SHARP L -ESTIMATES FOR THE WAVE EQUATION ON HEISENBERG TYPE GROUPS

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SHARP Lp -ESTIMATES FOR THE WAVE EQUATION
ON HEISENBERG TYPE GROUPS
LECTURE NOTES ORLEANS, APRIL 2008
HTTP://ANALYSIS.MATH.UNI-KIEL.DE/MUELLER/
DETLEF MÜLLER
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.
Date: April 8, 2008.
1
2
DETLEF MÜLLER
Contents
1. Introduction
2
1.1. Connections with spectral multipliers and further facts
about the wave equation
7
2. The sub-Riemannian geometry of a Heisenberg type group
10
3. The Schrödinger and the wave propagators on a Heisenberg
type group
14
3.1. Twisted convolution and the metaplectic group
16
3.2. A subordination formula
18
4. Estimation of Aλ,k,l,± if k ≥ 1
22
√
5. Estimation of (1 + L)−(d−1)/4 ei
L
δ
25
5.1. Anisotropic re-scaling for k fixed
27
6. L2 -estimates for components of the wave propagator
27
7. Estimation for p = 1
28
8. Proof of Theorem 1.1
31
9. Appendix: The Fourier transform on a group of Heisenberg
type
34
References
38
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) ,
THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
3
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
(1.1)
for every µ ∈ g∗2 \ {0}.
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-Campbell-Hausdorff 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 leftinvariant vector field on G given by the Lie-derivative
Xf (g) :=
d
f (g exp(tX))|t=0 ,
dt
where exp : g → G denotes the exponential mapping, which agrees
with the identity mapping in our case.
Let us next fix an orthonormal basis X1 , . . . , X2m of g1 , and let us
define the non-elliptic sub-Laplacian
L := −
2m
X
j=1
Xj2
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DETLEF MÜLLER
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 canonical Euclidean
P
2m+n
product z · w = 2m+n
z
, and Jµ will be identified with
j wj on R
j=1
a skew-symmetric 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 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}, (x, u) ∈ G, r > 0,
then the volume |Qr (x, u)| is given by
where
|Qr (x, u)| = |Q1 (0, 0)| r D ,
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}, (x, u) ∈ G, r > 0,
with respect to the Euclidean norm
|(x, u)| := (|x|2 + |u|2)1/2 .
In the special case n = 1 we may assume that Jµ = µJ, µ ∈ R, where
0
Im
J :=
−Im 0
THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
5
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′ + 21 hJz, z ′ i).
A basis of the Lie algebra hm of Hm is then given by X1 , . . . , X2m , U,
∂
is central.
where U := ∂u
Consider the following Cauchy problem for the wave equation on
G × R associated to L:
∂2u
∂u (1.2)
−
(−L)u
=
0,
u
=
f,
= g,
∂t2
∂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.
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 (noticeR that L is essentially selfadjoint on C0∞ (G) [33]) 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 m(L)f = f ∗ Km =
R
f (y)Km(y −1·) dy, where a priori the convolution kernel Km is a tem-
G
pered 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 ,
one is √naturally led√to study the mapping properties of operators such
sin(t L)
eit L
√
as (1+L)
as operators on Lp (G) into itself.
α/2 or
L(1+L)α/2
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DETLEF MÜLLER
For the classical wave equation on Euclidian space, sharp estimates
for the corresponding operators have been established by Peral [35] and
Miyachi [21].
√
d
−α/2 it −∆
In particular, if ∆ denotes the Laplacian
on
e
R , then (1−∆)
1 1
p
d
is bounded on L (R ), if α ≥ (d − 1) p − 2 , for 1 < p < ∞. Moreover,
(d−1)/2
(1 − ∆)− 2
into L1 (Rd ).
√
eit
−∆
is bounded from the real Hardy space H 1 (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.
The problem in studying the wave equation associated to the subLaplacian on the Heisenberg group is the lack of strict hyperbolicity,
since L is degenerate-elliptic, 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.
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.
THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
7
√
i L
e
p
Theorem 1.1. (1+L)
α/2 extends to a bounded operator on L (G), for
.
