Let (X, d, µ) be a RCD * (K, N) space with K ∈ R and N ∈ [1, ∞). We derive the upper and lower bounds of the heat kernel on (X, d, µ) by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in time. For applications, we study the large time behavior of the heat kernel, the stability of solutions to the heat equation, and show the L p boundedness of (local) Riesz transforms.

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New lower bounds of the first nonzero eigenvalue of the weighted p-Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the m-Bakry-Émery Ricci curvature, the Escober-Lichnerowicz-Reilly type estimates are proved; under the assumption of nonnegative ∞-Bakry-Émery Ricci curvature and the m-Bakry-Émery Ricci curvature bounded from below by a non-positive constant, the Li-Yau type lower bound estimates are given. The weighted p-Bochner formula and the weighted p-Reilly formula are derived as the key tools for the establishment of the above results. Mathematics Subject Classification (2010). Primary 58J05, 58J50; Secondary 35J92.