# Optimal estimates for harmonic functions in the unit ball

Optimal estimates for harmonic functions in the unit ball We find the sharp constants C p and the sharp functions C p  = C p (x) in the inequality $$|u(x)|\leq \frac{C_{p}}{(1-|x|^{2})^{(n-1)/p}} \|u\|_{h^{p}(B^{n})}, u\in h^{p}(B^{n}), x\in B^{n},$$ in terms of Gauss hypergeometric and Euler functions. This extends and improves some results of Axler et al. (Harmonic function theory, New York, 1992), where they obtained similar results which are sharp only in the cases p = 2 and p = 1. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Optimal estimates for harmonic functions in the unit ball

, Volume 16 (4) – Sep 28, 2011
12 pages

/lp/springer_journal/optimal-estimates-for-harmonic-functions-in-the-unit-ball-zHMyg21jwu
Publisher
Springer Journals
Subject
Mathematics; Potential Theory; Operator Theory; Fourier Analysis; Econometrics; Calculus of Variations and Optimal Control; Optimization
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-011-0145-5
Publisher site
See Article on Publisher Site

### Abstract

We find the sharp constants C p and the sharp functions C p  = C p (x) in the inequality $$|u(x)|\leq \frac{C_{p}}{(1-|x|^{2})^{(n-1)/p}} \|u\|_{h^{p}(B^{n})}, u\in h^{p}(B^{n}), x\in B^{n},$$ in terms of Gauss hypergeometric and Euler functions. This extends and improves some results of Axler et al. (Harmonic function theory, New York, 1992), where they obtained similar results which are sharp only in the cases p = 2 and p = 1.

### Journal

PositivitySpringer Journals

Published: Sep 28, 2011

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