Weighted weak-type inequality for martingales

Adam Osękowski¹

¹ Department of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 65 (2017) no. 2, pp. 165-175.

Résumé

Let $X=(X_t)_{t\geq 0}$ be a bounded martingale and let $Y=(Y_t)_{t\geq 0}$ be differentially subordinate to $X$. We prove that if $1\leq p \lt \infty $ and $W=(W_t)_{t\geq 0}$ is an $A_p$ weight of characteristic $[W]_{A_p}$, then $$ \| Y\| _{L^{p,\infty }(W)}\leq C_p[W]_{A_p}\| X\| _{L^\infty (W)}.$$ The linear dependence on $[W]_{A_p}$ is shown to be the best possible. The proof exploits a weighted exponential bound which is of independent interest. As an application, a related estimate for the Haar system is established.

Détail

DOI : 10.4064/ba8096-11-2017

Keywords: geq bounded martingale geq differentially subordinate prove leq infty geq weight characteristic infty leq infty linear dependence shown best possible proof exploits weighted exponential bound which independent interest application related estimate haar system established