SHARP Lp-ESTIMATES FOR THE WAVE EQUATION … Lp-ESTIMATES FOR THE WAVE EQUATION ON HEISENBERG TYPE GROUPS LECTURE NOTES ORLEANS, APRIL 2008 HTTP://ANALYSIS.MATH.UNI-KIEL.DE/MUELLER

SHARP Lp-ESTIMATES FOR THE WAVE EQUATIONON HEISENBERG TYPE GROUPS

LECTURE NOTES ORLEANS, APRIL 2008

HTTP://ANALYSIS.MATH.UNI-KIEL.DE/MUELLER/

DETLEF MULLER

Abstract. In these lectures, which are based on recent joint workwith A. Seeger [23], I shall present sharp analogues of classical es-timates by Peral and Miyachi for solutions of the standard waveequation on Euclidean space in the context of the wave equationassociated to the sub-Laplacian on a Heisenberg type group. Somerelated questions, such as spectral multipliers for the sub-Laplacianor Strichartz-estimates, will be briefly addressed. Our results im-prove on earlier joint work of mine with E.M. Stein. The newapproach that we use has the additional advantage of bringing outmore clearly the connections of the problem with the underlyingsub-Riemannian geometry.

Date: April 8, 2008.1

2 DETLEF MULLER

Contents

1. Introduction 2

1.1. Connections with spectral multipliers and further factsabout the wave equation 7

2. The sub-Riemannian geometry of a Heisenberg type group 10

3. The Schrodinger and the wave propagators on a Heisenbergtype 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)/4ei√Lδ 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 Heisenbergtype 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 innerproduct 〈 , 〉 such that g1 and g2 or orthogonal subspaces and thefollowing 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 ) = 〈µ, [V,W ]〉 = 〈Jµ(V ),W 〉(here, we also used the natural identification of g∗

2 with g2 via the innerproduct). Then

(1.1) J2µ = −|µ|2I

for every µ ∈ g∗2 \ 0.

Note that this implies in particular that [g1, g1] = g2.

As the corresponding connected, simply connected Heisenberg typeLie group G we shall then choose the linear manifold g, endowed withthe 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 sim-ple 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

Xf(g) :=d

dtf(g exp(tX))|t=0,

where exp : g → G denotes the exponential mapping, which agreeswith the identity mapping in our case.

Let us next fix an orthonormal basis X1, . . . , X2m of g1, and let usdefine the non-elliptic sub-Laplacian

L := −2m∑

j=1

X2j

4 DETLEF MULLER

on G. Since the vector fields Xj together with their commutators spanthe tangent space to G at every point, L is still hypoelliptic and pro-vides an example of a non-elliptic “sum of squares operator ” in thesense of Hormander ([13]). Moreover, L takes over in many respects ofanalysis on G the role which the Laplacian plays on Euclidean space.

To simplify the notation, we shall also fix an orthonormal basisU1, . . . , Un of g2, and shall in the sequel identify g = g1 + g2 and Gwith R2m × Rn by means of the basis X1, . . . , X2m, U1, . . . , Un of g.Then our inner product on g will agree with the canonical Euclideanproduct z · w =

∑2m+nj=1 zjwj on R2m+n, and Jµ will be identified with

a skew-symmetric 2m × 2m matrix. Moreover, the Lebesgue measuredx 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 theautomorphic dilations

δr(x, u) := (rx, r2u), r > 0,

on G, and the Koranyi norm

‖(x, u)‖K := (|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 : ‖(y, v)−1(x, u)‖K < r, (x, u) ∈ G, r > 0,

then the volume |Qr(x, u)| is given by

|Qr(x, u)| = |Q1(0, 0)| rD,where

D := 2m+ 2n

is the homogeneous dimension of G. We shall also have to work withthe 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

J :=

(0 Im

−Im 0

)


This is the case of the Heisenberg group Hm, which is R2m×R, endowedwith the product

(z, u) · (z′, u′) = (z + z′, u+ u′ + 12〈Jz, z′〉).

A basis of the Lie algebra hm of Hm is then given by X1, . . . , X2m, U ,where U := ∂

∂uis central.

Consider the following Cauchy problem for the wave equation onG× R associated to L:

(1.2)∂2u

∂t2− (−L)u = 0, u

∣∣∣t=0

= f,∂u

∂t

∣∣∣t=0

= g,

where t ∈ R denotes time.

The solution to this problem is formally given by

u(x, t) =

(sin(t

√L)√

Lg

)(x) + (cos(t

√L)f)(x),

(x, t) ∈ G× R.

In fact, the above expression for u makes perfect sense at least forf, g ∈ L2(G), if one defines the functions of L involved by the spectraltheorem (notice that L is essentially selfadjoint on C∞

0 (G) [33]) asfollows: If L =

∫∞0s dEs denotes the spectral resolution of L on L2(G),

then every Borel function m on R+ gives rise to a (possibly unbounded)operator

m(L) :=

∫ ∞

0

m(s) dEs

on L2(G). Let us also note that if m is a bounded spectral multi-plier, then, by left–invariance and the Schwartz kernel theorem, it iseasy to see that m(L) is a convolution operator m(L)f = f ∗ Km =∫G

f(y)Km(y−1·) dy, where a priori the convolution kernel Km is a tem-

pered distribution. We also write Km = m(L)δ.

If one decides to measure smoothness properties of the solutionu(x, t) for fixed time t in terms of Sobolev norms of the form

‖f‖Lpα

:= ‖(1 + L)α/2f‖Lp,

one is naturally led to study the mapping properties of operators such

as eit√

L

(1+L)α/2 or sin(t√L)√

L(1+L)α/2 as operators on Lp(G) into itself.

6 DETLEF MULLER

For the classical wave equation on Euclidian space, sharp estimatesfor the corresponding operators have been established by Peral [35] andMiyachi [21].

In particular, if ∆ denotes the Laplacian on Rd, then (1−∆)−α/2eit√−∆

is bounded on Lp(Rd), if α ≥ (d−1)∣∣∣1p − 1

2

∣∣∣, for 1 < p <∞. Moreover,

(1 − ∆)−(d−1)/2

2 eit√−∆ is bounded from the real Hardy space H1(Rd)

into L1(Rd).

Local analogues of these results hold true for solutions to strictlyhyperbolic 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 inte-gral operators, and solutions to strictly hyperbolic equations can beexpressed in terms of such operators.

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 degenerate-elliptic, and classical Fourier integral operatortechnics do not seem to be available any more. However, as we shallsee, it is still possible to represent solutions to (1.2) as infinite sums ofFourier integral operators with degenerate phases and amplitudes.

Interesting information about solutions to (1.2) have been obtainedby Nachman [32]. Among other things, Nachman showed that thewave operator on Hm admits a fundamental solution supported in a“forward light cone”, whose singularities lie along the cone Γ formedby the characteristics through the origin. Moreover, he computed theasymptotic behaviour of the fundamental solution as one approachesa generic singular point on Γ. His method does, however, not provideuniform estimates on these singularities, so that it cannot be used toprove 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 correspond-ing cone in the Euclidian case. This is related to the underlying, morecomplex sub-Riemannian geometry, which we shall briefly discuss.

Nevertheless, the following theorem is true, which improves on earlierwork in [31] on the Heisenberg group, by also including the endpointregularity, and which is a direct analogue of the results by Peral andMiyachi in the Euclidean setting.


Theorem 1.1. ei√

L

(1+L)α/2 extends to a bounded operator on Lp(G), for

1 < p <∞, when α ≥ (d− 1)|1p− 1

2|, and for p = 1, if α > d−1

2.

The proof will be based on some “Hardy space type result” for theendpoint p = 1, which will become evident later. The main differ-ence compared to the Euclidean setting seems, however, that there isnot just one single Hardy space, but a whole sequence of local Hardyspaces associated with the problem, whose definitions are based onsome interesting interplay between two different homogeneities, namelythose coming from the isotropic dilations of the underlying Euclideanstructure of G and the one coming from the non-isotropic automorphicdilations on G (compare Remark 7.3).

Remark 1.2. The same result holds for sin√L√

L(1+L)α−1

2, or with the fac-

tors (1 +L)−α/2 (respectively (1 +L)−(α−1)/2) replaced by (1 +√L)−α,

(respectively (1 +√L)−(α−1)).

Notice that the restriction to time t = 1 in our theorem is inessential,since L is homogeneous of degree 2 with respect to the automorphicdilations (z, t) 7→ (rz, r2t), r > 0.

