Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales

Osękowski, Adam

Canadian mathematical bulletin, Tome 55 (2012) no. 3, pp. 597-610

Résumé

We determine the best constants ${{C}_{p,\infty }}$ and ${{C}_{1,p}},\,1\,<\,p\,<\,\infty $ , for which the following holds. If $u,v$ are orthogonal harmonic functions on a Euclidean domain such that $v$ is differentially subordinate to $u$ , then $${{\left\| v \right\|}_{p}}\le {{C}_{p}}{{,}_{\infty }}{{\left\| u \right\|}_{\infty }},\,\,\,\,\,\,\,\,\,\,\,{{\left\| v \right\|}_{1}}\le {{C}_{1,p}}{{\left\| u \right\|}_{p}}.$$ In particular, the inequalities are still sharp for the conjugate harmonic 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.

Détail

DOI : 10.4153/CMB-2011-113-8

Mots-clés : 31B05, 60G44, 60G40, harmonic function, conjugate harmonic functions, orthogonal harmonic functions, martingale, orthogonal martingales, norm inequality, optimal stopping problem