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$L^{p}$ spectral multipliers on the free group $N_{3,2}$

100%

Martini A. , Müller D.

tom 217

nr 1

41-55

Let L be a homogeneous sublaplacian on the 6-dimensional free 2-step nilpotent Lie group $N_{3,2}$ 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.

A class of solvable non-homogeneous differential operators on the Heisenberg group

100%

Müller D. , Zhang Z.

Studia Mathematica

2001

tom 148

nr 1

87-96

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 way. Previously known methods to study such non-homogeneous operators, as in [9] or [6], do not apply to these operators, and it is the main purpose of this article to introduce a new method, which should be applicable also in much wider settings.

Wave equation and multiplier estimates on ax + b groups

100%

Müller D. , Thiele C.

Studia Mathematica

2007

tom 179

nr 2

117-148

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 $e^{it√L} ψ(√L/λ)$ 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 estimates for solutions to the wave equation associated to L. There appears no dispersive effect with respect to the $L^{∞}$-norms for large times in our estimates, so that it seems unlikely that non-trivial Strichartz type estimates hold.

On weighted estimates for the Kakeya maximal operator

70%

Müller D.

1990

tom 60-61

nr 2

457-475