth x + coth y
We are thus led to define, for non-singular S,
1
A(t) := J coth(tS/2),
(3.5)
2
which is well-defined at least for |t| > 0 small.
Then
eA(t1 ) ×eA(t2 ) = 2n (det(A(t1 )+A(t2 )))−1/2 eA(t1 +t2 ) .
And, from
coth x + coth y =
sinh(x + y)
,
sinh x sinh y
we obtain (ignoring again the determination of roots)
[det sinh((t1 + t2 )S/2)]1/2 [det(A(t1 ) + A(t2 ))]−1/2
= [det sinh(t1 S/2)]1/2 [det sinh(t2 S/2)]1/2 .
0-34
Together this shows that
γt,S := 2−n [det sinh(tS/2)]−1/2 eA(t)
(3.6)
forms a (local) 1-parameter group under twisted convolution, and it is not hard to check that its infinitesi
imal generator is 4π
∆1S .
Finally, if we choose S := J, so that A = −I
and ∆S = L, we obtain Proposition 3.1, at least
for |µ| = 1; the general case follows by a scaling
argument.
Remark 3.2. It is important to note that Howe’s
construction of the metaplectic group by means of
twisted convolution operators can be extended to
the so-called oscillator semigroup. If we denote by
sp+ (C, m) the cone of all elements S ∈ sp(C, m)
such that the associated complex, symmetric matrix
A = SJ has positive semi-definite real part, then
accordingly explicit
formulas for the one-parameter
µ
semigroups et∆S have been derived in [28].
0-35
3.2
A subordination formula
Fix a bump function χ0 ∈ C0∞ (R) supported in the
interval [1/2, 2]. If a ∈ S ν is a symbol of order ν ∈ R
on T ∗ R = R2 , i.e., if
|∂sα ∂σβ a(s, σ)| ≤ Cα,β (1 + |σ|)ν−β ,
(s, σ) ∈ R2 ,
for every α, β ∈ N, then we say that a is supported
over the subset I ⊂ R if supp a(·, σ) ⊂ I for every
σ ∈ R.
Proposition 3.3. There exist a symbol a0 ∈ S 0
supported over the interval [1/16, 4] and a Schwartz
function ρ ∈ S(R2 ) such that, for x ≥ 0,
√x √
tx
it x
χ0
e
, λt
(3.7)
= ρ
λ
λZ
√
ts
λt
λt a0 (s, λt) ei 4s ei λ x ds,
+
for every λ ≥ 1 and t > 0 such that λt ≥ 1.
Proof. This can easily be obtained by the method
of stationary phase.
0-36
µ
The explicit formulas in Proposition 3.1 for eitL
in combination with Proposition 3.3 allow us to describe spectrally
localized parts of the wave propaga√
tors e±it L δ. Of course, there is no loss of generality
in assuming that t ≥ 0 and that the sign in the exponent is positive. Moreover, since L is homogeneous
of degree 1 with respect to the automorphic dilations
δr , we may reduce ourselves to considering the case
where t = 1.
Let us recall here that it is well-known that if
ψ ∈ S(R), then the convolution kernel ψ(L)δ will
also be a Schwartz function (on G), whose L1 (G)norm will be controlled by some Schwartz norm on
S(R) (see, [17]; also [27] for a connection with finite
propagation speed).
0-37
Proposition 3.4. Let χ0 ∈ C0∞ (R) be a bump function supported in the interval [1/2, 2], and assume
that λ ≥ 1. Then there exist Schwartz functions ψλ ∈
S(G) and Kλ ∈ S(G) such that
√L √
ei L δ = ψλ + Kλ ,
χ0
(3.8)
λ
and so that the following hold true:
(i) For every N ∈ N there is constant CN ≥ 0 so that
||ψλ ||1 ≤ CN λ−N
for every λ ≥ 1.
(ii) The partial Fourier transform Kλµ ∈ S(R2m ), (µ 6=
0), of Kλ is given by an oscillatory integral of the
form
Z
√
λ
µ
µ
i 4t
, ϕi dt, ϕ ∈ S(R2m ),
hKλ , ϕi = λ a0 (t, λ) e hγt/λ
where a0 ∈ S 0 is a symbol of order 0 supported over
the interval [1/16, 4] and where γtµ is given by Proposition 3.1.
0-38
Proof. Apart from (i), all the claims follow easily
from the subordination formula (3.7)
R in Proposition
3.3, the spectral resolution L = R+ s dEs and Fubini’s theorem in combination with Proposition 3.1.
To prove (i), notice that we may choose ψλ :=
2
ρ L
,
λ
δ,
with
a
Schwartz
function
ρ
∈
S(R
). In
λ
particular, every Schwartz norm of x 7→ ρ λx , λ de-
cays like = O(λ−N ) for every N ∈ N. By the previous remark, it is then also clear that the convolution
kernel ψλ of the spectral multiplier operator ρ L
λ,λ
will satisfy the estimates in (i).
In view of this proposition, it will in the sequel
suffice to estimate the convolution operators defined
by the kernels Kλ . Moreover, for simplicity, we may
and shall assume that
Z
√
λ
µ
µ
i 4t
, ϕi dt, ϕ ∈ S(R2m ),
hKλ , ϕi = λ χ0 (t) e hγt/λ
(3.9)
where χ0 ∈ C0∞ (R) is supported in the interval [1/16, 4].