1 < p < ∞, when α ≥ (d − 1)| p1 − 12 |, and for p = 1, if α > d−1
2
The proof will be based on some “Hardy space type result” for the
endpoint p = 1, which will become evident later. The main difference compared to the Euclidean setting seems, however, that 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 (compare Remark 7.3).
Remark 1.2. The same result holds for
tors (1 + L)−α/2 (respectively
√ −(α−1) (1 + L)
(respectively (1 + L)
).
√
sin
√
L
, or with the fac√
) replaced by (1 + L)−α ,
L(1+L)
−(α−1)/2
α−1
2
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.
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 Sobolev-space 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]):
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DETLEF MÜLLER
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. 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:
√
i L
e
Proposition 1.5. Assume that (1+L)
α/2 extends to a bounded operator
1
on L (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) .
Proof. Observe first that (1 + L)−ε δ ∈ L1 (G) for any ε > 0. This
follows from the formula
Z ∞
Z ∞
1
1
−ε
ε−1 −t(1+L)
(1 + L) δ =
t e
δ dt =
tε−1 e−t pt dt
Γ(ε) 0
Γ(ε) 0
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
Z ∞
√
√
√
−γ
ψ̂(t)eit L g dt.
Kϕ = ψ( L)((1 + L) δ) = ψ( L)g = c
−∞
THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
9
And, our assumption in combination with some easy multiplier estimates (cf. [31]) and the homogeneity of L yields
√
keit
L
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 gk1dt.
But, (1 + L)α/2 g = (1 + L)α/2−γ δ ∈ L1 , hence, by Hölder’s inequality
and Plancherel’s theorem,
Z ∞
1/2 Z ∞
1/2
β 2
kKϕ ||1 .
|ψ̂(t)(1+|t|) | dt
·
(1+|t|)2(α−β) dt
. kψkH β .
−∞
−∞
Q.E.D.
+ 12 = d2 in this subordination prinIndeed, we may choose β > d−1
2
ciple. 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 Calderón-Zygmund 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.
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].
10
DETLEF MÜLLER
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) we write hX, Y ig := gp (X, Y )
1/2
and kXk = hX, Xig . M, endowed with such a structure, is called a
sub-Riemannian manifold.
An absolutely continuous curve γ : [0, T ] → M in M is then called
horizontal, if
d
γ̇(t) = 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
The Carnot–Carathéodory distance (or sub-Riemannian distance) on
(M, D, g) is then defined by
dCC (p, q) := inf{L(γ) : γ is a horinzontal curve joining p with q},
p, q ∈ M, where we use the convention that inf ∅ := ∞.
We shall here be interested in the special situation where D =
span {X1 , . . . , Xk }, and 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 iff there are functions aj (t) such that γ̇(t) =
1/2
RT P
Pk
2
a
(t)X
(γ(t))
for
a.e.
t
∈
[0,
T
],
and
L(γ)
=
a
(t)
dt.
j
j
j
j=1
j
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.
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 ,
THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
11
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:
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) := 21 kξk2g
is called the Hamiltonian of the sub-Riemannian manifold M. Observe
that this is, apart from the factor 21 , just the principal symbol of the
associated sub-Laplacian. Since T ∗ M is a symplectic manifold
in a
P
canonical 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
ẋ(t) =
∂H
(x(t), ξ(t)),
∂ξ
∂H
ξ˙ = −
(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 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
12
DETLEF MÜLLER
it is known [11] that groups of Heisenberg type (and more generally,
Metivier groups) do not admit non-trivial 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 = 12 |ξ − 21 Jµ x|2 ,
if we write coordinates for T ∗ G as ((x, u), (ξ, µ)) ∈ (R2m ×Rn )×(R2m ×
Rn ). The Hamilton-Jacobi equations are then given by
∂H
∂H
,
u̇ =
∂ξ
∂µ
∂H
∂H
ξ˙ = −
,
µ̇ = −
.
