BEST CONSTANT IN ZYGMUND'S INEQUALITY AND RELATED ESTIMATES FOR ORTHOGONAL HARMONIC FUNCTIONS AND MARTINGALES

Title & Authors
BEST CONSTANT IN ZYGMUND'S INEQUALITY AND RELATED ESTIMATES FOR ORTHOGONAL HARMONIC FUNCTIONS AND MARTINGALES

Abstract
For any $\small{K}$ > $\small{2/{\pi}}$ we determine the optimal constant $\small{L(K)}$ for which the following holds. If $\small{u}$, $\small{tilde{u}}$ are conjugate harmonic functions on the unit disc with $\small{\tilde{u}(0)=0}$, then $\small{ {\int}_{-\pi}^{\pi}{\mid}\tilde{u}(e^{i\phi}){\mid}\frac{d{\phi}}{2{\pi}}{\leq}K{\int}_{-\pi}^{\pi}{\mid}u(e^{i{\phi}}){\mid}{\log}^+{\mid}u(e^{i{\phi}}){\mid}\frac{d{\phi}}{2{\pi}}+L(K).}$ We also establish a related estimate for orthogonal harmonic functions given on Euclidean domains as well as an extension concerning orthogonal martingales under differential subordination.
Keywords
harmonic function;martingale;LlogL inequality;differential sub-ordination;best constants;
Language
English
Cited by
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