1.1. Connections with spectral multipliers and further factsabout the wave equation. If m is a bounded spectral multiplier,then clearly the operator m(L) is bounded on L2(G). An importantquestion is then under which additional conditions on the spectral mul-tiplier m the operatorm(L) extends from L2∩Lp(M) to an Lp-boundedoperator, for a given p 6= 2. If so, m is called an Lp-multiplier for L.

Fix a non-trivial cut-off function χ ∈ C∞0 (R) supported in the inter-

val [1, 2], and define for α > 0

‖m‖sloc,α := supr>0

‖χm(r·)‖Hα,

where Hα = Hα(R) denotes the classical Sobolev-space of order α.Thus, ‖m‖sloc,α < ∞, if m is locally in Hα, uniformly on every scale.Notice also that, up to equivalence, ‖·‖sloc,α is independent of the choiceof the cut-off function χ. Then the following analogue of the classicalMihlin-Hormander multiplier theorem holds true (see M. Christ [3],and also Mauceri-Meda [19]):

8 DETLEF MULLER

Theorem 1.3. If G is a stratified Lie group of homogeneous dimensionD, and if ‖m‖sloc,α < ∞ for some α > D/2, then m(L) is bounded onLp(G) for 1 < p <∞, and of weak type (1,1).

Observe that, in comparizon to the classical case G = Rd, the ho-mogeneous dimension D takes over the role of the Euclidian dimensiond. Because of this fact, which is an outgrowth of the homogeneity ofL with respect to the automorphic dilations, the condition α > D/2in Theorem 1.3 appeared natural and was expected to be sharp fora while. The following result, which was found in joint work withE.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 conditionα > 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 Prin-ciple, respectively variants of it:

Proposition 1.5. Assume that ei√

L

(1+L)α/2 extends to a bounded operator

on L1(G), and let β > α + 1/2. Then there is a constant C > 0, suchthat for any multiplier ϕ ∈ Hβ(R) supported in [1, 2], the correspondingconvolution kernel Kϕ = ϕ(L)δ is in L1(G), and

‖Kϕ‖L1(G) ≤ C‖ϕ‖Hβ(R).

Proof. Observe first that (1 + L)−εδ ∈ L1(G) for any ε > 0. Thisfollows from the formula

(1 + L)−εδ =1

Γ(ε)

∫ ∞

0

tε−1e−t(1+L)δ dt =1

Γ(ε)

∫ ∞

0

tε−1e−tpt dt

and the fact that the heat kernel pt := e−tLδ is a probability measureon G. Write

ϕ(s) = ψ(√s)(1 + s)−γ , with γ > α/2,

and put g := (1 + L)−γδ ∈ L1(G). Then ||ψ||Hβ ≃ ||ϕ||Hβ , and

Kϕ = ψ(√L)((1 + L)−γδ) = ψ(

√L)g = c

∫ ∞

−∞ψ(t)eit

√Lg dt.


And, our assumption in combination with some easy multiplier esti-mates (cf. [31]) and the homogeneity of L yields

‖eit√Lg‖1 . ‖(1 + t2L)α/2g‖1 . ‖(1 + t2)α/2(1 + L)α/2g‖1,

hence

‖Kϕ‖1 .

∫ ∞

−∞|ψ(t)|(1 + |t|)α‖(1 + L)α/2g‖1dt.

But, (1 + L)α/2g = (1 + L)α/2−γδ ∈ L1, hence, by Holder’s inequalityand Plancherel’s theorem,

‖Kϕ||1 .(∫ ∞

−∞|ψ(t)(1+|t|)β|2dt

)1/2

·(∫ ∞

−∞(1+|t|)2(α−β)dt

)1/2

. ‖ψ‖Hβ .

Q.E.D.

Indeed, we may choose β > d−12

+ 12

= d2

in this subordination prin-ciple. This is just the required regularity of the multiplier in Theorem1.4, and one can in fact deduce this theorem from Theorem 1.1 bymeans of a refinement of the above subordination principle and stan-dard arguments from Calderon-Zygmund theory.

In contrast to the close analogy between the wave equation on G andon Euclidean space expressed in Theorem 1.1, there are other aspectswhere the solutions behave quite differently compared to the classicalsetting.

One instance of this is the “almost” failure of dispersive estimatesfor solutions to (1.2) on Heisenberg groups. Indeed, in [1], Bahouri,Gerard and Xu prove that solutions to (1.2) on Hm with sufficientlysmooth initial data in L1 satisfy

‖u(t)‖∞ ≤ C|t|−1/2,

and that the exponent −1/2 is optimal, whereas for the Laplacian onRd a dispersive estimate holds with exponent −(d− 1)/2. Correspond-ingly, the Strichartz estimates proven in [1] are much weaker than theEuclidean analogues would suggest.

A related result is the “almost” failure of (spectral) restriction the-orems for Hm proven in [25].

10 DETLEF MULLER

2. The sub-Riemannian geometry of a Heisenberg type

group

Let M be a smooth manifold, and let D ⊂ TM be a smooth distri-bution, i.e., a smooth subbundel of TM. A sub-Riemannian structure(D, g) onM is given by a smooth distribution D and a Riemannian met-ric g on D. For p ∈ M and X, Y ∈ D(p) we write 〈X, Y 〉g := gp(X, Y )

and ‖X‖ = 〈X,X〉1/2g . M, endowed with such a structure, is called asub-Riemannian manifold.

An absolutely continuous curve γ : [0, T ] → M in M is then calledhorizontal, if

γ(t) =d

dtγ(t) ∈ D(γ(t))

for almost every t ∈ [0, T ]. As in Riemannian geometry, the length of ahorizontal curve γ is then defined by

L(γ) :=

∫ T

0

‖γ(t)‖g dt.

The Carnot–Caratheodory distance (or sub-Riemannian distance) on(M,D, g) is then defined by

dCC(p, q) := infL(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 onM 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 γ(t) =∑k

j=1 aj(t)Xj(γ(t)) for a.e. t ∈ [0, T ], and L(γ) =T∫0

(∑j aj(t)

2)1/2

dt.

Moreover, if these vector fields are bracket generating, i.e., if theysatisfy Hormander’s condition, then by the Chow-Rashevskii theoremdCC(p, q) < ∞ for every p, q ∈ M, and dCC is a metric on M thatinduces the topology of M.

Let us now consider the sub-Riemannian structure on a group G ofHeisenberg type, where D = span X1, . . . ,X2m. Here, the Carnot–Caratheodory distance is left-invariant, i.e.,

dCC(x, y) = ‖x−1y‖CC,


where ‖x‖CC := dCC(0, x) is again a homogeneous norm on G. Observethat there exists a constant C ≥ 1 such that

(2.1) C−1‖x‖CC ≤ ‖x‖K ≤ C‖x‖CC ,since any two homogeneous norms are equivalent [7]. In analogy withwave equations on Riemannian manifolds, one expects that singularitiesof solutions propagate along “geodesics” of the underlying geometry.However, it turns out that the notion of geodesic of a sub-Riemannianmanifold is not as clear as one might expect. The perhaps best ap-proach to the problem of determining length minimizing curves in thiscontext 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 geomet-rical optics ideas:

If x ∈ M, we endow the dual space D(x)∗ with the dual metric g∗xto gx on D(x) ⊂ T ∗M. This allows to define the length of a cotangentvector (x, ξ) ∈ T ∗M as ‖ξ‖g := ‖ξ|D(x)‖g∗x . The function H : T ∗M → R

given by

H(x, ξ) := 12‖ξ‖2

g

is called the Hamiltonian of the sub-Riemannian manifold M. Observethat this is, apart from the factor 1

2, just the principal symbol of the

associated sub-Laplacian. Since T ∗M is a symplectic manifold in acanonical way with respect to the symplectic form ωM :=

∑j dξj ∧ dxj

(in canonical coordinates), we may associate toH the Hamiltonian vec-tor field H, i.e., ωM(X,H) = dH(X) for every X ∈ TM. The integralcurves (x(t), ξ(t)) of H are called bicharacteristics. They satisfy theHamilton-Jacobi equations

x(t) =∂H

∂ξ(x(t), ξ(t)), ξ = −∂H

∂x(x(t), ξ(t)),

and H is constant along every bicharacteristic. A normal geodesic ofthe sub-Riemannian manifold M will then be the space projection x(t)of a bicharacteristic (x(t), ξ(t)) along which the Hamiltonian H doesnot vanish. This notion is in accordance with the idea that the wavefront set of a solution to our wave equation should evolve along thebicharacteristics of the wave equation.