0-39
∞
We next choose
η
∈
C
(R) such that supp η0 ⊂
0
0
P
π
3π
[− 3π
k∈Z η0 (s + k 2 ) = 1 for every s ∈ R,
8 , 8 ] and
and put
hAµλ,k , ϕi
Z
λ
µ
i 4t
1/2
, ϕi dt,
χ0 (t) η0 (2πt|µ|/λ − kπ) e hγt/λ
:= λ
µ
hBλ,k
, ϕi
Z
λ
µ
, ϕi dt.
:= λ1/2 χ0 (t) η0 (2πt|µ|/λ − kπ − π2 ) ei 4t hγt/λ
By Aλ,k and Bλ,k we shall denote the tempered distributions defined on G whose partial Fourier transµ
forms along the center are given by Aµλ,k and Bλ,k
,
respectively. Then
Kλ =
∞
X
(Aλ,k + Bλ,k )
(3.10)
k=0
in the sense of distributions.
For the sake of this exposition, we shall restrict
ourselves to considering the Aλ,k . Since
µ
µ
b
, ϕi = hγd
hγt/λ
t/λ , ϕi,
0-40
µ
where γd
t/λ is given by Proposition 3.1, it is clear that
Aµλ,k is a well-defined tempered distribution. Howµ
has a singularity
ever, since the integral kernel γt/λ
in t where 2πt|µ|/λ = kπ if k 6= 0, we shall here perform a priori a dyadic decomposition with respect
to t. More precisely, if k ≥ 1, let us fix another
bump function χ1 ∈P
C0∞ (R) supported in the interl
val [1/2, 2] so that ∞
χ
(2
s) = 1 for every s ∈
1
l=0
µ
]0, 3π
],
and
decompose
the
distribution
A
λ,k as
8
hAµλ,k , ϕi =
∞
X
l=0
(hAµλ,k,l,+ , ϕi + hAµλ,k,l,− , ϕi), (3.11)
where Aλ,k,l,± is defined to be the smooth function
Aµλ,k,l,± (x) :=
R
λ1/2 χ0 (t)χ1 ± 2l (2πt|µ|/λ − kπ) η0 (2πt|µ|/λ − kπ)
m λ
|µ|
i 4t −m −i π
|µ| cot(2πt|µ|/λ)|x|2
−m
2
e 2
e
dt
2
(sin(2πt|µ|/λ)
By Fourier inversion in the central variable u, we
have
Z
Aλ,k,l,± (x, u) =
Aµλ,k,l,± (x)e2πiu·µ dµ,
Rn \{0}
0-41
In the integration w.r. to µ, we introduce polar coordinates µ = rω, r > 0, ω ∈ S n−1 . Then
dµ = r n−1 dσn−1 (ω), where dσn−1 denotes the surface measure on the sphere S n−1 . It is well-known
[14] by the method of stationary phase that
Z
eiu·ω dσn−1 (ω) = ei|u| a+ (|u|) + e−2πi|u| a− (|u|),
S n−1
where a+ , a− ∈ S −(n−1)/2 are symbols of order −(n−
1)/2.
Since the integral defining Aµλ,k,l,± is absolutely
convergent in t, applying this and suitable changes
of coordinates, we then find that
Aλ,k,l,± (x, u) = Aλ,k,l,±,a+ (x, |u|)+Aλ,k,l,±,a− (x, −|u|),
(3.12)
where for any symbol a ∈ S −(n−1)/2 the function
Aλ,k,l,±,a (x, v) on R2m × R is given by
Aλ,k,l,±,a (x, v)
R
R
= λm+n+1/2
χ(t) χ1 ± 2l (s − kπ) η(s − kπ)
1
sm+n−1 sin(s)−m eiλts( s −cot(s)|x|
0-42
2
+4v)
a(λst4|v|) dtds.
Since k ≥ 1, this can be re-written in the form
Aλ,k,l,±,a (x, v)
R
R
χ1 (±2l s)
t
m+n+1/2 m+n−1
=λ
k
χ s/k+π sinm (s) η(s)
iλkt
e
2
1
s+kπ −cot(s)|x| +4v
a(λkt4|v|) dsdt,
where the functions χ and η have similar properties
as χ1 and η0 .
A similar expressions can be given for k = 0.
4
Estimation of Aλ,k,l,± if k ≥ 1
We next need to establish estimates for the kernels
Aλ,k,l,± , into which the kernels Aλ,k decompose, and
their derivatives. In some regions, the estimates that
can be obtained for these kernels will be useful only
if λ & k. However, this problem can be fixed in an
easy way, and we shall not go into details here but
prefer to give a somewhat simplified account of the
actual estimates that we can obtain, for the sake of
clarity of exposition.