∂x
∂u
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 − 12 Jµ x. Then
H = 12 hη, ηi, so that |η| is constant along the bicharacteristic. Also
µ = const., and
ẋ =
ẋ = η, ξ˙ = − 21 Jµ η
η̇ = ξ˙ − 12 Jµ ẋ = −Jµ η,
hence
η(t) = e−tJµ α,
if we put ξ(0) = η(0) := α. This implies x(t) =
Jµ2 = −|µ|2 I, we have
(2.2)
1−e−tJµ
α,
Jµ
and since
sin( 2t |µ|) − t Jµ
e 2 α.
x(t) = 2
|µ|
For simplicity, let us now assume that n = 1, i.e., that we are dealing
with a Heisenberg group. Then Jµ = µJ, with µ ∈ R, and we have
sin( 2t |µ|) −tJµ
t
he
α, Je− 2 Jµ αi
u̇ =
=−
|µ|
t
sin( 2 |µ|)
t
hα, Je 2 Jµ αi.
= −
|µ|
hη, − 21 Jxi
THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
13
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
µ
t
u̇ =
|α|2 sin2 ( |µ|),
2
|µ|
2
if µ 6= 0. Consequently,
(
sin( 2t |µ|) cos( 2t |µ|)
) |α|2 |µ|µ2 ,
( 2t −
|µ|
(2.3)
u(t) =
0, if µ = 0.
if µ 6= 0
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=P
0, and it has constant speed kγ̇kg = |ẋ(t)| = |η| = |α|, since
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 arbitrarily within the unit sphere in R2m and µ in Rn .
Therefore,
(2.4)
sin s s − sin s cos s Σ1 = {(x, u) ∈ G : |x| = , 4|u| = for some s ∈ R.}
2
s
s
We then expect that Σ1 is exactly the singular support of any of the
wave propagators
√
√
√
sin(t L)
√
δ, cos(t L)δ or eit L 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
14
DETLEF MÜLLER
The “outer hull” of the set Σ1 is just the unit sphere with respect to
the Carnot-Carathéodory distance.
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 propagators
√
sin(t L)
√
L)δ, then, for t ≥ 0,
δ,
cos(t
L
(2.5)
supp Pt ⊂ {(x, u) ∈ G : k(x, u)kCC ≤ t}
3. 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, 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 WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
the center by
µ
f (x) :=
Z
Rn
15
f (x, u)e−2πiµ·u du (x ∈ R2m ).
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 µtwisted 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.,
2m
X
(Lf )µ = Lµ f µ = −
(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 the one-parameter
µ
Schrödinger group eitL , t ∈ R, generated by Lµ can be computed
explicitly:
Proposition 3.1. For f ∈ S(G),
µ
eitL f = f ∗µ γtµ ,
t ∈ R,
where γtµ ∈ S ′ (R2m ) is a tempered distribution given by the imaginary
Gaussian
m
π
|µ|
2
µ
−m
γt (x) = 2
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
2π
1
tan(2πt|µ|)|ξ|2
i |µ|
µ
γc
e
t (ξ) =
(cos(2πt|µ|))m
whenever cos(2πt|µ|) 6= 0.
16
DETLEF MÜLLER
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
(3.1)
S := −AJ,
0
Im
is as in the definition of the product in Hm .
−Im 0
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
where J :=
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
(3.2)
[∆S1 , ∆S2 ] = −2U∆[S1 ,S2 ] ,
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
(3.3)
S 7→
i
∆µ
4πµ S
is a representation of sp(m, R) by (formally) skew-adjoint 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
THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
17
group Mp (m, R) can in fact be represented by twisted convolution operators of the form f 7→ f × γ, where the γ’s are suitable measures
which, generically, are multiples of purely imaginary Gaussians
eA (z) := e−iπ
tz·A·z
,
with real, symmetric 2m × 2m matrices A. In particular, one has
t
1
ei 4π ∆S f = f × γt,S ,
(3.4)
t ∈ R.
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 = 21 JS1 , B = 21 JS2 , with S1 , S2 ∈ sp(m, R), and assuming that S1
and S2 commute, one finds that
e 1 JS1 × e 1 JS2 = 2n (det(S1 + S2 ))−1/2 e 1 J[S1 S2 +I)(S1 +S2 )−1 ] .
