Optimal estimates for harmonic functions in the unit ball

Details Optimal estimates for harmonic functions in the unit ball Optimal estimates for harmonic functions in the unit ball

2011-02-19

arxiv.org/abs/1102.3995v1

Mathematics

Analysis of PDEs

9 pages

Scientific paper

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, Bourdon and Ramey (\cite{ABR}), where they obtained similar results which are sharp only in the cases $p=2$ and $p=1$.

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