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 Sⁿ[EQUATION PRESENTED] where L_s denotes a fractional Laplace operator of order s (0, n), q [1, 2) ∪ (2, q+], q+ = 2n n.s is a critical exponent, and dμ is the uniform probability measure on Sn. These inequalities are established with optimal constants using spectral properties of fractional operators. Their consequences for fractional heat flows are considered. If q > 2, these inequalities interpolate between fractional Sobolev and subcritical fractional logarithmic Sobolev inequalities, which correspond to the limit case as q → 2. For q < 2, the inequalities interpolate between fractional logarithmic Sobolev and fractional Poincare inequalities. In the subcritical range q < q+, the method also provides us with remainder terms which can be considered as an improved version of the optimal inequalities. The case s o (.n, 0) is also considered. Finally, weighted inequalities involving the fractional Laplacian are obtained in the Euclidean space, by using the stereographic projection.