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

100%

Müller D., Zhang Z.

Studia Mathematica

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2001

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tom 148

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nr 1

87-96

EN

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.

2

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

100%

Martini A., Müller D.

Studia Mathematica

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2013

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tom 217

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nr 1

41-55

EN

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.

3

Wave equation and multiplier estimates on ax + b groups

100%

Müller D., Thiele C.

Studia Mathematica

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2007

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tom 179

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nr 2

117-148

EN

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.

4

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On weighted estimates for the Kakeya maximal operator

70%

Müller D.

Colloquium Mathematicum

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1990

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tom 60-61

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nr 2

457-475

Ograniczanie wyników

3 Studia Mathematica

1 Colloquium Mathematicum

4 Müller D.

1 Martini A.

1 Thiele C.

1 Zhang Z.

1 2013

1 2007

1 2001

1 1990

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

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

Wave equation and multiplier estimates on ax + b groups

On weighted estimates for the Kakeya maximal operator