726 research outputs found
Problems of Harmonic Analysis related to finite type hypersurfaces in R^3, and Newton polyhedra
This article, which grew out of my lecture at the conference "Analysis and
Applications: A Conference in Honor of Elias M. Stein" in May 2011, is intended
to give an overview on a collection of results which have been obtained jointly
with I.I. Ikromov, and in parts also with M. Kempe, and at the same time to
give a kind of guided tour through the rather comprehensive proofs of the major
results that I shall address. All of our work is highly influenced by the
pioneering ideas developed by E.M. Stein.Comment: 44 pages, a picture
Spectral multiplier theorems of Euclidean type on new classes of 2-step stratified groups
From a theorem of Christ and Mauceri and Meda it follows that, for a
homogeneous sublaplacian on a -step stratified group with Lie
algebra , an operator of the form is of weak type
and bounded on for if the spectral multiplier
satisfies a scale-invariant smoothness condition of order , where is the homogeneous
dimension of . Here we show that the condition can be pushed down to , where is the topological dimension of ,
provided that or .Comment: 33 page
spectral multipliers on the free group
Let be the homogeneous sublaplacian on the 6-dimensional free 2-step
nilpotent group on 3 generators. We prove a theorem of
Mihlin-H\"ormander type for the functional calculus of , where the order of
differentiability is required on the multiplier
Sharp -bounds for the wave equation on groups of Heisenberg type
Consider the wave equation associated with the Kohn Laplacian on groups of
Heisenberg type. We construct parametrices using oscillatory integral
representations and use them to prove sharp and Hardy space regularity
results
Spectral multipliers on -step groups: topological versus homogeneous dimension
Let be a -step stratified group of topological dimension and
homogeneous dimension . Let be a homogeneous sub-Laplacian on . By a
theorem due to Christ and to Mauceri and Meda, an operator of the form
is of weak type and bounded on for all
whenever the multiplier satisfies a scale-invariant smoothness condition of
order . It is known that, for several -step groups and
sub-Laplacians, the threshold in the smoothness condition is not sharp
and in many cases it is possible to push it down to . Here we show that,
for all -step groups and sub-Laplacians, the sharp threshold is strictly
less than , but not less than .Comment: 17 page
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