Optimal Functional Inequalities for Fractional Operators on the Sphere and Applications

This paper is devoted to the family of optimal functional inequalities on the n-dimensional sphere [Formula: see text], namely [Formula: see text] where [Formula: see text] denotes a fractional Laplace operator of order [Formula: see text], [Formula: see text], [Formula: see text] is a critical exponent, and [Formula: see text] is the uniform probability measure on [Formula: see text]. These inequalities are established with optimal constants using spectral properties of fractional operators. Their consequences for fractional heat flows are considered. If [Formula: see text], these inequalities interpolate between fractional Sobolev and subcritical fractional logarithmic Sobolev inequalities, which correspond to the limit case as [Formula: see text]. For [Formula: see text], the inequalities interpolate between fractional logarithmic Sobolev and fractional Poincaré inequalities. In the subcritical range [Formula: see text], the method also provides us with remainder terms which can be considered as an improved version of the optimal inequalities. The case [Formula: see text] is also considered. Finally, weighted inequalities involving the fractional Laplacian are obtained in the Euclidean space, by using the stereographic projection.