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Best constant in the weighted Hardy inequality: the spatial and spherical version. (English) Zbl 1127.26013

The sharp constant is obtained for the Hardy-Stein-Weiss inequality for fractional Riesz potential operator in the space \(L^p(\mathbb R^n,\rho)\) with the power \(\rho=| x| ^\beta\). The sharp constant is found for a similar weight inequality for fractional powers of the Beltrami-Laplace operator on the unit sphere. It is given a non-Hilbert version of a Hardy-Rellich-type inequality and a best constant, then an integral form of it in terms of Riesz potential. One of the best constants is calculated as a ratio of gamma functions, where the author deals with the norm of an integral operator in \(L^p(\mathbb R^n)\) with kernel homogeneous and invariant with respect to rotations, or with Catalan formula for integrals.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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