Pointwise Characterizations of Even Order Sobolev Spaces via Derivatives of Ball Averages

Let $\ell \in \mathbb{N}$ and $p\in (1,\infty ]$ . In this article, the authors establish several equivalent characterizations of Sobolev spaces $W^{2\ell +2,p}(\mathbb{R}^{n})$ in terms of derivatives of ball averages. The novelty in the results of this article is that these equivalent characterizations reveal some new connections between the smoothness indices of Sobolev spaces and the derivatives on the radius of ball averages and also that, to obtain the corresponding results for higher order Sobolev spaces, the authors first establish the combinatorial equality: for any $\ell \in \mathbb{N}$ and $k\in \{0,\ldots ,\ell -1\}$ , $\sum _{j=0}^{2\ell }(-1)^{j}\binom{2\ell }{j}|\ell -j|^{2k}=0$ .