2
2
2
This reminds of the addition law for the hyperbolic cotangent, namely
coth x coth y + 1
.
coth x + coth y
We are thus led to define, for non-singular S,
1
(3.5)
A(t) := J coth(tS/2),
2
which is well-defined at least for |t| > 0 small.
coth(x + y) =
Then
eA(t1 ) × eA(t2 ) = 2n (det(A(t1 ) + A(t2 )))−1/2 eA(t1 +t2 ) .
And, from
sinh(x + y)
,
sinh x sinh y
we obtain (ignoring again the determination of roots)
coth x + coth y =
[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 .
18
DETLEF MÜLLER
Together this shows that
(3.6)
γt,S := 2−n [det sinh(tS/2)]−1/2 eA(t)
forms a (local) 1-parameter group under twisted convolution, and it is
not hard to check that its infinitesimal generator is 4πi ∆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 semigroups
t∆µ
S
e , t ≥ 0, have been derived in [28].
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
e
χ0
= ρ
, λt
λ
λZ
√
λt
ts
+
(3.7)
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.
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 propagators 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.
THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
19
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).
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 √
χ0
ei L δ = ψλ + Kλ ,
(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
, ϕi dt, ϕ ∈ S(R2m ),
hKλ , ϕi = λ a0 (t, λ) ei 4t 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.
Proof. Apart from (i), all the claims follow easily from the subordination
formula (3.7) in Proposition 3.3, the spectral resolution
R
L = R+ s dEs and Fubini’s theorem in combination with Proposition
3.1.
To prove (i), notice that we may choose ψλ := ρ Lλ , λ δ, with a
Schwartz
ρ ∈ S(R2 ). In particular, every Schwartz norm of
function
x 7→ ρ λx , λ decays 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 Lλ , λ will satisfy the estimates in (i).
20
DETLEF MÜLLER
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
(3.9)
hKλ , ϕi = λ χ0 (t) ei 4t hγt/λ
, ϕi dt, ϕ ∈ S(R2m ),
where χ0 ∈ C0∞ (R) is supported in the interval [1/16, 4]. Indeed, this
will be justified since our estimates will only depend on some C k -norm
of χ0 .
, 3π
] and
η0 ∈ C0∞ (R) such that supp η0 ⊂ [− 3π
8
8
PWe next choose
π
η
(s
+
k
)
=
1
for
every
s
∈
R,
and
put
0
k∈Z
2
Z
λ
µ
µ
1/2
, ϕi dt,
χ0 (t) η0 (2πt|µ|/λ − kπ) ei 4t hγt/λ
hAλ,k , ϕi := λ
Z
λ
µ
µ
1/2
, ϕi dt.
χ0 (t) η0 (2πt|µ|/λ − kπ − π2 ) ei 4t hγt/λ
hBλ,k , ϕi := λ
By Aλ,k and Bλ,k we shall denote the tempered distributions defined
on G whose partial Fourier transforms along the center are given by
µ
Aµλ,k and Bλ,k
, respectively. Then
(3.10)
Kλ =
∞
X
(Aλ,k + Bλ,k )
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
hγt/λ
, ϕi = hγd
b
t/λ , ϕi,
µ
µ
where γd
t/λ is given by Proposition 3.1, it is clear that Aλ,k is a wellµ
defined tempered distribution. However, since the integral kernel γt/λ
has a singularity 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
χ1 ∈ C0∞ (R) supported in
P∞ function
the interval [1/2, 2] so that l=0 χ1 (2l s) = 1 for every s ∈]0, 3π
], and
8
decompose the distribution Aµλ,k as
(3.11)
hAµλ,k , ϕi
=
∞
X
l=0
(hAµλ,k,l,+, ϕi + hAµλ,k,l,−, ϕi),
THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
21
where Aλ,k,l,± is defined to be the smooth function
R
Aµλ,k,l,±(x) := λ1/2 χ0 (t)χ1 ± 2l (2πt|µ|/λ − kπ) η0 (2πt|µ|/λ − kπ)
m λ
π
2
|µ|
2−m (sin(2πt|µ|/λ)
ei 4t 2−m e−i 2 |µ| cot(2πt|µ|/λ)|x| dt
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}
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|), u ∈ Rn ,
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
(3.12)
Aλ,k,l,± (x, u) = Aλ,k,l,±,a+ (x, |u|) + Aλ,k,l,±,a− (x, −|u|),
where for any symbol a ∈ S −(n−1)/2 the function Aλ,k,l,±,a(x, v) on R2m ×
R is given by
RR
Aλ,k,l,±,a(x, v) = λm+n+1/2
χ(t) χ1 ± 2l (s − kπ) η(s − kπ)sm+n−1
1
sin(s)−m eiλts( s −cot(s)|x|
2 +4v)
a(λst4|v|) dtds.
