Hilbert transform along measurable vector fields constant on Lipschitz curves : L2 boundedness

Guo, Shaoming

doi:10.2140/apde.2015.8.1263

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Hilbert transform along measurable vector fields constant on Lipschitz curves: $L^2$ boundedness

Shaoming Guo

Vol. 8 (2015), No. 5, 1263–1288

DOI: 10.2140/apde.2015.8.1263

Abstract

We prove the $L^{2}$ boundedness of the directional Hilbert transform in the plane relative to measurable vector fields which are constant on suitable Lipschitz curves. One novelty of our proof lies in the definition of the adapted Littlewood–Paley projection (see Definition 3.3). The other novelty is that we will use Jones’ beta numbers to control certain commutator in the critical Lipschitz regularity (see Lemma 5.5).

Keywords

singular integrals, differentiation theory, Jones' beta numbers, Littlewood–Paley theory on Lipschitz curves, Carleson embedding theorem

Mathematical Subject Classification 2010

Primary: 42B20, 42B25

Milestones

Received: 10 January 2015

Revised: 8 March 2015

Accepted: 15 April 2015

Published: 28 July 2015

Authors

Shaoming Guo
Mathematisches Institut University of Bonn Endenicher Allee 60 D-53115 Bonn Germany

Vol. 8, No. 5, 2015