Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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BEST CONSTANT IN ZYGMUND'S INEQUALITY AND RELATED ESTIMATES FOR ORTHOGONAL HARMONIC FUNCTIONS AND MARTINGALES

  • Osekowski, Adam (Department of Mathematics, Informatics and Mechanics University of Warsaw)
  • Received : 2011.01.14
  • Published : 2012.05.01
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Abstract

For any $K$ > $2/{\pi}$ we determine the optimal constant $L(K)$ for which the following holds. If $u$, $tilde{u}$ are conjugate harmonic functions on the unit disc with $\tilde{u}(0)=0$, then $$ {\int}_{-\pi}^{\pi}{\mid}\tilde{u}(e^{i\phi}){\mid}\frac{d{\phi}}{2{\pi}}{\leq}K{\int}_{-\pi}^{\pi}{\mid}u(e^{i{\phi}}){\mid}{\log}^+{\mid}u(e^{i{\phi}}){\mid}\frac{d{\phi}}{2{\pi}}+L(K).$$ 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.

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References

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