It is known that every length minimizing curve in a sub-Riemannianmanifold M is either such a normal geodesic, or an “abnormal ge-odesic” [18]. I won’t give a definition of the latter ones here, since

12 DETLEF MULLER

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 geodesicsdo exist, even in nilpotent Lie groups (see [11] for an example of a groupof step 4, and [34] for an example of step 2). This fact has been ignoredin some cases in the literature, which had led to some confusion.

On a Heisenberg type group G, one easily finds that the Hamiltonianis given by

H = 12|ξ − 1

2Jµx|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

x =∂H

∂ξ, u =

∂H

∂µ

ξ = −∂H∂x

, µ = −∂H∂u

.

By left-invariance, it suffices to condsider bicharacteristics starting atthe origin, i.e., x(0) = 0, u(0) = 0. We also put η := ξ − 1

2Jµx. Then

H = 12〈η, η〉, so that |η| is constant along the bicharacteristic. Also

µ = const., and

x = η, ξ = −12Jµη

η = ξ − 12Jµx = −Jµη,

hence

η(t) = e−tJµα,

if we put ξ(0) = η(0) := α. This implies x(t) = 1−e−tJµ

Jµα, and since

J2µ = −|µ|2I, we have

(2.2) x(t) = 2sin( t

2|µ|)

|µ| e−t2Jµα.

For simplicity, let us now assume that n = 1, i.e., that we are dealingwith a Heisenberg group. Then Jµ = µJ, with µ ∈ R, and we have

u = 〈η,−12Jx〉 = −sin( t

2|µ|)

|µ| 〈e−tJµα, Je−t2Jµα〉

= −sin( t2|µ|)

|µ| 〈α, Je t2Jµα〉.


Looking at the Taylor series of et2Jµ and using the fact that J is skew

and satisfies J2 = −I, this yields

u =µ

|µ|2 |α|2 sin2(

t

2|µ|),

if µ 6= 0. Consequently,

(2.3) u(t) =

( t

2− sin( t

2|µ|) cos( t

2|µ|)

|µ| ) |α|2 µ|µ|2 , if µ 6= 0

0, if µ = 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 geodesiciff α 6= 0, and it has constant speed ‖γ‖g = |x(t)| = |η| = |α|, sinceγ(t) =

∑j xj(t)Xj(γ(t)).

Denote by Σ1 the set of endpoints of all geodesics of length 1 (seefigure 1). These endpoints are given by (2.2),(2.3), with t = 1, whereα can be chosen arbitrarily within the unit sphere in R2m and µ in Rn.Therefore,(2.4)

Σ1 = (x, u) ∈ G : |x| =∣∣∣sin ss

∣∣∣, 4|u| =∣∣∣s− sin s cos s

s2

∣∣∣ for some s ∈ R.

We then expect that Σ1 is exactly the singular support of any of thewave propagators

sin(t√L)√

Lδ, cos(t

√L)δ or eit

√Lδ,

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 MULLER

The “outer hull” of the set Σ1 is just the unit sphere with respect tothe Carnot-Caratheodory distance.

Let us finally mention the well-known fact that solutions to our waveequation have finite propagation speed (see [20], or [27] for an elemen-tary proof for Lie groups), i.e., if Pt denotes any of the wave propagatorssin(t

√L)√

Lδ, cos(t

√L)δ, then, for t ≥ 0,

(2.5) supp Pt ⊂ (x, u) ∈ G : ‖(x, u)‖CC ≤ t

3. The Schrodinger and the wave propagators on a

Heisenberg type group

One principal problem in proving estimates for solutions to waveequations consists in finding sufficiently explicit expressions for theassociated wave propagators. For strictly hyperbolic equations, suchexpressions can be given by means of Fourier integral operators, at leastlocally in space and time. These technics, however, do not apply towave equations associated to non-elliptic Laplacians, and no sufficientlyexplicit expressions for the wave propagators are known to date in thesemore general situations, except for the case of Heisenberg type groupsand a few related examples. What allows us to give such expressionson Heisenberg type groups is the existence of explicit formulas for theSchrodinger propagators eitLδ on these groups.

There are several possible approaches to these formulas. One wouldconsist in appealing to the Gaveau-Hulanicki formula [9],[16] for theheat kernels e−tLδ, which can be derived by means of representationtheory and Mehler’s formula, and then trying to apply physicists ap-proach and replacing time t by complex time −it.

We shall here sketch another approach, which is more far-reachingand has turned out to be useful also in the study of questions of solv-ability of general second order left-invariant differential operators on2-step nilpotent Lie groups (see [28], also for further references to thistopic).

Let G be a group of Heisenberg type as before. Given f ∈ L1(G) andµ ∈ g∗

2 = R2m, we define the partial Fourier transform fµ of f along


the center by

fµ(x) :=

∫

Rn

f(x, u)e−2πiµ·u du (x ∈ R2m).

Moreover, if we define the µ–twisted convolution of two suitable func-tions ϕ and ψ on g1 = R2m by

(ϕ ∗µ ψ)(x) :=

∫

R2m

ϕ(x− y)ψ(y)e−iπωµ(x,y) dy,

then(f ∗ g)µ = fµ ∗µ gµ,

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 Xµ

j such that (Xjf)µ = Xµj f

µ,and the sub-Laplacian is transformed into the µ-twisted Laplacian Lµ,i.e.,

(Lf)µ = Lµfµ = −2m∑

j=1

(Xµj )2

(say for f ∈ S(G)). An easy computation shows that explicitly

Xµj =

∂

∂xj+ iπωµ(x,Xj).

What turns out to be important for us is that the one-parameterSchrodinger group eitL

µ, t ∈ R, generated by Lµ can be computed

explicitly:

Proposition 3.1. For f ∈ S(G),

eitLµ

f = f ∗µ γµt , t ∈ R,

where γµt ∈ S ′(R2m) is a tempered distribution given by the imaginaryGaussian

γµt (x) = 2−m( |µ|

(sin(2πt|µ|))m

e−iπ2|µ| cot(2πt|µ|)|x|2 ,

whenever sin(2πt|µ|) 6= 0. Moreover, by well-known formulas for theinverse Fourier transform of a generalized Gaussian [14], its Fouriertransform is

γµt (ξ) =1

(cos(2πt|µ|))m ei2π|µ| tan(2πt|µ|)|ξ|2

whenever cos(2πt|µ|) 6= 0.

16 DETLEF MULLER

3.1. Twisted convolution and the metaplectic group. It is notdifficult to reduce the proof of Proposition 3.1 to the case of the Heisen-berg group G = Hm, which we shall consider now. For any real, sym-metric matrix A := (ajk)j,k=1,...,2m let us put

(3.1) S := −AJ,

where J :=

(0 Im

−Im 0

)is as in the definition of the product in Hm.

It is easy to see that the mapping A 7→ S is a vector space isomorphismof the space of all real, symmetric 2m×2mmatrices onto the Lie algebra

sp(m,R) := S : tSJ + JS = 0.Let us define

∆S :=

2m∑

j,k=1

ajkXjXk, S ∈ sp(m,R),

where A is related to S by (3.1). Then one verifies easily that for anyS1, S2 ∈ sp(m,R), we have

(3.2) [∆S1 ,∆S2 ] = −2U∆[S1,S2],

where U denotes again the central basis element of hm. Applying apartial Fourier transform along the (here one-dimensional) center, thisleads to the following commutation relations among the µ-twisted con-volution operators ∆µ

S :

[∆µS1,∆µ

S2] = −4πiµ∆µ

[S1,S2].

Since ∆µS is formally self-adjoint, this means that the mapping

(3.3) S 7→ i4πµ

∆µS

is a representation of sp(m,R) by (formally) skew-adjoint operators onL2(R2m).