0-43
We shall only consider Aλ,k,l,+,a , since Aλ,k,l,−,a ,
can be estimated very much in the same way. Let us
use the short-hand notation Fl (x, v) := Aλ,k,l,+,a (x, v),
so that, by (3.13), for k ≥ 1 we have
Fl (x, v)
R
R
t
χ(l) (r)
= λm+n+1/2 k m+n−1 2(m−1)l
χ 2−l k−1
r+π
−l
iλkt ψ(2−l r)+4v
η(2 r)e
a(λkt4|v|) drdt,
where the function ψ(s) is given by
1
− |x|2 cot(s),
ψ(s) :=
s + kπ
and where
χ(l) (r) :=
χ1 (r)
(2l sin(2−l r))m
is again supported where r ∼ 1. Moreover, if 2−l r ∈
kπ
≤
supp η and k ≥ 1, then |2−l r| ≤ 3π
8
2 , so that
1
−1
∼
k
±2−l r + kπ
for every k ≥ 1, l ≥ 0 (the case of the negative sign
will actually become relevant for the Aλ,k,l,−,a only.)
0-44
Notice that all derivatives of χ(l) (r) are uniformly
bounded in l. We can therefore re-write
Fl (x, v)
=
·
RR
λm+n+1/2 k m+n−1 2(m−1)l
iλkt ψ(2−l r)+4v
χk,l (r, t)e
(4.1)
a(λkt4|v|) drdt,
where χk,l (r, t) is a smooth function supported where
r ∼ 1, t ∼ 1,
such that
|∂rα ∂tβ χk,l (r, t)| ≤ Cα,β for every α, β ∈ N,
(4.2)
uniformly in k and l.
Since
1
2
−2
+
|x|
sin(s)
,
(4.3)
2
(s + kπ)
2
2
−3
−
2|x|
cos(s)
sin(s)
,
3
(s + kπ)
ψ ′ (s) = −
ψ ′′ (s) =
we see that the critical points s = sc of ψ are given
by the equation sin(sc ) = ±|x|(sc + kπ), hence those
of ψ(2−l ·) by r = 2l sc .
0-45
If r is supposed to lie in our domain of integration, then |x| ∼ 2−l k −1 . For such x, there is then in
fact exactly one critical point sc = sc,k (|x|) ∼ 2−l
sufficiently close to our domain of integration, given
by
sin(sc ) = |x|(sc + kπ).
(4.4)
For later use, let us also put
γk (|x|) := ψ(sc,k (|x|)) =
1
−|x|2 cot(sc,k (|x|)).
sc,k (|x|) + kπ
Observe that, by (4.4), this can be re-written as
γk (|x|) =
sk (|x|) − sin sk (|x|) cos sk (|x|)
,
2
sk (|x|)
(4.5)
if we put sk (|x|) := sc,k (|x|) + kπ.
Assume that λ & 2l k.
In this case, one gains from the r-integration in
Fl , which we perform first. To this end, recall that
ψ(s) has now a unique critical point sc = sc,k (|x|) ∼
2−l if |x| ∼ 2−l k −1 .
We may thus apply the method of stationary
phase to the integration in r.
0-46
Noticing that the critical point of the phase ψl is
given by 2l sc,k (|x|), one eventually finds that
R
m+n+1/2 m+n−1 (m−1)l
Fl (x, v)
=λ
k
2
χ(t)a(λkt4|v|)
2 2l
2
b(k 2 |x| , λk
−1 −l
iλkt ψ(sc,k (|x|))+4v
2 t)e
drdt,
where χ(t) is again a smooth cut-off function supported where t ∼ 1, and where b ∈ S −1/2 is a symbol
of order −1/2 on T ∗ R, which may in fact depend
also on k and l, but which lie in a bounded set of
S −1/2 with respect to the natural Fréchet topology
on S −1/2 .
Consequently,
Aλ,k,l,+,a (x, ±|u|)
R
m+n+1/2 m+n−1 (m−1)l
= λ
k
2
χ(t)a(λkt4|u|)
2 2l
2
b(k 2 |x| , λk
−1 −l
iλkt γk (|x|)±4|u|
2 t)e
dt(4.6)
where γk (|x|) := ψ(sc,k (|x|)) is explicitly given by
(4.5). Now, integrations by parts in (4.6) easily yield
that, for every N ∈ N,
1
1
|Aλ,k,l,+,a (x, ±|u|)| ≤ CN λm+n k m+n− 2 2(m− 2 )l
−N
n−1
·(1 + λk|u|)− 2 1 + λk|γk (|x|) ± 4|u|| (4.7)
.
0-47
The actual details of these arguments are more involved, and a complete estimate of Aλ,k,l,+,a requires
a careful case analysis and domain decompositions.
1/k
1/(k+1)
1/k
Figure 2: curve γ
Remarks 4.1. (a) Observe that (4.7) shows that the
singularities of Aλ,k,l,+ (x, u) are located where |u| =
γk (|x|). In view of (4.4) and (4.5), these are thus
located where the pair of norms (|x|, 4|u|) lies on the
piece of the parametric curve
sin s s − sin s cos s γ : s 7→ , ,
2
s
s
corresponding to the range of parameters where s −
kπ ∼ 2−l .
0-48
This corresponds exactly to the set Σ1 , in view
of (2.4).
(b) Analogous estimates and results hold true for
Aλ,k,l,− (x, u). Here, we have sc (|x|) < 0, and the
singularities correspond to the range of parameters
where s − kπ ∼ −2−l .
