On a counterexample related to weighted weak type estimates for singular integrals

TL;DR

This paper demonstrates a counterexample showing that the Hilbert transform fails to map certain weighted Lebesgue spaces to weak Lebesgue spaces under specific growth conditions of the Young function.

Contribution

It provides a counterexample for weighted weak type estimates of the Hilbert transform, clarifying limitations of existing boundedness results.

Findings

Hilbert transform does not map $L^1(M_{\Phi}w)$ to $L^{1, } (w)$ for certain Young functions

Counterexample based on construction by Reguera and Thiele

Limits on weighted weak type estimates for singular integrals

Abstract

We show that the Hilbert transform does not map $L^1(M_{\Phi}w)$ to $L^{1,\infty}(w)$ for every Young function $\Phi$ growing more slowly than $t\log\log ({\rm e}^{\rm e}+t)$. Our proof is based on a construction of M.C. Reguera and C. Thiele.