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Weighted weak-type inequality for martingales

Volume 65 / 2017

Adam Osękowski Bulletin Polish Acad. Sci. Math. 65 (2017), 165-175 MSC: Primary 60G44; Secondary 60G42. DOI: 10.4064/ba8096-11-2017 Published online: 27 November 2017


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.


  • Adam OsękowskiDepartment of Mathematics, Informatics and Mechanics
    University of Warsaw
    Banacha 2
    02-097 Warszawa, Poland

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