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Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales

  Published:2011-06-08
 Printed: Sep 2012
  • Adam Osękowski,
    Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
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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
MSC Classifications: 31B05, 60G44, 60G40 show english descriptions Harmonic, subharmonic, superharmonic functions
Martingales with continuous parameter
Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
31B05 - Harmonic, subharmonic, superharmonic functions
60G44 - Martingales with continuous parameter
60G40 - Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
 

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