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

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