Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales

Canadian mathematical bulletin

Adam Osękowski¹ 

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

关键词: harmonic function;    conjugate harmonic functions;    orthogonal harmonic functions;    martingale;    orthogonal martingales;    norm inequality;    optimal stopping problem;   

DOI  :  10.4153/CMB-2011-113-8

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

We determine the best constants $C_{p,infty}$ and $C_{1,p}$,$1 < p < infty$, for which the following holds. If $u$, $v$ areorthogonal harmonic functions on a Euclidean domain such that $v$ isdifferentially subordinate to $u$, then $$ |v|_p leq C_{p,infty}|u|_infty,quad|v|_1 leq C_{1,p} |u|_p.$$In particular, the inequalities are still sharp for the conjugateharmonic functions on the unit disc of $mathbb R^2$. Sharp probabilistic versions of these estimates are also studied. As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund.

【 授权许可】

Unknown