Optimal control of singular Fourier multipliers by maximal operators

Bennett, Jonathan

doi:10.2140/apde.2014.7.1317

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Optimal control of singular Fourier multipliers by maximal operators

Jonathan Bennett

Vol. 7 (2014), No. 6, 1317–1338

DOI: 10.2140/apde.2014.7.1317

Abstract

We control a broad class of singular (or “rough”) Fourier multipliers by geometrically defined maximal operators via general weighted $L^{2} (ℝ)$ norm inequalities. The multipliers involved are related to those of Coifman, Rubio de Francia and Semmes, satisfying certain weak Marcinkiewicz-type conditions that permit highly oscillatory factors of the form $e^{i | ξ |^{α}}$ for both $α$ positive and negative. The maximal functions that arise are of some independent interest, involving fractional averages associated with tangential approach regions (related to those of Nagel and Stein), and more novel “improper fractional averages” associated with “escape” regions. Some applications are given to the theory of $L^{p}$ – $L^{q}$ multipliers, oscillatory integrals and dispersive PDE, along with natural extensions to higher dimensions.

Dedicated to the memory of Adela Moyua, 1956–2013.

Keywords

Fourier multipliers, maximal operators, weighted inequalities

Mathematical Subject Classification 2010

Primary: 42B15, 42B25, 42B20

Secondary: 42B37

Milestones

Received: 7 June 2013

Accepted: 12 July 2014

Published: 18 October 2014

Authors

Jonathan Bennett
School of Mathematics The University of Birmingham The Watson Building Edgbaston Birmingham B15 2TT United Kingdom

Vol. 7, No. 6, 2014