Logarithmic L^p bounds for maximal directional singular integrals in the plane

TL;DR

This paper establishes logarithmic bounds for maximal directional singular integrals in the plane across various sets of directions, using harmonic analysis and time-frequency methods, with sharp results for specific structured sets.

Contribution

It provides new logarithmic bounds for maximal singular integrals over arbitrary direction sets and sharp bounds for lacunary and Vargas sets, advancing understanding in harmonic analysis.

Findings

Logarithmic bounds for arbitrary direction sets in L^p

Sharp bounds for lacunary and Vargas sets

Development of an L^p almost orthogonality principle

Abstract

We discuss the L^p-boundedness of maximal singular integrals in the plane over a finite set V of N directions. Logarithmic bounds are established for a set V of arbitrary structure in the 2<=p<infinity range. Sharp bounds are proved for lacunary and Vargas sets of directions. The latter include the case of uniformly distributed directions and the finite truncations of the Cantor set. We make use of both classical harmonic analysis methods and product-BMO based time-frequency analysis techniques. As a further application of the latter, we derive an L^p almost orthogonality principle for Fourier restrictions to cones.