Functional calculus of Lie groups and wave propagation.
A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces.
Singular kernels supported by homogeneous submanifolds.
On Riesz means of eigenfunction expansions for the Kohn-Laplacian.
Twisted convolutions with Calderón-Zygmund kernels.
On weighted estimates for the Kakeya maximal operator
Estimates of one-dimensional oscillatory integrals
We study one-dimensional oscillator integrals which arise as Fourier-Stieltjes transforms of smooth, compactly supported measures on smooth curves in Euclidean spaces and determine their decay at infinity, provided the curves satisfy certain geometric conditions.
A continuous Helson surface in
For some time it has been known that there exist continuous Helson curves in . This result, which is related to Lusin’s rearrangement problem, had been proved first by Kahane in 1968 with the aid of Baire category arguments. Later McGehee and Woodward extended this result, giving a concrete construction of a Helson -manifold in for . We present a construction of a Helson 2-manifold in . With modification, our method should even suffice to prove that there are Helson hypersurfaces in any .
Sub-Laplacians of holomorphic -type on exponential Lie groups
In this survey article, I shall give an overview on some recent developments concerning the -functional calculus for sub-Laplacians on exponential solvable Lie groups. In particular, I shall give an outline on some recent joint work with W. Hebisch and J. Ludwig on sub-Laplacians which are of holomorphic -type, in the sense that every -spectral multiplier for will be holomorphic in some domain.
Inequalities for spherically symmetric solutions of the wave equation.
Solvability for a class of non-homogeneous differential operators on two-step nilpotent groups.
A class of solvable non-homogeneous differential operators on the Heisenberg group
In [8], we studied the problem of local solvability of complex coefficient second order left-invariant differential operators on the Heisenberg group ℍₙ, whose principal parts are "positive combinations of generalized and degenerate generalized sub-Laplacians", and which are homogeneous under the Heisenberg dilations. In this note, we shall consider the same class of operators, but in the presence of left invariant lower order terms, and shall discuss local solvability for these operators in a complete...
Wave equation and multiplier estimates on ax + b groups
Let L be the distinguished Laplacian on certain semidirect products of ℝ by ℝⁿ which are of ax + b type. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form for arbitrary time t and arbitrary λ > 0, where ψ is a smooth bump function supported in [-2,2] if λ ≤ 1 and in [1,2] if λ ≥ 1. As a corollary, we reprove a basic multiplier estimate of Hebisch and Steger [Math. Z. 245 (2003)] for this particular class of groups, and derive Sobolev...
spectral multipliers on the free group
Let L be a homogeneous sublaplacian on the 6-dimensional free 2-step nilpotent Lie group on three generators. We prove a theorem of Mikhlin-Hörmander type for the functional calculus of L, where the order of differentiability s > 6/2 is required on the multiplier.
Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups, II.
Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups, I.
L-estimates for the wave equation on the Heisenberg group.
Let £ denote the sub-Laplacian on the Heisenberg group H. We prove that e / (1 - £) extends to a bounded operator on L(H), for 1 ≤ p ≤ ∞, when α > (d - 1) |1/p - 1/2|.
A.e. convergence of spectral sums on Lie groups
Let be a right-invariant sub-Laplacian on a connected Lie group and let denote the associated “spherical partial sums,” where is the spectral resolution of We prove that converges a.e. to as under the assumption
Eigenvalues of the Hodge Laplacian on the Heisenberg group.
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