Radial fractional Laplace operators and Hessian inequalities

In this paper we deduce a formula for the fractional Laplace operator on radially symmetric functions useful for some applications. We give a criterion of subharmonicity associated with the fractional laplace operator , and apply it to a problem related to the Hessian inequalityof Sobolev type and the k-Hessian operator on under some restrictions on a k-convex function u. In particular, we show that the class of u for which the above inequality was established in Ferrari et al. [5] contains the extremal functions for the Hessian Sobolev inequality of X.-J. Wang (1994) [15]. This is proved using logarithmic convexity of the Gaussian ratio of hypergeometric functions which might be of independent interest.