On a counterexample related to 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 $t\log \log ({\mathrm{e}}^{\mathrm{e}}+t)$. Our proof is based on a construction of M.C. Reguera and C. Thiele.