  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.
