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We develop the theory of tamed spaces which are Dirichlet spaces with distributionvalued 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-Émery 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. MATTHIAS ERBAR, CHIARA RIGONI, KARL-THEODOR STURM, AND LUCA TAMANINI 4.2. A manifold which is tamed but not 2-tamed 36 4.3. Manifolds with boundary and potentially singular curvature 36 4.4. A tamed domain with boundary that is not semiconvex 40 4.5. A Tamed Manifold with Highly Irregular Boundary 41 5. Functional Inequalities for Tamed Spaces 45 6. Self-Improvement of the Taming Condition 47 6.1. Measure-Valued Taming Operator and Bochner Inequality 47 6.2. Self-Improvement of the L 2 -Taming Condition 51 7. Sub-tamed Spaces 55 7.1. Reflected Dirichlet Spaces and Sub-taming 56 7.2. Doubling of Dirichlet Spaces and Sub-taming 58 7.3. Doubling of Riemannian Surfaces 64 References 66

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