Dimension-free Harnack inequalities on \(\mathrm{RCD}(K, \infty )\) spaces (Q501819)

In [Probab. Theory Relat. Fields 109, No. 3, 417--424 (1997; Zbl 0887.35012)], \textit{F.-Y. Wang} introduced the dimension-free Harnack inequality for the diffusion (in particular, heat) semigroup on a smooth Riemannian manifold \((M,d)\) (with the Ricci curvature bounded from below by a constant \(K\in\mathbb{R}\)), which can be formulated as \[ \left|(P_tf)(x)\right|^p\leq \left(P_t|f|^p \right)(y)\exp\left\{\frac{pKd(x,y)^2}{2(p-1)(e^{2Kt}-1)} \right\}, \] where \(f\in C_b(M)\), \(x,y\in M,\) \(p>1\). The author extends this result to the case of the heat semigroup on non-smooth metric measure spaces with Riemannian curvature bounded from below, the so-called \(\mathrm{RCD}(K,\infty)\)-spaces. The proof extends the ideas of the original proof of Wang [loc.\,cit.]. The obtained dimension-free Harnack inequality on \(\mathrm{RCD}(K,\infty)\)-spaces leads to the corresponding extensions of the log-Harnack, entropy-cost and log-Sobolev inequalities. The paper contains a detailed introduction and all needed preliminaries.