5
√
−(d−1)/4 i L
Estimation of (1+L)
e
Let us put
1 1
α(p) := (d − 1) − ,
p 2
1 ≤ p < ∞.
√
−α(p)/2 i L
δ on Lp (G) in
We need to estimate (1+L)
e
Theorem 1.1. By means of a dyadic decomposition,
this can easily be reduced to estimating the operator
∞
√L √
X
i L
2−α(p)j χ̃
e
,
(5.1)
j
2
j=0
for a suitable cut-off function χ̃ supported in [1/16, 4].
0-49
δ
In view of (3.10), this means that we have to
estimate the convolution operators on Lp (G) given
by the integral kernels
p,±
Ek,J
(x, u) :=
∞
J X
X
2−α(p)j A2j ,k,l,± (x, u)
j=0 l=0
and the corresponding kernels, with Aλ,k,l,± replaced
by Bλ,k,l,± . These kernels will converge, in the sense
of distributions, to distributions Ekp,± as J → ∞,
and we shall have to show that the estimates sum
in k. Our estimates will follow from some rather elementary estimate for the case p = 2, and a deeper
estimate for p = 1 on a suitable local Hardy space
h1k , whose definition will depend on k, by means of
complex interpolation. The corresponding interpolation argument is standard (cf. [35],[21]).
Let’s here concentrate on the case p = 1, and the
kernels
+
Ek,J
:=
1,+
Ek,J
=
∞
J X
X
−
2
(d−1)
j
2
A2j ,k,l,+ .
j=0 l=0
What can be proved in the end for these kernels is a
result, which, somewhat simplified, reads as follows:
0-50
Proposition 5.1. Let k ≥ 1. There exist kernel
±
functions Kλ,k,l
∈ S(G), for λ ≥ 2l k, and Rk,J ∈
L1 (G) so that
+
Ek,J
=
∞
X
J
X
(K2+j ,k,l +K2−j ,k,l ) +Rk,J , (5.2)
l=0 j=l+log2 k
and such that the following hold true:
±
is supported in the set
(a) The kernel Kλ,k,l
{(x, u) ∈ G : |x| ∼ 2−l k −1 and λk|u| & 1},
and can be estimated, for every α ∈ N2m , β ∈ Nn , by
n+1
1
1
±
|∂xα ∂uβ Kλ,k,l
(x, u)| . λ|α| (λk)|β| λ 2 k m+n− 2 2(m− 2 )l
−N
n−1
(1 + λk|u|)− 2 1 + λk|γk (|x|) ± 4|u||
,
(5.3)
for every N ∈ N, with constants CN,α,β which independent of λ, k and l, and where γk (|x|) is given by
(4.5). In particular,
±
k∂xα ∂uβ Kλ,k,l
k1
|α|
.λ
(λk)
0-51
|β|
l
(2 k)
−m− 12
.
(5.4)
(b) The sequence {Rk,J }J converges for J → ∞ in L1 (G)
towards a function Rk ∈ L1 (G), and
1
kRk k1 ≤ Ck −m− 2 ,
(5.5)
with a constant C not depending on k.
5.1
Anisotropic re-scaling for k fixed
Let k ≥ 1, and denote by
F̃ (x, u) := k −D F (k −1 x, k −2 u)
the L1 -norm preserving scaling of a function F on G
by δk−1 . Let us also put
2
−Γk (v) := k γk (k
0-52
−1
k
v) − .
π
We can then make the interesting observation
that, by (5.3), (5.4), the non-isotropically re-scaled
±
^
are supported in the set
kernels K
λ,k,l
−l
{(x, u) ∈ G : |x| ∼ 2
λ
and |u| & 1},
k
and satisfy the isotropic estimates
±
^
|∂xα ∂uβ K
λ,k,l (x, u)|
.
|α|+|β|
λ
k
(1 + λk |u|)−
n−1
2
and
±
^
k∂xα ∂uβ K
λ,k,l k1
n+1
2
1
1
k −(m+n+ 2 ) 2(m− 2 )l (5.6)
−N
1 + λk πk − Γk (|x|) ± 4|u|
.
λ
λ |α|+|β|
k
l
(2 k)
−m− 12
.
(5.7)
Moreover, one easily verifies that
Γk (v) = π
−2
arcsin(πv) + π
−1
0-53
v
p
1 − (πv)2 + O
1
k
.
6
L2 -estimates for components
of the wave propagator
Denote by Aλ,k,l,± the convolution operator
Aλ,k,l,± f := f ∗ Aλ,k,l,±
on L2 (G). It follows from Proposition 3.1 that
R
Aµλ,k,l,±
χ0 (t)χ1 (±2l (2πt|µ|/λ − kπ))
µ
λ
η0 (2πt|µ|/λ − kπ)ei 4t eit/λ L dt δ.
1/2
= λ
Comparing with (9.13) and (9.12) in the Appendix,
which obtained by means of Plancherel’s formula on
G, we then find that the operator norm of Aλ,k,l,±
on L2 (G) is given by
kAλ,k,l,± k = λ1/2 sup{|αλ,k,l,± (ν, q)| : ν ≥ 0, q ∈ N},
(6.1)
where
Z
αλ,k,l,± (ν, q) := χ0 (t)χ1 (±2l (t λν − kπ))
η0 (t λν
ν
λ
+t λ (m+2q)}
i{ 4t
− kπ)e
0-54
dt.