Let us consider the case µ = 1 (it is not difficult to reduce considera-tions 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) canbe exponentiated to a unitary representation of the metaplectic groupMp(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 op-erators of the form f 7→ f × γ, where the γ’s are suitable measureswhich, generically, are multiples of purely imaginary Gaussians

eA(z) := e−iπtz·A·z,

with real, symmetric 2m× 2m matrices A. In particular, one has

eit

4π∆1

Sf = f × γt,S, t ∈ R.(3.4)

The distributions γt,S have been determined explicitly in [22]. Toindicate how this can be accomplished, let us argue on a completelyformal basis:

If eA, eB are two Gaussians as before such that det(A+B) 6= 0, onecomputes that

eA × eB = [det(A+B)]−1/2eA−(A−J/2)(A+B)−1(A+J/2),

where a suitable determination of the root has to be chosen. ChoosingA = 1

2JS1, B = 1

2JS2, with S1, S2 ∈ sp(m,R), and assuming that S1

and S2 commute, one finds that

e 12JS1

× e 12JS2

= 2n(det(S1 + S2))−1/2 e 1

2J [S1S2+I)(S1+S2)−1].

This reminds of the addition law for the hyperbolic cotangent, namely

coth(x+ y) =cothx coth y + 1

coth x+ coth y.

We are thus led to define, for non-singular S,

A(t) :=1

2J coth(tS/2),(3.5)

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(t1S/2)]1/2 [det sinh(t2S/2)]1/2.

18 DETLEF MULLER

Together this shows that

γt,S := 2−n[det sinh(tS/2)]−1/2eA(t)(3.6)

forms a (local) 1-parameter group under twisted convolution, and it isnot hard to check that its infinitesimal generator is i

4π∆1S.

Finally, if we choose S := J, so that A = −I and ∆S = L, weobtain Proposition 3.1, at least for |µ| = 1; the general case follows bya scaling argument.

Remark 3.2. It is important to note that Howe’s construction of themetaplectic group by means of twisted convolution operators can be ex-tended 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 com-plex, symmetric matrix A = SJ has positive semi-definite real part,then accordingly explicit formulas for the one-parameter semigroupset∆

µS , t ≥ 0, have been derived in [28].

3.2. A subordination formula. Fix a bump function χ0 ∈ C∞0 (R)

supported in the interval [1/2, 2]. If a ∈ Sν is a symbol of order ν ∈ R

on T ∗R = R2, i.e., if

|∂αs ∂βσa(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 subsetI ⊂ R if supp a(·, σ) ⊂ I for every σ ∈ R.

Proposition 3.3. There exist a symbol a0 ∈ S0 supported over theinterval [1/16, 4] and a Schwartz function ρ ∈ S(R2) such that, forx ≥ 0,

χ0

(√xλ

)eit

√x = ρ

(txλ, λt)

+√λt

∫a0(s, λt) e

iλt4s ei

tsλx ds,(3.7)

for every λ ≥ 1 and t > 0 such that λt ≥ 1.

Proof. This can easily be obtained by the method of stationary phase.

The explicit formulas in Proposition 3.1 for eitLµ

in combinationwith Proposition 3.3 allow us to describe spectrally localized parts of

the wave propagators e±it√Lδ. 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 auto-morphic dilations δr, we may reduce ourselves to considering the casewhere t = 1.

Let us recall here that it is well-known that if ψ ∈ S(R), then theconvolution kernel ψ(L)δ will also be a Schwartz function (onG), whoseL1(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 ∈ C∞0 (R) be a bump function supported in

the interval [1/2, 2], and assume that λ ≥ 1. Then there exist Schwartzfunctions ψλ ∈ S(G) and Kλ ∈ S(G) such that

χ0

(√Lλ

)ei

√Lδ = ψλ +Kλ,(3.8)

and so that the following hold true:

(i) For every N ∈ N there is constant CN ≥ 0 so that

||ψλ||1 ≤ CNλ−N

for every λ ≥ 1.(ii) The partial Fourier transform Kµ

λ ∈ S(R2m), (µ 6= 0), of Kλ isgiven by an oscillatory integral of the form

〈Kµλ , ϕ〉 =

√λ

∫a0(t, λ) ei

λ4t 〈γµt/λ, ϕ〉 dt, ϕ ∈ S(R2m),

where a0 ∈ S0 is a symbol of order 0 supported over the interval[1/16, 4] and where γµt is given by Proposition 3.1.

Proof. Apart from (i), all the claims follow easily from the sub-ordination formula (3.7) in Proposition 3.3, the spectral resolutionL =

∫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 function ρ ∈ S(R2). In particular, every Schwartz norm of

x 7→ ρ(xλ, λ)

decays like = O(λ−N) for every N ∈ N. By the previ-

ous 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 MULLER

In view of this proposition, it will in the sequel suffice to estimatethe convolution operators defined by the kernels Kλ. Moreover, forsimplicity, we may and shall assume that

(3.9) 〈Kµλ , ϕ〉 =

√λ

∫χ0(t) e

i λ4t 〈γµt/λ, ϕ〉 dt, ϕ ∈ S(R2m),

where χ0 ∈ C∞0 (R) is supported in the interval [1/16, 4]. Indeed, this

will be justified since our estimates will only depend on some Ck-normof χ0.

We next choose η0 ∈ C∞0 (R) such that supp η0 ⊂ [−3π

8, 3π

8] and∑

k∈Zη0(s+ k π

2) = 1 for every s ∈ R, and put

〈Aµλ,k, ϕ〉 := λ1/2

∫χ0(t) η0(2πt|µ|/λ− kπ) ei

λ4t 〈γµt/λ, ϕ〉 dt,

〈Bµλ,k, ϕ〉 := λ1/2

∫χ0(t) η0(2πt|µ|/λ− kπ − π

2) ei

λ4t 〈γµt/λ, ϕ〉 dt.

By Aλ,k and Bλ,k we shall denote the tempered distributions definedon G whose partial Fourier transforms along the center are given byAµλ,k and Bµ

λ,k, respectively. Then

(3.10) Kλ =

∞∑

k=0

(Aλ,k +Bλ,k)

in the sense of distributions.

For the sake of this exposition, we shall restrict ourselves to consid-ering the Aλ,k. Since

〈γµt/λ, ϕ〉 = 〈γµt/λ, ϕ〉,

where γµt/λ is given by Proposition 3.1, it is clear that Aµλ,k is a well-

defined tempered distribution. However, since the integral kernel γµt/λhas a singularity in t where 2πt|µ|/λ = kπ if k 6= 0, we shall here per-form a priori a dyadic decomposition with respect to t. More precisely,if k ≥ 1, let us fix another bump function χ1 ∈ C∞

0 (R) supported inthe interval [1/2, 2] so that

∑∞l=0 χ1(2

ls) = 1 for every s ∈]0, 3π8

], anddecompose the distribution Aµλ,k as

(3.11) 〈Aµλ,k, ϕ〉 =∞∑

l=0

(〈Aµλ,k,l,+, ϕ〉 + 〈Aµλ,k,l,−, ϕ〉),


where Aλ,k,l,± is defined to be the smooth function

Aµλ,k,l,±(x) := λ1/2∫χ0(t)χ1

(± 2l(2πt|µ|/λ− kπ)

)η0(2πt|µ|/λ− kπ)

2−m(

|µ|(sin(2πt|µ|/λ)

)mei

λ4t 2−m e−i

π2|µ| cot(2πt|µ|/λ)|x|2 dt

By Fourier inversion in the central variable u, we have

Aλ,k,l,±(x, u) =

∫

Rn\0Aµλ,k,l,±(x)e2πiu·µ dµ,

In the integration w.r. to µ, we introduce polar coordinates µ =rω, r > 0, ω ∈ Sn−1. Then dµ = rn−1dσn−1(ω), where dσn−1 denotesthe surface measure on the sphere Sn−1. It is well-known [14] by themethod of stationary phase that∫

Sn−1

eiu·ωdσn−1(ω) = ei|u|a+(|u|) + e−2πi|u|a−(|u|), u ∈ Rn,

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, ap-plying 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

Aλ,k,l,±,a(x, v) = λm+n+1/2∫ ∫

χ(t)χ1

(± 2l(s− kπ)

)η(s− kπ)sm+n−1

sin(s)−m eiλts(1s−cot(s)|x|2+4v) a(λst4|v|) dtds.

Since k ≥ 1, this can be re-written in the form

Aλ,k,l,±,a(x, v) = λm+n+1/2km+n−1∫ ∫

χ(

ts/k+π

)χ1(±2ls)sinm(s)

η(s)

eiλkt

(1

s+kπ−cot(s)|x|2+4v

)a(λkt4|v|) dsdt,(3.13)

where the functions χ and η have similar properties as χ1 and η0.

