DIRECTIONAL SQUARE FUNCTIONS

Quantitative formulations of Fefferman's counterexample for the ball multiplier are naturally linked to square function estimates for conical and directional multipliers. We develop a novel framework for these square function estimates, based on a directional embedding theorem for Carleson sequences and multiparameter 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 extend to the weighted setting, yielding previously unknown weighted estimates for directional maximal functions and singular integrals.

Automated summary

This article proposes a novel framework for square function estimates based on the directional embedding theorem for Carleson sequences and multiparameter time-frequency analysis techniques. Sharp or quantified bounds for square functions of conical multipliers and multipliers adapted to rectangles pointing along N directions are proven. Furthermore, a new logarithmic bound for the Fourier restriction to an N-gon is derived, and previously unknown weighted estimates for directional maximal functions and singular integrals in the weighted setting are obtained.