The method of stationary phase then implies that
kAλ,k,l,± k ≤ C,
uniformly in λ, k and l. By the same method, we also
get, for any α ≥ 0,
k
∞ X
X
l=0 j≥j0
2−αj A2j ,k,l,± k ≤ C 2−αj0 ,
(6.2)
uniformly in l and j0 .
7
Estimation for p = 1
We introduce a non-standard atomic local Hardy
space h1 on G as follows:
An atom centered at the origin is an L2 -function
a supported in a Euclidean ball Br (0) of radius r ≤ 1
satisfying the normalizing condition
kak2 ≤ r −d/2
(7.1)
and, if r < 1, in addition the moment condition
Z
a(x) dx = 0.
(7.2)
G
0-55
Notice that in particular kak1 . 1. An atom centered at z ∈ G is the left-translate by z of an atom
centered at the origin. The local Hardy space h1 (G)
consists of all L1 -functions which admit an atomic
decomposition
X
λj aj ,
(7.3)
f=
j
P
with atoms aj and coefficients λj such that j |λj | <
∞. WePdefine the norm kf kh1 as the infimum of the
sums j |λj | < ∞ over all possible atomic decompositions (7.3) of f.
In comparison, let us denote by h1E := h1 (R2m ×
Rn ) the classical Euclidean local Hardy space introduced by Goldberg [10]. Since atoms in this space
which are supported in a ball of radius r ≥ 1 need
not satisfy (7.2), such atoms can be decomposed into
atoms supported in balls of radius 1, so that one can
give the same atomic definition for h1E as for h1 ,
only with the non-commutative group translation in
G replaced by Euclidean translation.
Consider the cylindrical sets Z(y) := {(x, u) ∈
G : |x − y| ≤ 10}, for y ∈ R2m . Note the following
easy, but important observations:
0-56
Lemma 7.1. a) Any atom a supported in Z(0) is
indeed also a Euclidean atom in h1E (possibly up to a
fixed factor depending only on G), and vice versa.
1
b) If HE
:= H 1 (R2m × Rn ) denote the classical
Euclidean Hardy space, and if η ∈ C0∞ (R2m ), then
1
continuously into
multiplication with η ⊗ 1 maps HE
h1E .
b) is known for compactly supported smooth multipliers, and the proof carries over easily.
Since G can be covered by sets Z(yj ) with bounded
overlap, if we choose a suitable ε-separated set of
points {yj }j for a suitable constant ε > 0, one can
use a) and b) in combination with classical arguments [6] to show that our non-standard Hardy space
h1 (G) interpolates with L2 (G) by the complex method.
We shall not do this here, but indicate the main
idea in the interpolation argument explained later.
Proposition 7.2. There is a constant C > 0, such
that
1,±
]
−m
kf ∗ E
k
≤
Ck
kf kh1
1
k
for every f ∈ h1 (G).
0-57
Proof. We again consider only Ek1,+ = Ek+ . In view
of the atomic decomposition of f and because of leftinvariance, one can show that it suffices to assume
that f = a is an atom centered at the origin and
supported in a ball Br (0). We shall also assume that
r < 1, so that a has vanishing integral, since the case
r = 1 is somewhat easier. In view of Proposition 5.1,
we have to understand in particular the functions
±
^
a∗K
.
2j ,k,l
To simplify the argument a bit, assume that n =
1, so that |u| = ±u. Let’s also assume that we con±
±
^
^
where u > 0, and that K
sider the part of K
λ,k,l
λ,k,l
were not only essentially, but actually supported in
the set where
k
k
− Γk (|x|) + 4u| ≤ .
π
λ
Translating these kernels along the center by πk , in
view of (5.7) we can then make the (slightly oversimplified) assumption that
±
^
(7.4)
supp K
λ,k,l
⊂ {(x, u) ∈ G : |x| ∼ 2−l and u − 41 Γk (|x|) ≤ λk },
0-58
and that
±
^
|∂xα ∂uβ K
(7.5)
λ,k,l (x, u)|
|α|+|β|
λ
−(m+n+ 12 ) (m− 21 )l
≤ CN,α,β k
λk
2
Define for ρ > 0 the set
Ωρ := {(x, u) ∈ G : |x| . 1 and u− 41 Γk (|x|) ≤ Cρ}
where C is a sufficiently large constant, and choose
j0 so that
k2−j0 = r.
Obrserve that with this choice of j0
supp a ∗ (
XX
j≥j0
l
±
^
K
2j ,k,l ) ⊂ Ωr ,
since supp a ⊂ Br (0). We therefore split
Ek1,± = Ek,r + Fk,r ,
where
Ek,r :=
Fk,r :=
P∞ P
l=0
P∞
l=0
j≥j0
P
j<j0
2−α(1)j A2j ,k,l,± ,
2−α(1)j A2j ,k,l,± .
0-59
(7.6)
g
We estimate a ∗ E
k,r separately on the set Ωr and
it’s complement. By Hölder’s inequality and (6.2),
we get
g
ka ∗ E
k,r kL1 (Ωr )
. |Ωr |1/2 kak2 2−α(1)j0
. r
1/2 −(m+1/2) −
r
2
(d−1)
j0
2
≤ k −m .
