Vol. 8, No. 5, 2015

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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
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