Tamed spaces -- Dirichlet spaces with distribution-valued Ricci bounds

TL;DR

This paper develops a theory of tamed Dirichlet spaces with distribution-valued Ricci bounds, analyzing their properties through singular perturbations, generalized curvature conditions, and applications to manifolds with singularities.

Contribution

It introduces the concept of tamed spaces with distributional Ricci bounds, extending curvature-dimension conditions to singular settings and establishing their equivalence to heat semigroup gradient estimates.

Findings

Distributional Ricci bounds are equivalent to gradient estimates for heat semigroups.

The framework generalizes Bakry-Émery conditions to singular and distributional contexts.

Examples include Riemannian manifolds with interior or boundary singularities.

Abstract

We develop the theory of tamed spaces which are Dirichlet spaces with distribution-valued lower bounds on the Ricci curvature and investigate these from an Eulerian point of view. To this end we analyze in detail singular perturbations of Dirichlet form by a broad class of distributions. The distributional Ricci bound is then formulated in terms of an integrated version of the Bochner inequality using the perturbed energy form and generalizing the well-known Bakry-\'Emery curvature-dimension condition. Among other things we show the equivalence of distributional Ricci bounds to gradient estimates for the heat semigroup in terms of the Feynman-Kac semigroup induced by the taming distribution as well as consequences in terms of functional inequalities. We give many examples of tamed spaces including in particular Riemannian manifolds with either interior singularities or singular boundary behavior.