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Estimates of one-dimensional oscillatory integrals

Detlef Muller — 1983

Annales de l'institut Fourier

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 𝐑 3

Detlef Müller — 1984

Annales de l'institut Fourier

For some time it has been known that there exist continuous Helson curves in R 2 . 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 k -manifold in R n k for n k + 1 . We present a construction of a Helson 2-manifold in R 3 . With modification, our method should even suffice to prove that there are Helson hypersurfaces in any R n .

Sub-Laplacians of holomorphic L p -type on exponential Lie groups

Detlef Müller — 2002

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this survey article, I shall give an overview on some recent developments concerning the L p -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 L p -type, in the sense that every L p -spectral multiplier for p 2 will be holomorphic in some domain.

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

Detlef MüllerZhenqiu Zhang — 2001

Studia Mathematica

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

Detlef MüllerChristoph Thiele — 2007

Studia Mathematica

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 i t 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...

A.e. convergence of spectral sums on Lie groups

Christopher MeaneyDetlef MüllerElena Prestini — 2007

Annales de l’institut Fourier

Let be a right-invariant sub-Laplacian on a connected Lie group G , and let S R f : = 0 R d E λ f , R 0 , denote the associated “spherical partial sums,” where = 0 λ d E λ is the spectral resolution of . We prove that S R f ( x ) converges a.e. to f ( x ) as R under the assumption log ( 2 + ) f L 2 ( G ) .

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