On Sharpness of $L\log L$ Criterion for Weak Type $(1,1)$ boundedness of rough operators

TL;DR

This paper establishes that the $L ext{log}L$ condition is the optimal size criterion for ensuring weak type (1,1) boundedness of certain rough singular integral operators on the sphere, assuming $L^2$ boundedness.

Contribution

It proves that the $L ext{log}L$ hypothesis is the sharpest size condition for weak type (1,1) boundedness of rough operators on the sphere, given $L^2$ boundedness.

Findings

$L ext{log}L$ is the strongest size condition for boundedness.

The result applies to homogeneous rough functions on the sphere.

The paper confirms the optimality of the $L ext{log}L$ criterion.

Abstract

In this note, we show that the $L\log L$ hypothesis is the strongest size condition on a homogeneous rough function on the sphere which ensures the weak type $(1,1)$ boundedness of the corresponding singular integral $T_\Omega$, provided $T_\Omega$ is bounded in $L^2$.