On higher order Morrey's inequalities over CD{\boldmath$(K,N)$}-spaces

Abstract

In this paper, we establish a higher order Morrey's inequality in the framework of $\mathsf{CD}(K,N)$-spaces for $K\in \mathbb{R}$ and $N\in \mathbb{N}$. We do so by first introducing an alternate version of second order Sobolev spaces in general metric measure spaces, which contains amply many functions even when $p>N$. These spaces contain the traditional $H^{2,p}(X)$ spaces on $\mathsf{RCD}(K,N)$-spaces.

arXiv comment33 pages, some results are now valid in the setting of CD spaces, title change. Comments welcome!