Radial fractional Laplace operators and Hessian inequalities

In this paper we deduce a formula for the fractional Laplace operator (−Δ)s(−Δ)s on radially symmetric functions useful for some applications. We give a criterion of subharmonicity associated with (−Δ)s(−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 FkFk is the k  -Hessian operator on RnRn, 1⩽k