Abstract
We prove higher summability and regularity of \(\Gamma \big (f\big )\) for functions \(f\) in spaces satisfying the Bakry–Émery condition \(\mathsf{BE}(K,\infty )\). As a byproduct, we obtain various equivalent weak formulations of \(\mathsf{BE}(K,N)\) and we prove the Local-to-Global property of the \(\mathsf{RCD}^*(K,N)\) condition in locally compact metric measure spaces \((X,\mathsf{d},\mathfrak m)\), without assuming a priori the non-branching condition on the metric space.
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The authors acknowledge the support of the ERC ADG GeMeThNES and warmly thank Tapio Rajala for his comments on a preliminary version of this paper.
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Communicated by Marco Abate.
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Ambrosio, L., Mondino, A. & Savaré, G. On the Bakry–Émery Condition, the Gradient Estimates and the Local-to-Global Property of \(\mathsf{RCD}^*(K,N)\) Metric Measure Spaces. J Geom Anal 26, 24–56 (2016). https://doi.org/10.1007/s12220-014-9537-7
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DOI: https://doi.org/10.1007/s12220-014-9537-7