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 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
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Contents
1
2
3
4
Introduction
1.1
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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±ifk≥1
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. 0-36
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5
6
7
8
9
Estimation of(1 +L)−(d−1)4ei
5.1
Anisotropic re-scaling forkfi
Lδ
xed
.
.
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. 0-52
L2-estimates for components of the wave propagator 0-54
Estimation forp= 1
Proof of Theorem 1.1
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Appendix: The Fourier transform on a group of Heisenberg type 0-65
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1
Introduction
Let g=g1⊕g2 with dimg1= 2mand dimg2=nbe a Lie algebra ofHeisenberg type,where
[gg]⊂g2⊂z(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 forµ∈g∗2\ {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 ofg2∗ withg2 Thenvia the inner product).
Jµ2=−|µ|2I
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(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 identifyX∈gwith the cor-responding left-invariant vector field onGgiven by the Lie-derivative
X f(g) :=ftdd(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 basisX1 X2m ofg1and let us define the non-ellipticpaalicnasub-L
2m L:=−XXj2 j=1
onGSince 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 Unofg Then 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 onGBy d:= 2m+n we shall denote the topological dimension ofGWe
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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 dimensionofGWe shall also have to work with the Euclidean balls
Br(x u) :={(y v)∈G:|(y−x v−u)|< r}
with respect to the Euclidean norm
|(x u)|:= (|x|2+|u|2)12
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