A similar expressions can be given for k = 0.

22 DETLEF MULLER

4. Estimation of Aλ,k,l,± if k ≥ 1

We next need to establish estimates for the kernels Aλ,k,l,±, into whichthe kernels Aλ,k decompose, and their derivatives. In some regions, theestimates that can be obtained for these kernels will be useful only ifλ & k. However, this problem can be fixed in an easy way, and weshall not go into details here but prefer to give a somewhat simplifiedaccount of the actual estimates that we can obtain, for the sake ofclarity of exposition.

We shall only consider Aλ,k,l,+,a, since Aλ,k,l,−,a, can be estimated verymuch 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) = λm+n+1/2km+n−12(m−1)l

∫ ∫χ( t

2−lk−1r + π

)χ(l)(r)

η(2−lr)eiλkt

(ψ(2−lr)+4v

)a(λkt4|v|) drdt,

where the function ψ(s) is given by

ψ(s) :=1

s+ kπ− |x|2 cot(s),

and where

χ(l)(r) :=χ1(r)

(2l sin(2−lr))m

is again supported where r ∼ 1. Moreover, if 2−lr ∈ supp η and k ≥ 1,then |2−lr| ≤ 3π

8≤ kπ

2, so that

1

±2−lr + kπ∼ k−1

for every k ≥ 1, l ≥ 0 (the case of the negative sign will actually becomerelevant for the Aλ,k,l,−,a only.)

Notice that all derivatives of χ(l)(r) are uniformly bounded in l. Wecan therefore re-write

Fl(x, v) = λm+n+1/2km+n−12(m−1)l

·∫ ∫

χk,l(r, t)eiλkt

(ψ(2−lr)+4v

)a(λkt4|v|) drdt,(4.1)

where χk,l(r, t) is a smooth function supported where

r ∼ 1, t ∼ 1,


such that

(4.2) |∂αr ∂βt χk,l(r, t)| ≤ Cα,β for every α, β ∈ N,

uniformly in k and l.

Since

ψ′(s) = − 1

(s+ kπ)2+ |x|2 sin(s)−2,(4.3)

ψ′′(s) =2

(s+ kπ)3− 2|x|2 cos(s) sin(s)−3,

we see that the critical points s = sc of ψ are given by the equationsin(sc) = ±|x|(sc + kπ), hence those of ψ(2−l·) by r = 2lsc. If r issupposed to lie in our domain of integration, then |x| ∼ 2−lk−1. Forsuch 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

(4.4) sin(sc) = |x|(sc + kπ).

For later use, let us also put

γk(|x|) := ψ(sc,k(|x|)) =1

sc,k(|x|) + kπ− |x|2 cot(sc,k(|x|)).

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 & 2lk.

In this case, one gains from the r-integration in Fl, which we performfirst. To this end, recall that ψ(s) has now a unique critical pointsc = sc,k(|x|) ∼ 2−l if |x| ∼ 2−lk−1.

We may thus apply the method of stationary phase to the integra-tion in r. Noticing that the critical point of the phase ψl is given by2lsc,k(|x|), one eventually finds that

Fl(x, v) = λm+n+1/2km+n−12(m−1)l

∫χ(t)a(λkt4|v|)

b(k222l|x|2, λk−12−lt)eiλkt

(ψ(sc,k(|x|))+4v

)drdt,

24 DETLEF MULLER

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 infact depend also on k and l, but which lie in a bounded set of S−1/2

with respect to the natural Frechet topology on S−1/2.

Consequently,

Aλ,k,l,+,a(x,±|u|) = λm+n+1/2km+n−12(m−1)l

∫χ(t)a(λkt4|u|)

b(k222l|x|2, λk−12−lt)eiλkt

(γk(|x|)±4|u|

)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,

|Aλ,k,l,+,a(x,±|u|)| ≤ CNλm+nkm+n− 1

2 2(m− 12)l

·(1 + λk|u|)−n−12

(1 + λk|γk(|x|) ± 4|u||

)−N.(4.7)

The actual details of these arguments are more involved, and a com-plete estimate of Aλ,k,l,+,a requires a careful case analysis and domaindecompositions.

Remarks 4.1. (a) Observe that (4.7) shows that the singularities ofAλ,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 thepiece of the parametric curve

γ : s 7→(∣∣∣sin s

s

∣∣∣,∣∣∣s− sin s cos 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 ofparameters where s− kπ ∼ −2−l.


1/k

1/(k+1)

1/k

Figure 2. curve γ

5. Estimation of (1 + L)−(d−1)/4ei√Lδ

Let us put

α(p) := (d− 1)∣∣∣1p− 1

2

∣∣∣, 1 ≤ p <∞.

We need to estimate (1 + L)−α(p)/2ei√Lδ on Lp(G) in Theorem 1.1.

By means of a dyadic decomposition, this can easily be reduced toestimating the operator

(5.1)

∞∑

j=0

2−α(p)j χ(√L

2j

)ei

√L,

for a suitable cut-off function χ 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

Ep,±k,J (x, u) :=

J∑

j=0

∞∑

l=0

2−α(p)jA2j ,k,l,±(x, u)

and the corresponding kernels, with Aλ,k,l,± replaced by Bλ,k,l,±. Thesekernels will converge, in the sense of distributions, to distributions Ep,±

k

as J → ∞, and we shall have to show that the estimates sum in k.

26 DETLEF MULLER

Our estimates will follow from some rather elementary estimate forthe case p = 2, and a deeper estimate for p = 1 on a suitable localHardy space h1

k, whose definition will depend on k, by means of complexinterpolation. The corresponding interpolation argument is standard(cf. [35],[21]).

Let’s here concentrate on the case p = 1, and the kernels

E+k,J := E1,+

k,J =J∑

j=0

∞∑

l=0

2−(d−1)

2jA2j ,k,l,+.

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 K±λ,k,l ∈

S(G), for λ ≥ 2lk, and Rk,J ∈ L1(G) so that

(5.2) E+k,J =

∞∑

l=0

J∑

j=l+log2 k

(K+2j ,k,l

+K−2j ,k,l

) +Rk,J ,

and such that the following hold true:

(a) The kernel K±λ,k,l is supported in the set

(x, u) ∈ G : |x| ∼ 2−lk−1 and λk|u| & 1,

and can be estimated, for every α ∈ N2m, β ∈ Nn, by

|∂αx∂βuK±λ,k,l(x, u)| ≤ CN,α,β λ

|α|(λk)|β| λn+1

2 km+n− 12 2(m− 1

2)l

(1 + λk|u|)−n−12

(1 + λk|γk(|x|) ± 4|u||

)−N,(5.3)

for every N ∈ N, with constants CN,α,β which independent ofλ, k and l, and where γk(|x|) is given by (4.5). In particular,

(5.4) ‖∂αx∂βuK±λ,k,l‖1 . λ|α|(λk)|β| (2lk)−m− 1

2 .

(b) The sequence Rk,JJ converges for J → ∞ in L1(G) towardsa function Rk ∈ L1(G), and

(5.5) ‖Rk‖1 ≤ Ck−m− 12 ,

with a constant C not depending on k.


5.1. Anisotropic re-scaling for k fixed. Let k ≥ 1, and denote by

F (x, u) := k−DF (k−1x, k−2u)

the L1-norm preserving scaling of a function F on G by δk−1. Let usalso put

−Γk(v) := k2γk(k−1v) − k

π.

We can then make the interesting observation that, by (5.3), (5.4), the

non-isotropically re-scaled kernels K±λ,k,l are supported in the set

(x, u) ∈ G : |x| ∼ 2−l andλ

k|u| & 1,

and satisfy the isotropic estimates

|∂αx ∂βuK±λ,k,l(x, u)| ≤ CN,α,β

(λk

)|α|+|β|λ

n+12 k−(m+n+ 1

2)2(m− 1

2)l

(1 +λ

k|u|)−n−1

2

(1 +

λ

k

∣∣∣kπ− Γk(|x|) ± 4|u|

∣∣∣)−N

(5.6)

and

(5.7) ‖∂αx∂βuK±λ,k,l‖1 .

(λk

)|α|+|β|(2lk)−m− 1

2 .

Moreover, one easily verifies that

(5.8) Γk(v) = π−2 arcsin(πv) + π−1v√

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

Aµλ,k,l,± = λ1/2( ∫

χ0(t)χ1(±2l(2πt|µ|/λ− kπ)) η0(2πt|µ|/λ− kπ)

eiλ4t eit/λ L

µdt)δ.

