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

Documents
80 pages
Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus

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

Informations

Publié par
Nombre de visites sur la page 17
Langue English
Signaler un problème
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
Sharp Lp-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
0-1
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 ofAlkλ±ifk1
0-2
.
.
.
.
.
.
. 0-36
0-43
5
6
7
8
9
Estimation of(1 +L)(d1)4ei
5.1
Anisotropic re-scaling fork
Lδ
xed
.
.
0-49
. 0-52
L2-estimates for components of the wave propagator 0-54
Estimation forp= 1
Proof of Theorem 1.1
0-55
0-62
Appendix: The Fourier transform on a group of Heisenberg type 0-65
0-3
1
Introduction
Let g=g1g2with dimg1= 2mand dimg2=nbe a Lie algebra ofHeisenberg type,where
[gg]g2z(g)
z(g) being the center ofg means that. Thisgis endowed with an inner producthisuch thatg1 andg2or orthogonal subspaces and the following holds true:
If we define forµg2\ {0}the skew formωµon g1by ωµ(V W) :=µ[V W]then there is a unique skew-symmetric linear endo-morphismJµofg1such that ωµ(V W) =hµ[V W]i=hJµ(V) Wi (here, we also used the natural identification ofg2withg2 Thenvia the inner product).
Jµ2=−|µ|2I
0-4
(1.1)
Note that this implies in particular that
[g1g1] =g2
As the corresponding connected, simply connected Heisenberg type Lie groupGwe shall then choose the linear manifoldgendowed with the Baker-Campbell-Hausdorff product
(
V1 U1) (V2 U2) := (V1+V2 U1+U2+2[V1 V2])
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 identifyXgwith the cor-responding left-invariant vector field onGgiven by the Lie-derivative
X f(g) :=ftdd(gexp(tX))|t=0
where exp :gGdenotes the exponential mapping, which agrees with the identity mapping in our case.
0-5
Let us next fix an orthonormal basisX1     X2m ofg1and let us define the non-ellipticpaalicnasub-L
2m L:=XXj2 j=1
onGSince the vector fieldsXjtogether 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 basisU1     Unofg2and shall in the sequel identifyg=g1+g2andGwithR2m×Rn by means of the basisX1     X2m U1     UnofgThen our inner product ongwill agree with the canonical Euclidean productzw=Pj2=m+1nzjwj onR2m+nandJµwill be identified with a skew-symmetric 2m×2mmatrix. Moreover, the Lebesgue measuredx duonR2m+nis a bi-invariant Haar mea-sure onGBy d:= 2m+n we shall denote the topological dimension ofGWe
0-6
also introduce the automorphic dilations
δr(x u) := (rx r2u)
onG, and the Koranyi norm
r >0
k(x u)kK:= (|x|4+|4u|2)14
Notice that this is a homogeneous norm with respect to the dilationsδrand thatLis 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}
then the volume|Qr(x u)|is given by
where
|Qr(x u)|=|Q1(00)|rD
D:= 2m+ 2n
is thehomogeneous dimensionofGWe shall also have to work with the Euclidean balls
Br(x u) :={(y v)G:|(yx vu)|< r}
with respect to the Euclidean norm
|(x u)|:= (|x|2+|u|2)12
0-7