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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


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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 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±ifk1 4. Estimation ofA 5. Estimation of (1 +L)(d1)4ei Lδ 5.1. Anisotropic re-scaling forkfixed 6.L2-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=g1g2with dimg1= 2mand dimg2=nbe a Lie algebra ofHeisenberg type, where [gg]g2z(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 producthisuch thatg1andg2or orthogonal subspaces and the following holds true: If we define forµg2\ {0}the skew formωµong1by ωµ(V W) :=µ[V W]then there is a unique skew-symmetric linear endomorphismJµofg1 such that ωµ(V W) =hµ[V W]i=hJµ(V) Wi (here, we also used the natural identification ofg2withg2via the inner product). Then
(1.1)
Jµ2=−|µ|2I
for everyµg2\ {0}. Note that this implies in particular that [g1g1] =g2As the corresponding connected, simply connectedHeisenberg 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 decomposition of a sim-ple Lie group of real rank one is always of Heisenberg type or Euclidean.
As usual, we shall identifyXgwith the corresponding left-invariant vector field onGgiven by the Lie-derivative
X f(g) :=ddtf(gexp(tX))|t=0where exp :gGdenotes the exponential mapping, which agrees with the identity mapping in our case.
Let us next fix an orthonormal basisX1     X2mofg1and let us define the non-ellipticsub-Laplacian
2m L:=XXj2 j=1
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¨ DETLEF MULLER
onGSince the vector fieldsXjtogether 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 U1     Unofg2and shall in the sequel identifyg=g1+g2andG withR2m×Rnby means of the basisX1     X2m U1     UnofgThen our inner product ongwill agree with the canonical Euclidean productzw=P2m+nwjonR2m+nandJµwill be identified with j=1zj a skew-symmetric 2m×2mmatrix. Moreover, the Lebesgue measure dx duonR2m+nis a bi-invariant Haar measure onGBy d:= 2m+n we shall denote the topological dimension ofGWe also introduce the automorphic dilations δr(x u) := (rx r2u) > r0onG, and the Koranyi norm k(x u)kK:= (|x|4+|4u|2)14Notice 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}(x u)G r >0then the volume|Qr(x u)|is given by |Qr(x u)|=|Q1(00)|rD
where D:= 2m+ 2n is thehomogeneous dimensionofGshall also have to work withWe the Euclidean balls Br(x u) :={(y v)G:|(yx vu)|< r}(x u)G r >0with respect to the Euclidean norm |(x u)|:= (|x|2+|u|2)12
In the special casen= 1 we may assume thatJµ=µJ µRwhere J:=0ImI0m