Comparing with (9.13) and (9.12) in the Appendix, which obtained bymeans of Plancherel’s formula on G, we then find that the operatornorm of Aλ,k,l,± on L2(G) is given by

(6.1) ‖Aλ,k,l,±‖ = λ1/2 sup|αλ,k,l,±(ν, q)| : ν ≥ 0, q ∈ N,

28 DETLEF MULLER

where

αλ,k,l,±(ν, q)

:=

∫χ0(t)χ1(±2l(t ν

λ− kπ)) η0(t

νλ− kπ)ei

λ4t

+tνλ

(m+2q) dt.

The method of stationary phase then implies that

‖Aλ,k,l,±‖ ≤ C,

uniformly in λ, k and l. By the same method, we also get, for any α ≥ 0,

(6.2) ‖∞∑

l=0

∑

j≥j0

2−αjA2j ,k,l,±‖ ≤ C 2−αj0,

uniformly in l and j0.

7. Estimation for p = 1

We introduce a non-standard atomic local Hardy space h1 on G asfollows:

An atom centered at the origin is an L2-function a supported in aEuclidean ball Br(0) of radius r ≤ 1 satisfying the normalizing condi-tion

(7.1) ‖a‖2 ≤ r−d/2

and, if r < 1, in addition the moment condition

(7.2)

∫

G

a(x) dx = 0.

Notice that in particular ‖a‖1 . 1. An atom centered at z ∈ G isthe left-translate by z of an atom centered at the origin. The localHardy space h1(G) consists of all L1-functions which admit an atomicdecomposition

(7.3) f =∑

j

λjaj,

with atoms aj and coefficients λj such that∑

j |λj| <∞. We define the

norm ‖f‖h1 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 notsatisfy (7.2), such atoms can be decomposed into atoms supported inballs of radius 1, so that one can give the same atomic definition forh1E 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, fory ∈ R2m. Note the following easy, but important observations:

Lemma 7.1. a) Any atom a supported in Z(0) is indeed also a Eu-clidean atom in h1

E(possibly up to a fixed factor depending only on G),and vice versa.

b) If H1E := H1(R2m × Rn) denote the classical Euclidean Hardy

space, and if η ∈ C∞0 (R2m), then multiplication with η ⊗ 1 maps H1

E

continuously into h1E .

b) is known for compactly supported smooth multipliers, and theproof carries over easily.

Since G can be covered by sets Z(yj) with bounded overlap, if wechoose a suitable ε-separated set of points yjj for a suitable constantε > 0, one can use a) and b) in combination with classical arguments[6] to show that our non-standard Hardy space h1(G) interpolates withL2(G) by the complex method.

We shall not do this here, but indicate the main idea in the interpo-lation argument explained later.

Proposition 7.2. There is a constant C > 0, such that

‖f ∗ E1,±k ‖1 ≤ Ck−m‖f‖h1

for every f ∈ h1(G).

Proof. We again consider only E1,+k = E+

k . In view of the atomicdecomposition of f and because of left-invariance, one can show thatit suffices to assume that f = a is an atom centered at the origin andsupported in a ball Br(0). We shall also assume that r < 1, so that ahas vanishing integral, since the case r = 1 is somewhat easier. In viewof Proposition 5.1, we have to understand in particular the functions

a ∗ K±2j ,k,l

.

30 DETLEF MULLER

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±

λ,k,l where u > 0, and

that K±λ,k,l were not only essentially, but actually supported in the set

where ∣∣∣kπ− Γk(|x|) + 4u|

∣∣∣ ≤ k

λ.

Translating these kernels along the center by kπ, in view of (5.6) we can

then make the (slightly oversimplified) assumption that

(7.4) supp K±λ,k,l ⊂ (x, u) ∈ G : |x| ∼ 2−l and

∣∣∣u− 14Γk(|x|)

∣∣∣ ≤ k

λ,

and that

(7.5) |∂αx∂βuK±λ,k,l(x, u)| ≤ CN,α,β

(λk

)|α|+|β|λ k−(m+n+ 1

2)2(m− 1

2)l

Define for ρ > 0 the set

Ωρ := (x, u) ∈ G : |x| . 1 and∣∣∣u− 1

4Γk(|x|)

∣∣∣ ≤ Cρwhere C is a sufficiently large constant, and choose j0 so that

k2−j0 = r.

Obrserve that with this choice of j0

(7.6) supp a ∗ (∑

l

∑

j≥j0

K±2j ,k,l

) ⊂ Ωr,

since supp a ⊂ Br(0). We therefore split

E1,±k = Ek,r + Fk,r,

where

Ek,r :=∑∞

l=0

∑j≥j0 2−α(1)jA2j ,k,l,±,

Fk,r :=∑∞

l=0

∑j<j0

2−α(1)jA2j ,k,l,±.

We estimate a ∗ Ek,r separately on the set Ωr and it’s complement. ByHolder’s inequality and (6.2), we get

‖a ∗ Ek,r‖L1(Ωr) . |Ωr|1/2‖a‖2 2−α(1)j0 . r1/2r−(m+1/2)2−(d−1)

2j0 ≤ k−m.

Moreover, since by Proposition 5.1 Ek,r =∑

l

∑j≥j0 K

±2j ,k,l+Rk,r, where

‖Rk,r‖1 . k−m−12 , we obtain from (7.6) that

‖a ∗ Ek,r‖L1(G\Ωr) ≤ ‖a ∗ Rk,r‖L1(G) . k−m−12 ,


so that

‖a ∗ Ek,r‖1 . k−m.(7.7)

Consider next a∗Fk,r. Again, by Proposition 5.1, we can write Fk,r =∑

l

∑j<j0

K±2j ,k,l

+ R′k,r, where ‖R′

k,r‖1 . k−m−12 . And, for j < j0, we

have, by (7.4),

supp a ∗ (∑

l

K±2j ,k,l

) ⊂ Ωk2−j .

Moreover, making use of the cancellation∫a dx = 0 and (7.5) in es-

timating the convolution product a ∗ K±2j ,k,l, we gain a factor of order

O( rk2−j ) compared to (5.7) and obtain

‖a ∗∑

l

K±2j ,k,l‖1 .

∞∑

l=0

r

k2−j(2lk)−m− 1

2 . 2jr

kk−m− 1

2

Summing in j, this gives

‖a ∗ Fk,r‖1 . k−m− 12 .(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) := kDf(kx, k2u), f ∈ h1(G).

Then Proposition 7.1 can be re-stated as

‖f ∗ E1,±k ‖1 ≤ Ck−m‖f‖h1

k

for every f ∈ h1k(G). Thus, h1

k(G) is a natural space associated withthe k-th piece E1,±

k of the wave propagator whose definition dependson the parameter k.

8. Proof of Theorem 1.1

Observe that by (6.2) convolution with the kernel

E2,±k :=

∞∑

j=0

∞∑

l=0

A2j ,k,l,±

is bounded on L2(G), and

(8.1) ‖f ∗ E2,±k ‖2 ≤ C‖f‖2,

32 DETLEF MULLER

where C is independent of k. Consider now the analytic family of op-erators

Tα : f 7→ f ∗∞∑

j=0

∞∑

l=0

2−αjA2j ,k,l,±, 0 ≤ Re α ≤ (d−1)2.

By ignoring some error terms, we may assume that all A2j ,k,l,± aresupported in Z(0) (even a smaller cylindrical set). This implies that iff is supported in a cylindrical set Z(y), then Tαf is supported in the“cylindrical double” 2Z(y) := (x, u) ∈ G : |x − y| ≤ 20, for everyy ∈ 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 ∈ C∞0 (R2m) so that η ⊗ 1

localizes to Z(0), we therefore consider the localized operators

T 0αf := Tα((η ⊗ 1)f).

By Lemma 7.1 and Proposition 7.3, T 0d−12

: H1E → L1 continuously, and

since the proof of Proposition 7.3 remains valid for T d−12

+iβ, we have

‖T 0d−12

+iβf‖1 ≤ Ck−m ‖f‖H1

E∀β ∈ R.

Similarly, by (8.1), we have

‖T 0iβf‖2 ≤ C‖f‖2 ∀β ∈ R.

We can thus apply the complex interpolation result by C. Feffermanand Stein in [6] to obtain

‖T 0α(p)f‖p ≤ Ck−( 2

p−1)m ‖f‖p, 1 < p ≤ 2.