Since k ≥ 1, this can be re-written in the form
R R t χ1 (±2l s)
Aλ,k,l,±,a(x, v) = λm+n+1/2 k m+n−1
χ s/k+π sinm (s) η(s)
(3.13)
iλkt
e
1
−cot(s)|x|2 +4v
s+kπ
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.
22
DETLEF MÜLLER
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.
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
Z Z t
m+n+1/2 m+n−1 (m−1)l
χ −l −1
Fl (x, v) = λ
k
2
χ(l) (r)
2 k r+π
iλkt ψ(2−l r)+4v
η(2−l 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
χ1 (r)
χ(l) (r) := l
(2 sin(2−l r))m
is again supported where r ∼ 1. Moreover, if 2−l r ∈ supp η and k ≥ 1,
then |2−l r| ≤ 3π
≤ kπ
, so that
8
2
1
∼ k −1
+ kπ
for every k ≥ 1, l ≥ 0 (the case of the negative sign will actually become
relevant for the Aλ,k,l,−,a only.)
±2−l r
Notice that all derivatives of χ(l) (r) are uniformly bounded in l. We
can therefore re-write
Fl (x, v) = λm+n+1/2 k m+n−1 2(m−1)l
Z Z
iλkt ψ(2−l r)+4v
(4.1)
·
χk,l (r, t)e
a(λkt4|v|) drdt,
where χk,l (r, t) is a smooth function supported where
r ∼ 1, t ∼ 1,
THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
23
such that
|∂rα ∂tβ χk,l (r, t)| ≤ Cα,β for every α, β ∈ N,
(4.2)
uniformly in k and l.
Since
(4.3)
1
+ |x|2 sin(s)−2 ,
(s + kπ)2
2
ψ ′′ (s) =
− 2|x|2 cos(s) sin(s)−3 ,
(s + kπ)3
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 . 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
(4.5)
γk (|x|) =
sk (|x|) − sin sk (|x|) cos sk (|x|)
,
sk (|x|)2
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. Noticing that the critical point of the phase ψl is given by
2l sc,k (|x|), one eventually finds that
Fl (x, v) = λ
m+n+1/2 m+n−1 (m−1)l
k
2
Z
χ(t)a(λkt4|v|)
iλkt ψ(sc,k (|x|))+4v
b(k 2 22l |x|2 , λk −1 2−l t)e
drdt,
24
DETLEF MÜLLER
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|) = λ
(4.6)
m+n+1/2 m+n−1 (m−1)l
k
2
Z
χ(t)a(λkt4|u|)
iλkt γk (|x|)±4|u|
b(k 2 22l |x|2 , λk −1 2−l t)e
dt,
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
(4.7)
.
·(1 + λk|u|)− 2 1 + λk|γk (|x|) ± 4|u||
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.
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→ , ,
s
s2
corresponding to the range of parameters where s − kπ ∼ 2−l .
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 .
THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
25
1/k
1/(k+1)
1/k
Figure 2. curve γ
√
5. Estimation of (1 + L)−(d−1)/4 ei
Let us put
1 1
α(p) := (d − 1) − ,
p 2
L
δ
1 ≤ p < ∞.
√
We need to estimate (1 + L)−α(p)/2 ei L δ on Lp (G) in Theorem 1.1.
