I will talk about a ``universal'' regularization property of heat semigroups motivated  by an original work of  Nick Dungey on graphs. Namely in the settings of Riemannian manifolds and discrete graphs, this property is the $L^p$ ($1 < p /le2$) boundedness for the gradient of the heat semigroup. On Dirichlet spaces, the counterpart is the continuity of the heat semigroup on heat semigroup-based Besov spaces. I will also discuss their applications on the study of Riesz transforms and critical exponents of Besov spaces.

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