Balogh, Z.M., Kristály, A. & Tripaldi, F. (2023) Journal of Functional Analysis [Matematică, Q1]

Publicat: 19 Decembrie 2023

Balogh, Z.M., Kristály, A. & Tripaldi, F. (2023) Sharp log-Sobolev inequalities in CD(0, N) spaces with applications. Journal of Functional Analysis, 286(2), 110217.

DOI: https://doi.org/10.1016/j.jfa.2023.110217

✓ Publisher: Elsevier
✓ Categories: Mathematics
✓ Article Influence Score (AIS): 1.635 (2022) / Q1

Abstract: Given p, N > 1, we prove the sharp L-p-log-Sobolev inequality on noncompact metric measure spaces satisfying the CD(0, N) condition, where the optimal constant involves the asymptotic volume ratio of the space. This proof is based on a sharp isoperimetric inequality in CD(0, N) spaces, symmetrization, and a careful scaling argument. As an application we establish a sharp hypercontractivity estimate for the Hopf-Lax semigroup in CD(0, N) spaces. The proof of this result uses Hamilton-Jacobi inequality and Sobolev regularity properties of the Hopf-Lax semigroup, which turn out to be essential in the present setting of nonsmooth and noncompact spaces. Moreover, a sharp Gaussian-type L-2-log-Sobolev inequality and a hypercontractivity estimate are obtained in RCD(0, N) spaces. Our results are new, even in the smooth setting of Riemannian/Finsler manifolds. In particular, an extension of the celebrated rigidity result of Ni (2004) [55] on Rieman nian manifolds will be a simple consequence of our sharp log-Sobolev inequality.

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