Directional square functions

Abstract

Quantitative formulations of Fefferman's counterexample for the ball multiplier are naturally linked to square function estimates for conical and directional multipliers. In this article we develop a novel framework for these square function estimates, based on a directional embedding theorem for Carleson sequences and multi-parameter time-frequency analysis techniques. As applications we prove sharp or quantified bounds for Rubio de Francia type square functions of conical multipliers and of multipliers adapted to rectangles pointing along $N$ directions. A suitable combination of these estimates yields a new and currently best-known logarithmic bound for the Fourier restriction to an $N$-gon, improving on previous results of A. Cordoba. Our directional Carleson embedding extends to the weighted setting, yielding previously unknown weighted estimates for directional maximal functions and singular integrals.

arXiv comment49 pages, 4 figures. Final version to appear in Analysis&PDE. arXiv admin note: text overlap with arXiv:1902.03644

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Cited by in corpus (2)

• Singular integrals along lacunary directions in ${\mathbb{R}}^n$
• Maximal subspace averages