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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 4 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 6 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]): 8 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 . References [1] Hajer Bahouri, Patrick Gérard, and Chao-Jiang Xu. Espaces de Besov et estimations de Strichartz généralisées 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. 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