In this talk, we will first introduce an extension of the classical Weyl’s lemma of very weakly harmonic functions to the setting of metric measure spaces satisfying a generalized Ricci lower bound. As its applications, we will give some new results for Lipschitz, $W^{1,p}$-estimates for linear PDEs on Euclidean spaces. This is a joint work with Yu Peng, Xiandong Sun and Xi-Ping Zhu.