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Search: All articles in the CMB digital archive with keyword harmonic functions

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1. CMB 2011 (vol 55 pp. 597)

Osękowski, Adam
Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales
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
Categories:31B05, 60G44, 60G40

2. CMB 1998 (vol 41 pp. 129)

Lee, Young Joo
Pluriharmonic symbols of commuting Toeplitz type operators on the weighted Bergman spaces
A class of Toeplitz type operators acting on the weighted Bergman spaces of the unit ball in the $n$-dimensional complex space is considered and two pluriharmonic symbols of commuting Toeplitz type operators are completely characterized.

Keywords:Pluriharmonic functions, Weighted Bergman spaces, Toeplitz type operators.
Categories:47B38, 32A37

3. CMB 1997 (vol 40 pp. 60)

Khavinson, Dmitry
Cauchy's problem for harmonic functions with entire data on a sphere
We give an elementary potential-theoretic proof of a theorem of G.~Johnsson: all solutions of Cauchy's problems for the Laplace equations with an entire data on a sphere extend harmonically to the whole space ${\bf R}^N$ except, perhaps, for the center of the sphere.

Keywords:harmonic functions, Cauchy's problem, homogeneous harmonics
Categories:35B60, 31B20

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