By means of a dyadic decomposition, this can easily be reduced to
estimating the operator
∞
√L √
X
−α(p)j
ei L ,
(5.1)
2
χ̃
j
2
j=0
for a suitable cut-off function χ̃ supported in [1/16, 4]. 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.
26
DETLEF MÜLLER
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:
±
Proposition 5.1. Let k ≥ 1. There exist kernel functions Kλ,k,l
∈
l
1
S(G), for λ ≥ 2 k, and Rk,J ∈ L (G) so that
(5.2)
+
Ek,J
=
∞
X
J
X
(K2+j ,k,l + K2−j ,k,l) + Rk,J ,
l=0 j=l+log2 k
and such that the following hold true:
±
(a) The kernel Kλ,k,l
is supported in the set
{(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
(5.3)
1
1
±
|∂xα ∂uβ Kλ,k,l
(x, u)| ≤ CN,α,β 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||
,
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,
(5.4)
1
±
k∂xα ∂uβ Kλ,k,l
k1 . λ|α| (λk)|β| (2l k)−m− 2 .
(b) The sequence {Rk,J }J converges for J → ∞ in L1 (G) towards
a function Rk ∈ L1 (G), and
(5.5)
1
kRk k1 ≤ Ck −m− 2 ,
with a constant C not depending on k.
THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
27
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
k
−Γk (v) := k 2 γk (k −1 v) − .
π
We can then make the interesting observation that, by (5.3), (5.4), the
±
]
non-isotropically re-scaled kernels K
are supported in the set
λ,k,l
{(x, u) ∈ G : |x| ∼ 2−l and
λ
|u| & 1},
k
and satisfy the isotropic estimates
λ |α|+|β| n+1
1
1
α β ]
±
|∂x ∂u Kλ,k,l (x, u)| ≤ CN,α,β
λ 2 k −(m+n+ 2 ) 2(m− 2 )l
k
−N
λ k
λ
− n−1
2
(5.6)
1 + − Γk (|x|) ± 4|u|
(1 + |u|)
k
k π
and
λ |α|+|β|
1
±
]
(5.7)
k∂xα ∂uβ K
k
.
(2l k)−m− 2 .
1
λ,k,l
k
Moreover, one easily verifies that
1
p
−2
−1
2
.
(5.8)
Γk (v) = π arcsin(πv) + π v 1 − (πv) + O
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,± = λ1/2
χ0 (t)χ1 (±2l (2πt|µ|/λ − kπ)) η0 (2πt|µ|/λ − kπ)
λ
µ
ei 4t eit/λ L dt δ.
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
(6.1)
kAλ,k,l,±k = λ1/2 sup{|αλ,k,l,±(ν, q)| : ν ≥ 0, q ∈ N},
28
DETLEF MÜLLER
where
αλ,k,l,±(ν, q)
Z
ν
λ
:=
χ0 (t)χ1 (±2l (t λν − kπ)) η0 (t λν − kπ)ei{ 4t +t λ (m+2q)} 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,
∞ X
X
2−αj A2j ,k,l,±k ≤ C 2−αj0 ,
(6.2)
k
l=0 j≥j0
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
(7.1)
kak2 ≤ r −d/2
and, if r < 1, in addition the moment condition
Z
(7.2)
a(x) dx = 0.
G
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
(7.3)
f=
λj aj ,
j
P
with atoms aj and coefficients λj such that j |λj | < ∞. We define the
P
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
THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
29
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:
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.
b) If HE1 := H 1 (R2m × Rn ) denote the classical Euclidean Hardy
space, and if η ∈ C0∞ (R2m ), then multiplication with η ⊗ 1 maps HE1
continuously into 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
for every f ∈ h1 (G).
1,±
−m
kf ∗ Eg
kf kh1
k k1 ≤ Ck
Proof. We again consider only Ek1,+ = Ek+ . In view of the atomic
decomposition of f and because of left-invariance, 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
30
DETLEF MÜLLER
To simplify the argument a bit, assume that n = 1, so that |u| = ±u.
