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

Caldarelli, Marcela; Lerner, Andrei; Ombrosi, Sheldy

doi:10.1090/proc/13496

On a counterexample related to weighted weak type estimates for singular integrals
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by Marcela Caldarelli, Andrei K. Lerner and Sheldy Ombrosi PDF

Proc. Amer. Math. Soc. 145 (2017), 3005-3012 Request permission

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

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