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Optimal functional inequalities for fractional operators on the sphere and applications. (English) Zbl 1353.35020

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

MSC:

35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
26D15 Inequalities for sums, series and integrals
35R11 Fractional partial differential equations
26D10 Inequalities involving derivatives and differential and integral operators
26A33 Fractional derivatives and integrals
35B33 Critical exponents in context of PDEs
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References:

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