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

We show that the Hilbert transform does not map $L^1(M_Φw)$ to $L^{1,\infty}(w)$ for every Young function $Φ$ 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.

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