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Search: MSC category 31B05 ( Harmonic, subharmonic, superharmonic functions )

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1. CMB 2016 (vol 60 pp. 146)

Khavinson, Dmitry; Lundberg, Erik; Render, Hermann
The Dirichlet Problem for the Slab with Entire Data and a Difference Equation for Harmonic Functions
It is shown that the Dirichlet problem for the slab $(a,b) \times \mathbb{R}^{d}$ with entire boundary data has an entire solution. The proof is based on a generalized Schwarz reflection principle. Moreover, it is shown that for a given entire harmonic function $g$ the inhomogeneous difference equation $h ( t+1,y) -h (t,y) =g ( t,y)$ has an entire harmonic solution $h$.

Keywords:reflection principle, entire harmonic function, analytic continuation
Categories:31B20, 31B05

2. 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

3. CMB 2003 (vol 46 pp. 252)

Miyamoto, Ikuko; Yanagishita, Minoru; Yoshida, Hidenobu
Beurling-Dahlberg-Sjögren Type Theorems for Minimally Thin Sets in a Cone
This paper shows that some characterizations of minimally thin sets connected with a domain having smooth boundary and a half-space in particular also hold for the minimally thin sets at a corner point of a special domain with corners, {\it i.e.}, the minimally thin set at $\infty$ of a cone.

Categories:31B05, 31B20

4. CMB 1998 (vol 41 pp. 257)

Bagby, Thomas; Gauthier, P. M.
Note on the support of Sobolev functions
We prove a topological restriction on the support of Sobolev functions.

Categories:46E35, 31B05

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