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