Abstract
This paper is devoted to the family of optimal functional inequalities on the n-dimensional sphere , namely ∥F∥Lq(𝕊n)2-∥F∥L2(𝕊n)2q-2≤𝖢q,s∫𝕊nFℒsF𝑑μ for all F∈Hs/2(𝕊n), 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.
Highlights
Introduction and main resultsLet us consider the unit sphere Sn with n ≥ 1 and assume that the measure dμ is the uniform probability measure, which is the measure induced on Sn by Lebesgue’s measure on Rn+1, up to a normalization constant
We find that k−1 βj (x) j=0 with βj(x) := n + j − x + j + x and observe that αk is positive
A striking feature of inequality (7) is that the optimal constant Cq,s is determined by a linear eigenvalue problem, the problem is definitely nonlinear
Summary
∀ F ∈ Hs/2(Sn) where Ls denotes a fractional Laplace operator of order s ∈ (0, n), q ∈ [1, 2) ∪ (2, q ], q = 2 n/(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. Weighted inequalities involving the fractional Laplacian are obtained in the Euclidean space, using a stereographic projection
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