Abstract
Abstract For exponents in the subcritical range, we revisit some optimal interpolation inequalities on the sphere with carré du champ methods and use the remainder terms to produce improved inequalities. The method provides us with lower estimates of the optimal constants in the symmetry breaking range and stability estimates for the optimal functions. Some of these results can be reformulated in the Euclidean space using the stereographic projection.
Highlights
Let us consider the sphere Sd endowed with the uniform probability measure d μ
An explicit lower bound for μ(λ) has been obtained in [15, Proposition 8]
It turns out that the whole range (8) for any d ≥ 1 can be covered as a consequence of Theorem 1 with a lower bound for μ(λ) which is increasing with respect to λ ≥ 1 and such that it takes the value 1 if λ = 1
Summary
Let us consider the sphere Sd endowed with the uniform probability measure d μ. In the limit case p = 2∗, with d ≥ 3, the inequality holds with optimal constant μ(λ) = min{λ, 1} and it is the Sobolev inequality on Sd when λ = 1. We will call (3) the Gagliardo-Nirenberg-Sobolev interpolation inequality. Emery in [2, 3], using the carré du champ method, where 2# is the Bakry-. We write down more precise estimates and draw some interesting consequences of (5), such as lower estimates for the best constants in (1) and (2) or improved weighted Gagliardo-Nirenberg inequalities in the Euclidean space Rd. The improved inequality (5), with φ(s) > s for s = 0, can be considered as a stability result for (3) in the sense that it can be rewritten as p d −. The right-hand side of the inequality is a measure of the distance to the optimal functions, which are the constant functions: see Appendix A for details
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