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HILBERT TRANSFORMS ALONG LIPSCHITZ DIRECTION FIELDS: A LACUNARY MODEL

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  09 February 2017

Shaoming Guo  and
Christoph Thiele
Shaoming Guo
Affiliation:
Institute of Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany email shaoguo@iu.edu Department of Mathematics, Indiana University, 831 E Third St, Bloomington, IN 47405, U.S.A.
Christoph Thiele
Affiliation:
Institute of Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany email thiele@math.uni-bonn.de
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Abstract

We prove bounds for the truncated directional Hilbert transform in $L^{p}(\mathbb{R}^{2})$ for any $1<p<\infty$ under a combination of a Lipschitz assumption and a lacunarity assumption. It is known that a lacunarity assumption alone is not sufficient to yield boundedness for $p=2$, and it is a major question in the field whether a Lipschitz assumption alone suffices, at least for some $p$.

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory
Type
Research Article
Information
Mathematika , Volume 63 , Issue 2 , 2017 , pp. 351 - 363
Copyright
Copyright © University College London 2017 

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