Weyl’s lemma on RCD(K, N) metric measure spaces

In this paper, we extend the classical Weyl’s lemma to RCD(K, N) metric measure spaces. As its applications, we show the local regularity of solutions for Poisson equations and a Liouville-type result for \(L^1\) $L^1$ very weak harmonic functions on RCD(K, N) spaces. Meanwhile, as an application of regularity theory on non-smooth settings, we obtain a gradient estimate for solutions to a class of elliptic equations with discontinuous coefficients.