SHARP Lp ESTIMATES FOR THE WAVE EQUATION ON HEISENBERG TYPE GROUPS

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Niveau: Supérieur, Doctorat, Bac+8
SHARP Lp-ESTIMATES FOR THE WAVE EQUATION ON HEISENBERG TYPE GROUPS LECTURE NOTES ORLEANS, APRIL 2008 DETLEF MULLER Abstract. In these lectures, which are based on recent joint work with A. Seeger [23], I shall present sharp analogues of classical es- timates 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 im- prove 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

• wave equation


• then


• euclidean norm


• laplacian plays


• ple lie


• product ?


• product


• shall denote


• lie group


• left- invariant vector

Sujets

• Wave equation
• Norm (mathematics)
• Product
• Lie group

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 23
• Langue: English

Extrait

SHARP

L p -ESTIMATES FOR THE WAVE EQUATION ON 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 work with A. Seeger [23], I shall present sharp analogues of classical es- timates 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 im- prove 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.

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1.

Introduction

¨ DETLEF MULLER

Contents

1.1. Connections with spectral multipliers and further facts about the wave equation

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

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3.TheSchr¨odingerandthewavepropagatorsonaHeisenberg type group 14

3.1. Twisted convolution and the metaplectic group

3.2. A subordination formula

λkl± ifk≥1 4. Estimation ofA 5. Estimation of (1 +L) −(d−1)4 e i L δ 5.1. Anisotropic re-scaling forkfixed 6.L 2 -estimates for components of the wave propagator

7. Estimation forp= 1

8. Proof of Theorem 1.1

9. Appendix: The Fourier transform on a group of Heisenberg type

References

1.Introduction

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Let g=g 1 ⊕g 2  with dimg 1 = 2mand dimg 2 =nbe a Lie algebra ofHeisenberg type, where [gg]⊂g 2 ⊂z(g)

THE WAVE EQUATION ON GROUPS OF HEISENBERG TYPE

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z(g) being the center ofg means that. Thisgis endowed with an inner producthisuch thatg 1 andg 2 or orthogonal subspaces and the following holds true: If we define forµ∈g ∗ 2 \ {0}the skew formω µ ong 1 by ω µ (V W) :=µ  [V W]   then there is a unique skew-symmetric linear endomorphismJ µ ofg 1 such that ω µ (V W) =hµ[V W]i=hJ µ (V) Wi (here, we also used the natural identification ofg ∗ 2 withg 2 via the inner product). Then

(1.1)

J µ 2 =−|µ| 2 I

for everyµ∈g 2 ∗ \ {0}. Note that this implies in particular that [g 1 g 1 ] =g 2  As the corresponding connected, simply connectedHeisenberg type Lie groupGwe shall then choose the linear manifoldgendowed with the Baker-Campbell-Hausdorff product (V 1  U 1 ) (V 2  U 2 ) := (V 1 +V 2  U 1 +U 2 + 2 [V 1  V 2 ]) and identity elemente= 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 identifyX∈gwith the corresponding left- invariant vector field onGgiven by the Lie-derivative

X f(g) := d dt f(gexp(tX))| t=0  where exp :g→Gdenotes the exponential mapping, which agrees with the identity mapping in our case.

Let us next fix an orthonormal basisX 1      X 2m ofg 1 and let us define the non-ellipticsub-Laplacian

2m L:=− X X j 2 j=1

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¨ DETLEF MULLER

onGSince the vector fieldsX j together with their commutators span the tangent space toGat every point,Lis still hypoelliptic and pro- vides an example of a non-elliptic “sum of squares operator ” in the senseofHo¨rmander([13]).Moreover,Ltakes over in many respects of analysis onGthe role which the Laplacian plays on Euclidean space.

To simplify the notation, we shall also fix an orthonormal basis U 1      U n ofg 2 and shall in the sequel identifyg=g 1 +g 2 andG withR 2m ×R n by means of the basisX 1      X 2m  U 1      U n ofg Then our inner product ongwill agree with the canonical Euclidean productzw= P 2m+n w j onR 2m+n andJ µ will be identified with j=1 z j a skew-symmetric 2m×2mmatrix. Moreover, the Lebesgue measure dx duonR 2m+n is a bi-invariant Haar measure onGBy d:= 2m+n we shall denote the topological dimension ofGWe also introduce the automorphic dilations δ r (x u) := (rx r 2 u) > r0 onG, and the Koranyi norm k(x u)k K := (|x| 4 +|4u| 2 ) 14  Notice that this is a homogeneous norm with respect to the dilationsδ r  and thatLis homogeneous of degree 2 with respect to these dilations. Moreover, if we denote the corresponding balls by Q r (x u) :={(y v)∈G:k(y v) −1 (x u)k K < r}(x u)∈G r >0 then the volume|Q r (x u)|is given by |Q r (x u)|=|Q 1 (00)|r D 

where D:= 2m+ 2n is thehomogeneous dimensionofGshall also have to work withWe the Euclidean balls B r (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 casen= 1 we may assume thatJ µ =µJ µ∈Rwhere J:=  − 0 I m I 0 m 