Optimal Functional Inequalities for Fractional Operators on the Sphere and Applications

Abstract

Abstract This paper is devoted to the family of optimal functional inequalities on the n-dimensional sphere 𝕊 n ${{\mathbb{S}}^{n}}$ , namely ∥ F ∥ L q ⁢ ( 𝕊 n ) 2 - ∥ F ∥ L 2 ⁢ ( 𝕊 n ) 2 q - 2 ≤ 𝖢 q , s ⁢ ∫ 𝕊 n F ⁢ ℒ s ⁢ F ⁢ 𝑑 μ   for all ⁢ F ∈ H s / 2 ⁢ ( 𝕊 n ) , $\frac{\lVert F\rVert_{\mathrm{L}^{q}({\mathbb{S}}^{n})}^{2}-\lVert F\rVert_{% \mathrm{L}^{2}({\mathbb{S}}^{n})}^{2}}{q-2}\leq\mathsf{C}_{q,s}\int_{{\mathbb{% S}}^{n}}{F\mathcal{L}_{s}F}\,d\mu\quad\text{for all }F\in\mathrm{H}^{s/2}({% \mathbb{S}}^{n}),$ where ℒ s ${\mathcal{L}_{s}}$ denotes a fractional Laplace operator of order s ∈ ( 0 , n ) ${s\in(0,n)}$ , q ∈ [ 1 , 2 ) ∪ ( 2 , q ⋆ ] ${q\in[1,2)\cup(2,q_{\star}]}$ , q ⋆ = 2 ⁢ n n - s ${q_{\star}=\frac{2n}{n-s}}$ is a critical exponent, and d ⁢ μ ${d\mu}$ is the uniform probability measure on 𝕊 n ${{\mathbb{S}}^{n}}$ . These inequalities are established with optimal constants using spectral properties of fractional operators. Their consequences for fractional heat flows are considered. If q > 2 ${q>2}$ , these inequalities interpolate between fractional Sobolev and subcritical fractional logarithmic Sobolev inequalities, which correspond to the limit case as q → 2 ${q\to 2}$ . For q < 2 ${q<2}$ , the inequalities interpolate between fractional logarithmic Sobolev and fractional Poincaré inequalities. In the subcritical range q < q ⋆ ${q<q_{\star}}$ , the method also provides us with remainder terms which can be considered as an improved version of the optimal inequalities. The case s ∈ ( - n , 0 ) ${s\in(-n,0)}$ is also considered. Finally, weighted inequalities involving the fractional Laplacian are obtained in the Euclidean space, by using the stereographic projection.