Dimension-free Harnack inequality suffices for sharp upper Gaussian heat kernel estimates on infinitesimally Hilbertian metric measure spaces, with local logarithmic Sobolev inequality as an intermediate step, claimed new even in RCD(K,∞) spaces.
Nonsmooth differential geometry - An approach tailored for spaces with Ricci curvature bounded from below
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abstract
We discuss in which sense general metric measure spaces possess a first order differential structure. Building on this, we then see that on spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting to define Hessian, covariant/exterior derivatives and Ricci curvature.
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2019 1verdicts
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From Harnack inequality to heat kernel estimates on metric measure spaces and applications
Dimension-free Harnack inequality suffices for sharp upper Gaussian heat kernel estimates on infinitesimally Hilbertian metric measure spaces, with local logarithmic Sobolev inequality as an intermediate step, claimed new even in RCD(K,∞) spaces.