Moreover, since by Proposition 5.1
Ek,r =
XX
l
where kRk,r k1 . k
j≥j0
1
−m− 2
±
^
g
K
2j ,k,l + Rk,r ,
, we obtain from (7.6) that
1
−m− 2
g
g
ka ∗ E
,
k,r kL1 (G\Ωr ) ≤ ka ∗ Rk,r kL1 (G) . k
so that
−m
g
ka ∗ E
.
k,r k1 . k
(7.7)
Consider next a∗ Fg
k,r . Again, by Proposition 5.1,
P P
±
^
′ , where
g
g
+
R
we can write Fk,r = l j<j0 K
j
k,r
2 ,k,l
′
k1
kRk,r
. k
1
−m− 2
. And, for j < j0 , we have, by
0-60
(7.4),
supp a ∗ (
X
l
±
^
−j
K
2j ,k,l ) ⊂ Ωk2 .
R
Moreover, making use of the cancellation a dx = 0
and (7.5) in estimating the convolution product a ∗
±
^
K
, we gain a factor of order O( r ) compared
k2−j
2j ,k,l
to (5.7) and obtain
∞
X
X
±
^
K
ka∗
2j ,k,l k1 .
l
l=0
r
l −m− 12
j r −m− 21
(2 k)
.2 k
k2−j
k
Summing in j, this gives
−m− 12
.
ka ∗ Fg
k
.
k
k,r 1
(7.8)
The estimates (7.7) and (7.8) imply the proposition.
Remark 7.3. Let us denote by h1k (G) the re-scaled
local Hardy space consisting of all function
f (x, u) := k D f˜(kx, k2 u), f˜ ∈ h1 (G).
Then Proposition 7.1 can be re-stated as
kf ∗ Ek1,± k1 ≤ Ck −m kf kh1k
for every f ∈ h1k (G). Thus, h1k (G) is a natural space
associated with the k-th piece Ek1,± of the wave propagator whose definition depends on the parameter k.
0-61
8
Proof of Theorem 1.1
Observe that by (6.2) convolution with the kernel
Ek2,± :=
∞
∞ X
X
A2j ,k,l,±
j=0 l=0
is bounded on L2 (G), and
kf ∗ Ek2,± k2 ≤ Ckf k2 ,
(8.1)
where C is independent of k. Consider now the analytic family of operators
∞
∞ X
X
2−αj A^
Tα : f 7→ f ∗
2j ,k,l,± ,
j=0 l=0
0 ≤ Re α ≤
(d−1)
2 .
By ignoring some error terms, we may assume that
all A^
2j ,k,l,± are supported in Z(0) (even a smaller
cylindrical set). This implies that if f is supported
in a cylindrical set Z(y), then Tα f is supported in the
“cylindrical double” 2Z(y) := {(x, u) ∈ G : |x − y| ≤
20}, for every y ∈ R2m , which allows to reduce to
functions f supported in a set Z(y), and we may
even assume y = 0, by translation invariance.
0-62
Choosing a suitable cut-off function η ∈ C0∞ (R2m )
so that η ⊗ 1 localizes to Z(0), we therefore consider
the localized operators
Tα0 f := Tα ((η ⊗ 1)f ).
1
0
: HE
→ L1
By Lemma 7.1 and Proposition 7.3, T d−1
2
continuously, and since the proof of Proposition 7.3
remains valid for T d−1 +iβ , we have
2
−m
0
f
k
≤
Ck
kf kHE1
kT d−1
1
+iβ
2
∀β ∈ R.
Similarly, by (8.1), we have
0
kTiβ
f k2 ≤ Ckf k2
∀β ∈ R.
We can thus apply the complex interpolation result
by C. Fefferman and Stein in [6] to obtain
0
kTα(p)
f kp
≤ Ck
2
−1)m
−( p
kf kp ,
1 < p ≤ 2.
In view of the afore-mentioned support property of
the convolution kernels involved, this immediately
implies
kf ∗
α(p),±
kp
Ek
≤ Ck
2
−( p
−1)m
0-63
kf kp ,
1 < p ≤ 2.
If m ≥ 2, these estimates sum in k for p sufficiently
close to 1, so that
k(1 + L)
√
−α(p)/2 i L
e
f kp ≤ Ckf kp
(8.2)
for these p. Interpolating these estimates with the
corresponding trivial estimate for p = 2 we then obtain Theorem 1.1 for 1 < p ≤ 2, and the case p > 2
follows by duality.
For m = 1, the result can be obtained by means
of a minor refinement of the arguments.
The L1 -estimate in Theorem 1.1 follows a lot
more easily from Proposition 5.1, since for α > d−1
2
d−1
we have an additional factor λ−(α− 2 ) in the corresponding estimate (5.4), so that the corresponding
norms kK2±j ,k,l k1 simply sum in j, k and l.