±
]
Let’s also assume that we consider the part of K
where u > 0, and
λ,k,l
±
]
that K
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 (5.6) we can
then make the (slightly oversimplified) assumption that
k
±
−l
1
]
u
−
Γ
(|x|)
(7.4) supp K
⊂
{(x,
u)
∈
G
:
|x|
∼
2
and
≤ },
λ,k,l
4 k
λ
and that
λ |α|+|β|
1
1
±
]
(7.5)
|∂xα ∂uβ K
(x,
u)|
≤
C
λ k −(m+n+ 2 ) 2(m− 2 )l
N,α,β
λ,k,l
k
Define for ρ > 0 the set
1
Ωρ := {(x, u) ∈ G : |x| . 1 and u − 4 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
XX
±
^
K
) ⊂ Ωr ,
(7.6)
supp a ∗ (
2j ,k,l
j≥j0
l
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,± .
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
(d−1)
1/2
g
ka ∗ E
kak2 2−α(1)j0 . r 1/2 r −(m+1/2) 2− 2 j0 ≤ k −m .
k,r kL1 (Ωr ) . |Ωr |
P P
±
^
g
Moreover, since by Proposition 5.1 Ek,r = l j≥j0 K
2j ,k,l +Rk,r , where
1
kRk,r k1 . k −m− 2 , we obtain from (7.6) that
1
−m−
g
g
2,
ka ∗ E
k,r kL1 (G\Ωr ) ≤ ka ∗ Rk,r kL1 (G) . k
THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
31
so that
−m
g
ka ∗ E
.
k,r k1 . k
(7.7)
g
g
Consider next a∗ F
k,r . Again, by Proposition 5.1, we can write Fk,r =
1
P P
−m−
±
′
^
′
g
2 . And, for j < j0 , we
l
j<j0 K2j ,k,l + Rk,r , where kRk,r k1 . k
have, by (7.4),
X
±
^
K
) ⊂ Ω −j .
supp a ∗ (
k2
2j ,k,l
l
R
Moreover, making use of the cancellation a dx = 0 and (7.5) in es±
^
timating the convolution product a ∗ K
, we gain a factor of order
O( k2r−j ) compared to (5.7) and obtain
ka ∗
X
l
±
^
K
2j ,k,l k1 .
Summing in j, this gives
(7.8)
∞
X
l=0
2j ,k,l
1
1
r
r
(2l k)−m− 2 . 2j k −m− 2
−j
k2
k
−m− 12
g
.
ka ∗ F
k,r k1 . k
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, k 2 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.
8. Proof of Theorem 1.1
Observe that by (6.2) convolution with the kernel
∞ X
∞
X
2,±
Ek :=
A2j ,k,l,±
j=0 l=0
is bounded on L2 (G), and
(8.1)
kf ∗ Ek2,± k2 ≤ Ckf k2 ,
32
DETLEF MÜLLER
where C is independent of k. Consider now the analytic family of operators
∞ X
∞
X
2−αj A^
0 ≤ Re α ≤ (d−1)
Tα : f 7→ f ∗
.
2j ,k,l,± ,
2
j=0 l=0
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.
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 ).
0
By Lemma 7.1 and Proposition 7.3, T d−1
: HE1 → L1 continuously, and
2
since the proof of Proposition 7.3 remains valid for T d−1 +iβ , we have
2
0
kT d−1 +iβ f k1 ≤ Ck
−m
2
kf kHE1
∀β ∈ R.
Similarly, by (8.1), we have
kTiβ0 f k2 ≤ Ckf k2
∀β ∈ R.
We can thus apply the complex interpolation result by C. Fefferman
and Stein in [6] to obtain
2
0
kTα(p)
f kp ≤ Ck −( p −1)m kf kp ,
1 < p ≤ 2.
In view of the afore-mentioned support property of the convolution
kernels involved, this immediately implies
α(p),±
kf ∗ Ek
2
kp ≤ Ck −( p −1)m kf kp ,
1 < p ≤ 2.
If m ≥ 2, these estimates sum in k for p sufficiently close to 1, so that
(8.2)
√
k(1 + L)−α(p)/2 ei
L
f kp ≤ Ckf kp
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 WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
33
The L1 -estimate in Theorem 1.1 follows a lot more easily from Propod−1
sition 5.1, since for α > d−1
we have an additional factor λ−(α− 2 )
2
in the corresponding estimate (5.4), so that the corresponding norms
kK2±j ,k,l k1 simply sum in j, k and l.
