Martingale transform and Square function: some end-point weak weighted estimates

Paata Ivanisvili, Alexander Volberg

Published 2017-11-28Version 1

Following the ideas of [8] we prove that there is a sequence of weights $w\in A^d_1$ such that $[w]^d_{A_1}\to \infty$, and martingale transforms $T$ such that $\|T: L^1(w) \to L^{1, \infty}(w)\| \ge c [w]^d_{A_1}\log [w]^d_{A_1}$ with an absolute positive $c$. We also show the existence of the sequence of weights such that $\|S\|_{w} \ge c \|M^d\|_{w}\sqrt{\log \|M^d\|_{w}}$. Finally we show that for any weight $w\in A^d_2$ and any characteristic function of a dyadic interval $\|S_w\chi_I\|_{L^{2, \infty}(w^{-1})} \le C \sqrt{[w]_{A^d_2}}\, \|\chi_I\|_w$. So on characteristic functions at least, no logarithmic correction is needed.