0-64
9
Appendix: The Fourier transform on a group of Heisenberg type
Let us first briefly recall some facts about the unitary
representation theory of a Heisenberg type group G
(compare [36],[5]). For the convenience of the reader,
we shall show how this representation theory can easily be reduced to the well-known case of the Heisenberg group Hm = R2m × R. In slightly modified notation, the product in Hm is given by
1
(z, t) · (z , t ) = (z + z , t + t + ω(z, z ′ )),
2
′
′
′
′
where ω denotes the canonical symplectic form
ω(z, w) := hJz, wi,
0
J :=
Im
−Im
,
0
(9.1)
on R2m . Let us split coordinates z = (x, y) ∈ Rm ×
Rm in R2m , and consider the associated natural basis
of left-invariant vector fields
X̃j :=
∂
∂
− 21 yj ,
∂xj
∂t
Ỹj :=
0-65
∂
∂
+ 12 xj , j = 1, . . . , m
∂yj
∂t
of the Lie algebra of Hm .
For τ ∈ R× := R \ {0}, the Schrödinger representation ρτ of Hm acts on the Hilbert space L2 (Rm ) as
follows:
1
[ρτ (x, y, t)h](ξ) := e2πiτ (t+y·ξ+ 2 y·x) h(ξ + x),
This is an irreducible, unitary representation, and
every irreducible unitary representation of Hm which
acts non-trivially on the center is in fact unitarily
equivalent to exactly one of these, by the Stone-von
Neumann theorem (a good reference to these and
related results is for instance [8]; see also [26]).
Next, if π is any unitary representation, say, of a
Heisenberg type group G, we denote by
Z
π(f ) :=
f (g)π(g) dg, f ∈ L1 (G),
G
the associated representation of the group algebra
L1 (G). Going back to the Heisenberg group, if f ∈
S(Hm ), then
it is well-known and easily seen that
R
ρτ (f ) = R2m f −τ (z)ρτ (z, 0) dz is a trace class oper-
0-66
ator on L2 (Rm ), and its trace is given by
Z
f (0, 0, t)e2πiτ t dt = |τ |−m f −τ (0, 0),
tr(ρτ (f )) = |τ |−m
R
(9.2)
for every τ ∈ R× .
From these facts, it is easy to derive the Plancherel
formula for our Heisenberg type group G. Given µ ∈
g∗2 = Rn , µ 6= 0, consider the matrix Jµ introduced
in Section 3. If |µ| = 1, then Jµ2 = −I, because
of (1.1). In particular, Jµ has only eigenvalues ±i,
and since it is orthogonal, it is easy to see that there
exists an orthonormal basis
Xµ,1 , . . . , Xµ,m , Yµ,1 , . . . , Yµ,m
of g1 = R2m which is symplectic with respect to
the form ωµ , i.e., ωµ is represented by the standard
symplectic matrix J in (9.1).
This means that, for every µ ∈ Rn \ {0}, there
µ ∈ O(2m, R) such
is an orthogonal matrix Rµ = R |µ|
that
Jµ = |µ|Rµ J tRµ .
(9.3)
Notice also that tRµ = Rµ−1 , and that condition (9.3)
is in fact equivalent to G being of Heisenberg type.
0-67
We remark that (9.3) easily implies that the subalgebra L1r (G) of L1 (G), consisting of all radial functions f (x, u) in the sense that they depend only on
|x| and u, is commutative. This fact is well-known
and easy to prove for Heisenberg groups ([8],[26]).
And, by means of (9.3), one easily checks that
(ϕ ∗µ ψ) ◦ Rµ = (ϕ ◦ Rµ ) ×|µ| (ψ ◦ Rµ )
for all functions ϕ, ψ ∈ L1 (R2m ). Therefore, if f, g ∈
L1r (G), then f µ ◦ Rµ , g µ ◦ Rµ are radial on R2m , and
thus f µ ◦Rµ ×|µ| g µ ◦Rµ = g µ ◦Rµ ×|µ| f µ ◦Rµ , which
is again radial. Consequently, f µ ∗µ g µ = g µ ∗µ f µ ,
for every µ, and thus
f ∗g =g∗f
for every f, g ∈ L1r (G).
(9.4)
The following lemma is easy to check and establishes a useful link between representations of G and
those of Hm .
Lemma 9.1. The mapping αµ : G → Hm , given by
αµ (z, u) := ( tRµ z, µ·u
|µ| ),
(z, u) ∈ R2m × Rn ,
is an epimorphism of Lie groups. In particular, G/ ker αµ
is isomorphic to Hm , where ker αµ = µ⊥ is just the
orthogonal complement of µ in the center Rn of G.
0-68
Given µ ∈ Rn \ {0}, we can now define an irreducible unitary representation πµ of G on L2 (Rm )
by simply putting
πµ := ρ|µ| ◦ αµ .
Observe that then πµ (0, u) = e2πiµ·u I. In fact, any
irreducible representation of G with central character e2πiµ·u factors through the kernel of αµ and
hence, by the Stone-von Neumann theorem, must
be equivalent to πµ .
One then computes that, for f ∈ S(G),
Z
πµ (f ) =
f −µ (Rµ z)ρ|µ| (z, 0) dz,
R2m
so that the trace formula (9.2) yields the analogous
trace formula
tr πµ (f ) = |µ|−m f −µ (0)
n
on G. The Fourier
inversion
formula
in
R
then leads
R
to f (0, 0) = µ∈Rn \{0} tr πµ (f ) |µ|m dµ. When applied to δg−1 ∗ f, we arrive at the Fourier inversion
formula
Z
f (g) =
tr (πµ (g)∗ πµ (f )) |µ|m dµ, g ∈ G.