In view of the afore-mentioned support property of the convolutionkernels involved, this immediately implies

‖f ∗ Eα(p),±k ‖p ≤ Ck−( 2

p−1)m ‖f‖p, 1 < p ≤ 2.

If m ≥ 2, these estimates sum in k for p sufficiently close to 1, so that

(8.2) ‖(1 + L)−α(p)/2ei√Lf‖p ≤ C‖f‖p

for these p. Interpolating these estimates with the corresponding trivialestimate for p = 2 we then obtain Theorem 1.1 for 1 < p ≤ 2, and thecase p > 2 follows by duality.

Form = 1, the result can be obtained by means of a minor refinementof the arguments.


The L1-estimate in Theorem 1.1 follows a lot more easily from Propo-

sition 5.1, since for α > d−12

we have an additional factor λ−(α− d−12

)

in the corresponding estimate (5.4), so that the corresponding norms‖K±

2j ,k,l‖1 simply sum in j, k and l.

34 DETLEF MULLER

9. Appendix: The Fourier transform on a group of

Heisenberg type

Let us first briefly recall some facts about the unitary representationtheory of a Heisenberg type group G (compare [36],[5]). For the con-venience of the reader, we shall show how this representation theorycan easily be reduced to the well-known case of the Heisenberg groupHm = R2m × R. In slightly modified notation, the product in Hm isgiven by

(z, t) · (z′, t′) = (z + z′, t+ t′ +1

2ω(z, z′)),

where ω denotes the canonical symplectic form

(9.1) ω(z, w) := 〈Jz, w〉, J :=(

0 −ImIm 0

),

on R2m. Let us split coordinates z = (x, y) ∈ Rm × Rm in R2m, andconsider the associated natural basis of left-invariant vector fields

Xj :=∂

∂xj− 1

2yj∂

∂t, Yj :=

∂

∂yj+ 1

2xj∂

∂t, j = 1, . . . , m, and T :=

∂

∂t,

of the Lie algebra of Hm.

For τ ∈ R× := R\0, the Schrodinger representation ρτ of Hm actson the Hilbert space L2(Rm) as follows:

[ρτ (x, y, t)h](ξ) := e2πiτ(t+y·ξ+12y·x)h(ξ + x), h ∈ L2(Rm).

This is an irreducible, unitary representation, and every irreducibleunitary representation of Hm which acts non-trivially on the center isin fact unitarily equivalent to exactly one of these, by the Stone-vonNeumann theorem (a good reference to these and related results is forinstance [8]; see also [26]).

Next, if π is any unitary representation, say, of a Heisenberg typegroup G, we denote by

π(f) :=

∫

G

f(g)π(g) dg, f ∈ L1(G),

the associated representation of the group algebra L1(G). Going backto the Heisenberg group, if f ∈ S(Hm), then it is well-known andeasily seen that ρτ (f) =

∫R2m f

−τ (z)ρτ (z, 0) dz is a trace class operatoron L2(Rm), and its trace is given by

(9.2) tr(ρτ (f)) = |τ |−m∫

R

f(0, 0, t)e2πiτt dt = |τ |−m f−τ (0, 0),


for every τ ∈ R×.

From these facts, it is easy to derive the Plancherel formula for ourHeisenberg type group G. Given µ ∈ g∗

2 = Rn, µ 6= 0, consider thematrix Jµ introduced in Section 3. If |µ| = 1, then J2

µ = −I, becauseof (1.1). In particular, Jµ has only eigenvalues ±i, and since it isorthogonal, 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., ωµis represented by the standard symplectic matrix J in (9.1).

This means that, for every µ ∈ Rn \ 0, there is an orthogonalmatrix Rµ = R µ

|µ|∈ O(2m,R) such that

(9.3) Jµ = |µ|RµJtRµ.

Notice also that tRµ = R−1µ , and that condition (9.3) is in fact equiva-

lent 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 theydepend only on |x| and u, is commutative. This fact is well-known andeasy 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 fµ

Rµ, gµRµ are radial on R2m, and thus fµRµ×|µ|g

µRµ = gµRµ×|µ|fµ Rµ, which is again radial. Consequently, fµ ∗µ gµ = gµ ∗µ fµ, forevery 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 linkbetween representations of G and those of Hm.

Lemma 9.1. The mapping αµ : G→ Hm, given by

αµ(z, u) := ( tRµz,µ·u|µ| ), (z, u) ∈ R2m × Rn,

is an epimorphism of Lie groups. In particular, G/ kerαµ is isomorphicto Hm, where kerαµ = µ⊥ is just the orthogonal complement of µ inthe center Rn of G.

36 DETLEF MULLER

Given µ ∈ Rn \ 0, we can now define an irreducible unitary repre-sentation πµ of G on L2(Rm) by simply putting

πµ := ρ|µ| αµ.Observe that then πµ(0, u) = e2πiµ·uI. In fact, any irreducible represen-tation of G with central character e2πiµ·u factors through the kernel ofαµ and hence, by the Stone-von Neumann theorem, must be equivalentto πµ.

One then computes that, for f ∈ S(G),

πµ(f) =

∫

R2m

f−µ(Rµz)ρ|µ|(z, 0) dz,

so that the trace formula (9.2) yields the analogous trace formula

tr πµ(f) = |µ|−m f−µ(0)

on G. The Fourier inversion formula in Rn then leads to f(0, 0) =∫µ∈Rn\0 trπµ(f) |µ|m dµ. When applied to δg−1 ∗ f, we arrive at the

Fourier inversion formula

(9.5) f(g) =

∫

µ∈Rn\0tr (πµ(g)

∗πµ(f)) |µ|m dµ, g ∈ G.

Applying this to f ∗ ∗ f at g = 0, where f ∗(g) := f(g−1), we obtain thePlancherel formula

(9.6) ‖f‖22 =

∫

µ∈Rn\0‖πµ(f)‖2

HS |µ|m dµ,

where ‖T‖2HS = tr (T ∗T ) denotes the Hilbert-Schmidt norm.

Let us next consider the group Fourier transform of our sub-LaplacianL on G.

We first observe that dαµ(X) = tRµX for every X ∈ g1 = R2m, ifwe view, for the time being, elements of the Lie algebra as tangentialvectors at the identity element. Moreover, by (9.3), we see that

tRµXµ,1, . . . ,tRµXµ,m,

tRµYµ,1, . . . ,tRµYµ,m

forms a symplectic basis with respect to the canonical symplectic formω on R2m. We may thus assume without loss of generality that thisbasis agrees with our basis X1, . . . , Xm, Y1, . . . , Ym of R2m, so that

dαµ(Xµ,j) = Xj, dαµ(Yµ,j) = Yj, j = 1, . . . , m.


By our construction of the representation πµ, we thus obtain for thederived representation dπµ of g that

(9.7) dπµ(Xµ,j) = dρ|µ|(Xj), dπµ(Yµ,j) = dρ|µ|(Yj), j = 1, . . . , m.

Let us define the sub-Laplacians Lµ on G and L on Hm by

Lµ := −m∑

j=1

(X2µ,j + Y 2

µ,j), L := −m∑

j=1

(X2j + Y 2

j ),

where from now on we consider elements of the Lie algebra again asleft-invariant differential operators. Then, by (9.6),

dπµ(Lµ) = dρ|µ|(L).

Moreover, since the basis Xµ,1, . . . , Xµ,m, Yµ,1, . . . , Yµ,m and our originalbases X1, . . . , X2m of g1 are both orthonormal bases, it is easy to verifythat the distributions Lδ0 and Lµδ0 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|(L).

But, it is well-known that dρ|µ|(L) = ∆ξ−(2π|µ|)2|ξ|2 is just a re-scaledHermite operator, and an orthonormal basis of L2(Rm) is given by thetensor products

h|µ|α := h|µ|α1⊗ · · · ⊗ h|µ|αm

, α ∈ Nm,

where hµk(x) := (2π|µ|)1/4hk((2π|µ|)1/2x). Here,

hk(x) = ck(−1)kex2/2 d

k

dxke−x

2

denotes the L2-normalized Hermite function of order k on R. Conse-quently,

(9.9) dπµ(L)h|µ|α = 2π|µ|(m+ 2|α|)h|µ|α , α ∈ Nm.

