Maximal directional singular integrals
Abstract
We survey maximal directional averages and directional singular integrals, and their connections with Kakeya--Nikodym phenomena, differentiation theory and the Carleson operator. The exposition is organized by the structure of the set of directions: lacunary and arbitrary finite sets, vector fields as in the conjectures of Zygmund and Stein, and, in higher dimensions, algebraic sets of directions and subspaces. The unifying tool is almost orthogonality, which rests on the ordering of the circle in the plane and on polynomial partitioning in higher dimensions. We explain where it yields sharp or nearly sharp bounds and where new ideas are needed, notably for multiscale averages along algebraic curves and for vector-valued single-annulus estimates, and we collect open problems.
Related
- Weak type (1,1) estimates for noncommutative maximal truncated rough singular integrals
- Weighted Norm Inequalities for Maximal Multi-sublinear Rough Singular Integrals
- SPARSE POINTWISE BOUNDS FOR MAXIMAL TRUNCATIONS OF ROUGH SINGULAR INTEGRALS AND SOBOLEV-TYPE INEQUALITIES
- Sparse bounds for maximal rough singular integrals
- The Singular Set of Minima of Multiple Integrals
BibTeX
@misc{plinio2026maximal,
title = {{Maximal directional singular integrals}},
author = {Francesco Di Plinio and Ioannis Parissis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02483},
eprint = {2610.02483},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02483},
}
RIS
TY - UNPB
TI - Maximal directional singular integrals
AU - Francesco Di Plinio
AU - Ioannis Parissis
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02483
UR - https://arxiv.org/abs/2610.02483
AB - We survey maximal directional averages and directional singular integrals, and their connections with Kakeya--Nikodym phenomena, differentiation theory and the Carleson operator. The exposition is organized by the structure of the set of directions: lacunary and arbitrary finite sets, vector fields as in the conjectures of Zygmund and Stein, and, in higher dimensions, algebraic sets of directions and subspaces. The unifying tool is almost orthogonality, which rests on the ordering of the circle in the plane and on polynomial partitioning in higher dimensions. We explain where it yields sharp or nearly sharp bounds and where new ideas are needed, notably for multiscale averages along algebraic curves and for vector-valued single-annulus estimates, and we collect open problems.
ER -