Radial fractional Laplace operators and Hessian inequalities

Abstract

In this paper we deduce a formula for the fractional Laplace operator (−Δ)s on radially symmetric functions useful for some applications. We give a criterion of subharmonicity associated with (−Δ)s, and apply it to a problem related to the Hessian inequality of Sobolev type:∫Rn|(−Δ)kk+1u|k+1dx⩽C∫Rn−uFk[u]dx, where Fk is the k-Hessian operator on Rn, 1⩽k<n2, 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.