34
DETLEF MÜLLER
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
0 −Im
(9.1)
ω(z, w) := hJz, wi,
J :=
,
Im
0
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 :=
− 12 yj , Ỹj :=
+ 12 xj , j = 1, . . . , m, and T := ,
∂xj
∂t
∂yj
∂t
∂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),
h ∈ L2 (Rm ).
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,
R if f ∈ S(Hm ), then it is well-known and
easily seen that ρτ (f ) = R2m f −τ (z)ρτ (z, 0) dz is a trace class operator
on L2 (Rm ), and its trace is given by
Z
−m
f (0, 0, t)e2πiτ t dt = |τ |−m f −τ (0, 0),
(9.2)
tr(ρτ (f )) = |τ |
R
THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
35
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 is an orthogonal
µ ∈ O(2m, R) such that
matrix Rµ = R |µ|
(9.3)
Jµ = |µ|Rµ J tRµ .
Notice also that tRµ = Rµ−1 , and that condition (9.3) is in fact equivalent to G being of Heisenberg type.
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
(9.4)
f ∗g = g∗f
for every f, g ∈ L1r (G).
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.
36
DETLEF MÜLLER
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
Ron G. The Fourier minversion formula in R then leads to f (0, 0) =
tr πµ (f ) |µ| dµ. When applied to δg−1 ∗ f, we arrive at the
µ∈Rn \{0}
Fourier inversion formula
Z
(9.5)
f (g) =
tr (πµ (g)∗πµ (f )) |µ|m dµ, g ∈ G.
µ∈Rn \{0}
Applying this to f ∗ ∗ f at g = 0, where f ∗ (g) := f (g −1), we obtain the
Plancherel formula
Z
2
(9.6)
kf k2 =
kπµ (f )k2HS |µ|m dµ,
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.
THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE
37
By our construction of the representation πµ , we thus obtain for the
derived representation dπµ of g that
(9.7)
dπµ (Xµ,j ) = dρ|µ| (X̃j ), dπµ (Yµ,j ) = dρ|µ| (Ỹj ),
j = 1, . . . , m.
Let us define the sub-Laplacians Lµ on G and L̃ on Hm by
Lµ := −
m
X
2
(Xµ,j
+
2
Yµ,j
),
j=1
m
X
L̃ := −
(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
(9.8)
dπµ (L) = dρ|µ| (L̃).
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|
h|µ|
α := hα1 ⊗ · · · ⊗ hαm ,
α ∈ Nm ,
where hµk (x) := (2π|µ|)1/4hk ((2π|µ|)1/2 x). Here,
dk −x2
e
dxk
denotes the L2 -normalized Hermite function of order k on R. Consequently,
hk (x) = ck (−1)k ex
(9.9)
2 /2
|µ|
dπµ (L)h|µ|
α = 2π|µ|(m + 2|α|)hα ,
It is also easy to see that
(9.10)
dπµ (Uj ) = 2πiµj I,
α ∈ Nm .
j = 1, . . . , n.
Now, the operators L, −iU1 , . . . , −iUn form a commuting set of selfadjoint 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 ).
38
DETLEF MÜLLER
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 ,
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
(9.11)
|µ|
πµ (ϕ(L, U)δ)h|µ|
α = ϕ(2π|µ|(m + 2|α|), 2πµ)hα
(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
(9.12)
kϕ(L, U)k = sup{|ϕ(|µ|(m + 2q), µ)| : µ ∈ Rn , q ∈ N}.
Finally, observe that
(9.13)
Kφµ = ϕ(Lµ , 2πµ)δ;
this follows for instance by applying the unitary representation induced
from the character e2πiµ·u on the center of G to Kϕ .
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D. Müller, Mathematisches Seminar, C.A.-Universität Kiel, LudewigMeyn-Str.4, D-24098 Kiel, Germany
E-mail address: [email protected]
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