Characterizations of Besov and Triebel-Lizorkin Spaces via Averages on Balls

Abstract

Let $\ell \in \mathbb{N}$ and $p\in (1,\infty ]$. In this article, the authors prove that the sequence $\{f-B_{\ell ,2^{-k}}f{\}}_{k\in \mathbb{Z}}$ consisting of the differences between $f$ and the ball average $B_{\ell ,2^{-k}}f$ characterizes the Besov space $\dot B^α_{p,q}(\rn)$ with $q\in (0,\infty ]$ and the Triebel-Lizorkin space $\dot F^α_{p,q}(\rn)$ with $q\in (1,\infty ]$ when the smoothness order $\alpha \in (0,2\ell )$. More precisely, it is proved that $f-B_{\ell ,2^{-k}}f$ plays the same role as the approximation to the identity ${\phi }_{2^{-k}}\ast f$ appearing in the definitions of $\dot B^α_{p,q}(\rn)$ and $\dot F^α_{p,q}(\rn)$. The corresponding results for inhomogeneous Besov and Triebel-Lizorkin spaces are also obtained. These results, for the first time, give a way to introduce Besov and Triebel-Lizorkin spaces with any smoothness order in $(0,2\ell )$ on spaces of homogeneous type, where $\ell \in \mathbb{N}$.

arXiv comment21 pages, Submitted