Abstract
In this paper we extend complex uniform convexity estimates for \mathbb{C} to \mathbb{R}^n and determine best constants. Furthermore, we provide the link to log-Sobolev inequalities and hypercontractivity estimates for ultraspherical measures.
Highlights
The starting point of this paper is the Bonami’s sharp complex convexity estimate
The proofs in [Wei80], respectively [Ale07] of (2) are complex analytic in nature; they don’t seem to work in higher dimensions, where S1 is replaced by the unit sphere in Rn and where m is replaced by σ, the normalized Haar measure on the unit sphere in Rn
First we prove a theorem on the unit circle for ultraspherical measures dνm(z) = cm| sin(θ)|mdθ for all real m > −1, where z
Summary
The starting point of this paper is the Bonami’s sharp complex convexity estimate The proofs in [Wei80], respectively [Ale07] of (2) are complex analytic in nature; they don’t seem to work in higher dimensions, where S1 is replaced by the unit sphere in Rn and where m is replaced by σ , the normalized Haar measure on the unit sphere in Rn. Recently, [LMS19] we recorded a proof of (2), based on Green’s identities and sub-harmonicity estimates such as,. Functional inequalities, hypercontractivity, ulstraspherical measure, best constants. Extension of (2) to higher dimensions, the main theorem of our paper, in an equivalent way can be restated as (4) implies (3) when (m, p, q, r) = (n − 2, p, 2, r) with n ≥ 2, n ∈ N, and 2 ≥ p > 0. Despite of partial progresses the description of all 4-tuples (m, p, q, r) for which the hypercontractivity (3) holds true remains an open question
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