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
directions. A suitable combination of these estimates yields a new and
currently best-known logarithmic bound for the Fourier restriction to an
-gon,
improving on previous results of A. Córdoba. Our directional Carleson embedding
extends to the weighted setting, yielding previously unknown weighted estimates for
directional maximal functions and singular integrals.
Keywords
directional operators, directional square functions, Rubio
de Francia inequalities, directional Carleson embedding
theorems, polygon multiplier