In these lectures which are based on re cent joint work with A Seeger 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 associ ated 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 ad dressed Our results improve on earlier joint work of mine with E M Stein The new ap proach that we use has the additional advan tage of bringing out more clearly the con nections of the problem with the underlying sub Riemannian geometry

Description

Niveau: Supérieur, Doctorat, Bac+8
Abstract In these lectures, which are based on re- cent 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 associ- ated 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 ad- dressed. Our results improve on earlier joint work of mine with E.M. Stein. The new ap- proach that we use has the additional advan- tage of bringing out more clearly the con- nections of the problem with the underlying sub-Riemannian geometry. 0-0

• wave equation


• present sharp analogues


• metaplec- tic group


• unique skew-symmetric


• sub- riemannian geometry

Sujets

• Müller
• Wave equation

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 17
• Langue: English

Extrait

Abstract

In these lectures, which are based on re- cent 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 associ- ated 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 ad- dressed. Our results improve on earlier joint work of mine with E.M. Stein. The new ap- proach that we use has the additional advan- tage of bringing out more clearly the con- nections of the problem with the underlying sub-Riemannian geometry.

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Sharp L p -estimates for the wave equation on Heisenberg type groups

Orleans, April 2008

http://analysis.math.uni- kiel.de/mueller/

DetlefMu¨lle

r

April 9, 2008

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Contents

1

2

3

4

Introduction

1.1

0-4

Connections with spectral multipliers and further facts about the wave equa- tion . . . . . . . . . . . . . . . . . . . 0-13

The sub-Riemannian geometry of a Heisen- berg type group 0-19

TheSchr¨odingerandthewavepropa- gators on a Heisenberg type group 0-27

3.1

3.2

Twisted convolution and the metaplec- tic group . . . . . . . . . . . . . . . . . 0-31

A subordination formula .

Estimation ofA lkλ± ifk≥1

0-2

.

.

.

.

.

.

. 0-36

0-43

5

6

7

8

9

Estimation of(1 +L) −(d−1)4 e i

5.1

Anisotropic re-scaling forkfi

L δ

xed

.

.

0-49

. 0-52

L 2 -estimates for components of the wave propagator 0-54

Estimation forp= 1

Proof of Theorem 1.1

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0-62

Appendix: The Fourier transform on a group of Heisenberg type 0-65

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1

Introduction

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)

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 on g 1 by ω µ (V W) :=µ  [V W]   then there is a unique skew-symmetric linear endo- morphismJ 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  Thenvia the inner product).

J µ 2 =−|µ| 2 I

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(1.1)

Note that this implies in particular that

[g 1 g 1 ] =g 2 

As the corresponding connected, simply connected Heisenberg 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 de- composition of a simple Lie group of real rank one is always of Heisenberg type or Euclidean.

As usual, we shall identifyX∈gwith the cor- responding left-invariant vector field onGgiven by the Lie-derivative

X f(g) :=f td d (gexp(tX))| t=0 

where exp :g→Gdenotes the exponential mapping, which agrees with the identity mapping in our case.

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

2m L:=− X X j 2 j=1

onGSince the vector fieldsX j together with their commutators span the tangent space toGat every point,Lis still hypoelliptic and provides 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 or- thonormal basisU 1      U n ofg 2 and shall in the sequel identifyg=g 1 +g 2 andGwithR 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 j 2 = m+ 1 n z j w j onR 2m+n andJ µ will be identified with a skew- symmetric 2m×2mmatrix. Moreover, the Lebesgue measuredx duonR 2m+n is a bi-invariant Haar mea- sure onGBy d:= 2m+n we shall denote the topological dimension ofGWe

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also introduce the automorphic dilations

δ r (x u) := (rx r 2 u)

onG, and the Koranyi norm

r >0

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}

then the volume|Q r (x u)|is given by

where

|Q r (x u)|=|Q 1 (00)|r D 

D:= 2m+ 2n

is thehomogeneous dimensionofGWe shall also have to work with the Euclidean balls

B r (x u) :={(y v)∈G:|(y−x v−u)|< r}

with respect to the Euclidean norm

|(x u)|:= (|x| 2 +|u| 2 ) 12 

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