The main goal of the present paper is to provide sharp hypercontractivity bounds of the heat flow (Hₜ) ₓ ₀ on RCD (0, N) metric measure spaces. The best constant in this estimate involves the asymptotic volume ratio, and its optimality is obtained by means of the sharp L²-logarithmic Sobolev inequality on RCD (0, N) spaces and a blow-down rescaling argument. Equality holds in this sharp estimate for a prescribed time t₀>0 and a non-zero extremizer f if and only if the RCD (0, N) space has an N-Euclidean cone structure and f is a Gaussian whose dilation factor is reciprocal to t₀, up to a multiplicative constant. Applications include an extension of Li's rigidity result, almost rigidities, as well as topological rigidities of non-collapsed RCD (0, N) spaces. Our results are new even on complete Riemannian manifolds with non-negative Ricci curvature.
Honda et al. (Wed,) studied this question.
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