It is also easy to see that

(9.10) dπµ(Uj) = 2πiµjI, j = 1, . . . , n.

Now, the operators L,−iU1, . . . ,−iUn form a commuting set of self-adjoint operators, with joint core S(G), so that they admit a jointspectral resolution, and we can thus give meaning to expressions likeϕ(L,−iU1, . . . ,−iUn) for each continuous function ϕ defined on thecorresponding joint spectrum. For simplicity of notation we write

U := (−iU1, . . . ,−iUn).

38 DETLEF MULLER

If ϕ is bounded, then ϕ(L,U) is a bounded, left invariant operator onL2(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 byϕ(L,U)δ. Moreover, if ϕ ∈ S(R × Rn), then ϕ(L,U)δ ∈ S(G) (com-pare [29], [30]). Since functional calculus is compatible with unitaryrepresentation theory, we obtain in this case from (9.9), (9.10) that

(9.11) πµ(ϕ(L,U)δ)h|µ|α = ϕ(2π|µ|(m+ 2|α|), 2πµ)h|µ|α

(this identity in combination with the Fourier inversion formula couldin fact be taken as the definition of ϕ(L,U)δ). In particular, thePlancherel theorem implies then that the operator norm on L2(G) isgiven by

(9.12) ‖ϕ(L,U)‖ = sup|ϕ(|µ|(m+ 2q), µ)| : µ ∈ Rn, q ∈ N.Finally, observe that

(9.13) Kµφ = ϕ(Lµ, 2πµ)δ;

this follows for instance by applying the unitary representation inducedfrom the character e2πiµ·u on the center of G to Kϕ.

References

[1] Hajer Bahouri, Patrick Gerard, and Chao-Jiang Xu. Espaces de Besov et esti-mations de Strichartz generalisees sur le groupe de Heisenberg. J. Anal. Math.,82:93–118, 2000.

[2] R. Michael Beals. Lp boundedness of Fourier integral operators. Mem. Amer.

Math. Soc., 38(264):viii+57, 1982.[3] Michael Christ. L

p bounds for spectral multipliers on nilpotent groups. Trans.Amer. Math. Soc., 328(1):73–81, 1991.

[4] Y. Colin de Verdiere and M. Frisch. Regularite lipschitzienne et solutions de

l’equation des ondes sur une variete riemannienne compacte. Ann. Sci. EcoleNorm. Sup. (4), 9(4):539–565, 1976.

[5] Ewa Damek and Fulvio Ricci. Harmonic analysis on solvable extensions ofH-type groups. J. Geom. Anal., 2(3):213–248, 1992.

[6] C. Fefferman and E. M. Stein. Hp spaces of several variables. Acta Math.,

129(3-4):137–193, 1972.[7] G. B. Folland and Elias M. Stein. Hardy spaces on homogeneous groups, vol-

ume 28 of Mathematical Notes. Princeton University Press, Princeton, N.J.,1982.

[8] Gerald B. Folland. Harmonic analysis in phase space, volume 122 of Annals ofMathematics Studies. Princeton University Press, Princeton, NJ, 1989.


[9] Bernard Gaveau. Principe de moindre action, propagation de la chaleur etestimees sous elliptiques sur certains groupes nilpotents. Acta Math., 139(1-2):95–153, 1977.

[10] David Goldberg. A local version of real Hardy spaces. Duke Math. J., 46(1):27–42, 1979.

[11] Chr. Gole and R. Karidi. A note on Carnot geodesics in nilpotent Lie groups.J. Dynam. Control Systems, 1(4):535–549, 1995.

[12] Waldemar Hebisch. Multiplier theorem on generalized Heisenberg groups. Col-loq. Math., 65(2):231–239, 1993.

[13] Lars Hormander. Hypoelliptic second order differential equations. Acta Math.,119:147–171, 1967.

[14] Lars Hormander. The analysis of linear partial differential operators. I, volume256 of Grundlehren der Mathematischen Wissenschaften [Fundamental Prin-ciples of Mathematical Sciences]. Springer-Verlag, Berlin, 1983. Distributiontheory and Fourier analysis.

[15] Roger Howe. The oscillator semigroup. In The mathematical heritage of Her-mann Weyl (Durham, NC, 1987), volume 48 of Proc. Sympos. Pure Math.,pages 61–132. Amer. Math. Soc., Providence, RI, 1988.

[16] A. Hulanicki. The distribution of energy in the Brownian motion in the Gauss-ian field and analytic-hypoellipticity of certain subelliptic operators on theHeisenberg group. Studia Math., 56(2):165–173, 1976.

[17] Andrzej Hulanicki. A functional calculus for Rockland operators on nilpotentLie groups. Studia Math., 78(3):253–266, 1984.

[18] Wensheng Liu and Hector J. Sussman. Shortest paths for sub-Riemannianmetrics on rank-two distributions. Mem. Amer. Math. Soc., 118(564):x+104,1995.

[19] Giancarlo Mauceri and Stefano Meda. Vector-valued multipliers on stratifiedgroups. Rev. Mat. Iberoamericana, 6(3-4):141–154, 1990.

[20] Richard Melrose. Propagation for the wave group of a positive subellipticsecond-order differential operator. In Hyperbolic equations and related topics(Katata/Kyoto, 1984), pages 181–192. Academic Press, Boston, MA, 1986.

[21] Akihiko Miyachi. On some estimates for the wave equation in Lp and H

p. J.Fac. Sci. Univ. Tokyo Sect. IA Math., 27(2):331–354, 1980.

[22] D. Muller and F. Ricci. Analysis of second order differential operators onHeisenberg groups. I. Invent. Math., 101(3):545–582, 1990.

[23] D. Muller and A. Seeger. Sharp Lp-estimates for the wave equation on Heisen-berg type groups. in preparation, 2008.

[24] D. Muller and E. M. Stein. On spectral multipliers for Heisenberg and relatedgroups. J. Math. Pures Appl. (9), 73(4):413–440, 1994.

[25] Detlef Muller. A restriction theorem for the Heisenberg group. Ann. of Math.(2), 131(3):567–587, 1990.

[26] Detlef Muller. Analysis of invariant PDO’s on the Heisenberg group.Lecture notes ICMS-Instructional conference, Edinburgh, April 1999,http://analysis.math.uni-kiel.de/mueller/, 1999.

[27] Detlef Muller. Marcinkiewicz multipliers and multi-parameter structure onHeisenberg groups. Lecture notes of a minicorso at Padova, June 2004,http://analysis.math.uni-kiel.de/mueller/, 2004.

40 DETLEF MULLER

[28] Detlef Muller. Local solvability of linear differential operators with double char-acteristics. II. Sufficient conditions for left-invariant differential operators oforder two on the Heisenberg group. J. Reine Angew. Math., 607:1–46, 2007.

[29] Detlef Muller, Fulvio Ricci, and Elias M. Stein. Marcinkiewicz multipliersand multi-parameter structure on Heisenberg (-type) groups. I. Invent. Math.,119(2):199–233, 1995.

[30] Detlef Muller, Fulvio Ricci, and Elias M. Stein. Marcinkiewicz multipliersand multi-parameter structure on Heisenberg (-type) groups. II. Math. Z.,221(2):267–291, 1996.

[31] Detlef Muller and Elias M. Stein. Lp-estimates for the wave equation on the

Heisenberg group. Rev. Mat. Iberoamericana, 15(2):297–334, 1999.[32] Adrian I. Nachman. The wave equation on the Heisenberg group. Comm. Par-

tial Differential Equations, 7(6):675–714, 1982.[33] Edward Nelson and W. Forrest Stinespring. Representation of elliptic operators

in an enveloping algebra. Amer. J. Math., 81:547–560, 1959.[34] Martin Paulat. Sub-Riemannian geometry and heat kernel estimates. Disser-

tation, C.A.University Kiel, 2008.[35] Juan C. Peral. L

p estimates for the wave equation. J. Funct. Anal., 36(1):114–145, 1980.

[36] Fulvio Ricci. Harmonic analysis on generalized Heisenberg groups. unpublishedpreprint.

[37] Andreas Seeger, Christopher D. Sogge, and Elias M. Stein. Regularity proper-ties of Fourier integral operators. Ann. of Math. (2), 134(2):231–251, 1991.

D. Muller, Mathematisches Seminar, C.A.-Universitat Kiel, Ludewig-

Meyn-Str.4, D-24098 Kiel, Germany

E-mail address : [email protected]

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