µ∈Rn \{0}
(9.5)
0-69
Applying this to f ∗ ∗ f at g = 0, where f ∗ (g) :=
f (g −1 ), we obtain the Plancherel formula
Z
kf k22 =
kπµ (f )k2HS |µ|m dµ,
(9.6)
µ∈Rn \{0}
where kT k2HS = tr (T ∗ T ) denotes the Hilbert-Schmidt
norm.
Let us next consider the group Fourier transform
of our sub-Laplacian L on G.
We first observe that dαµ (X) = tRµ X for every
X ∈ g1 = R2m , if we view, for the time being, elements of the Lie algebra as tangential vectors at the
identity element. Moreover, by (9.3), we see that
t
Rµ Xµ,1 , . . . , tRµ Xµ,m , tRµ Yµ,1 , . . . , tRµ Yµ,m
forms a symplectic basis with respect to the canonical symplectic form ω on R2m . We may thus assume
without loss of generality that this basis agrees with
our basis X̃1 , . . . , X̃m , Ỹ1 , . . . , Ỹm of R2m , so that
dαµ (Xµ,j ) = X̃j , dαµ (Yµ,j ) = Ỹj ,
j = 1, . . . , m.
By our construction of the representation πµ , we thus
0-70
obtain for the derived representation dπµ of g that
dπµ (Xµ,j ) = dρ|µ| (X̃j ), dπµ (Yµ,j ) = dρ|µ| (Ỹj ),
(9.7)
j = 1, . . . , m. Let us define the sub-Laplacians Lµ on
G and L̃ on Hm by
Lµ := −
m
X
2
2
(Xµ,j
+ Yµ,j
),
j=1
L̃ := −
m
X
(X̃j2 + Ỹj2 ),
j=1
where from now on we consider elements of the Lie
algebra again as left-invariant differential operators.
Then, by (9.6),
dπµ (Lµ ) = dρ|µ| (L̃).
Moreover, since the basis Xµ,1 , . . . , Xµ,m , Yµ,1 , . . . , Yµ,m
and our original bases X1 , . . . , X2m of g1 are both orthonormal bases, it is easy to verify that the distributions Lδ0 and Lµ δ0 do agree. Since Af = f ∗ (Aδ0 )
for every left-invariant differential operator A, we
thus have L = Lµ , hence
dπµ (L) = dρ|µ| (L̃).
(9.8)
But, it is well-known that dρ|µ| (L̃) = ∆ξ −(2π|µ|)2 |ξ|2
is just a re-scaled Hermite operator, and an orthonormal basis of L2 (Rm ) is given by the tensor products
|µ|
|µ|
|µ|
hα
:= hα
⊗
·
·
·
⊗
h
αm ,
1
0-71
α ∈ Nm ,
where hµk (x) := (2π|µ|)1/4 hk ((2π|µ|)1/2 x). Here,
k x2 /2
hk (x) = ck (−1) e
dk −x2
e
dxk
denotes the L2 -normalized Hermite function of order
k on R. Consequently,
|µ|
|µ|
dπµ (L)hα
= 2π|µ|(m + 2|α|)hα
,
α ∈ Nm . (9.9)
It is also easy to see that
dπµ (Uj ) = 2πiµj I,
j = 1, . . . , n.
(9.10)
Now, the operators L, −iU1 , . . . , −iUn form a commuting set of self-adjoint operators, with joint core
S(G), so that they admit a joint spectral resolution,
and we can thus give meaning to expressions like
ϕ(L, −iU1 , . . . , −iUn ) for each continuous function
ϕ defined on the corresponding joint spectrum. For
simplicity of notation we write
U := (−iU1 , . . . , −iUn ).
If ϕ is bounded, then ϕ(L, U ) is a bounded, left invariant operator on L2 (G), so that it is a convolution
operator
ϕ(L, U )f = f ∗ Kϕ ,
0-72
f ∈ S(G),
with a convolution kernel Kϕ ∈ S ′ (G) which will also
be denoted by ϕ(L, U )δ. Moreover, if ϕ ∈ S(R×Rn ),
then ϕ(L, U )δ ∈ S(G) (compare [29], [30]). Since
functional calculus is compatible with unitary representation theory, we obtain in this case from (9.9),
(9.10) that
|µ|
|µ|
πµ (ϕ(L, U )δ)hα
= ϕ(2π|µ|(m + 2|α|), 2πµ)hα
(9.11)
(this identity in combination with the Fourier inversion formula could in fact be taken as the definition of ϕ(L, U )δ). In particular, the Plancherel theorem implies then that the operator norm on L2 (G)
is given by
kϕ(L, U )k = sup{|ϕ(|µ|(m+2q), µ)| : µ ∈ Rn , q ∈ N}.
(9.12)
Finally, observe that
Kφµ = ϕ(Lµ , 2πµ)δ;
(9.13)
this follows for instance by applying the unitary representation induced from the character e2πiµ·u on the
center of G to Kϕ .
0-73
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