A Note on Lipschitz Continuity of Solutions of Poisson Equations in Metric Measure Spaces

Abstract

In this note we show how to adjust some proofs of Koskela et. al 2003 and Jiang 2011 in order to show that in certain spaces $(X,d,\mu )$, like $RCD(K,N)$-spaces, every Sobolev function with local $L^p$-Laplacian and $p>\dim \mu$ is locally Lipschitz continuous.

8 pages, corrected proof and included new Lemma for RCD(K,N)-spaces

References in corpus (1)

• Cheeger-harmonic functions in metric measure spaces revisited

Cited by in corpus (4)

• Harmonic functions on metric measure spaces
• The behavior of harmonic functions at singular points of $\mathsf{RCD}$ spaces
• Cheeger-harmonic functions in metric measure spaces revisited
• A Bishop type inequality on metric measure spaces with Ricci curvature bounded below