Abstract view
Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales


Published:20110608
Printed: Sep 2012
Adam OsÄ™kowski,
Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02097 Warsaw, Poland
Abstract
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
$$ \v\_p \leq C_{p,\infty}
\u\_\infty,\quad
\v\_1 \leq C_{1,p} \u\_p.
$$
In particular, the inequalities are still sharp for the 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.
Keywords: 
harmonic function, conjugate harmonic functions, orthogonal harmonic functions, martingale, orthogonal martingales, norm inequality, optimal stopping problem
harmonic function, conjugate harmonic functions, orthogonal harmonic functions, martingale, orthogonal martingales, norm inequality, optimal stopping problem
