A Mean Characterization of Weighted Anisotropic Besov and Triebel-Lizorkin Spaces

Abstract

In this article, the authors study weighted anisotropic Besov and Triebel-Lizorkin spaces associated with expansive dilations and $A_{\infty }$-weights. The authors show that elements of these spaces are locally integrable when the smoothness parameter $\alpha$ is positive. The authors also characterize these spaces for small values of $\alpha$ in terms of a mean square function recently introduced in the context of Sobolev spaces in [Math. Ann. 354 (2012), 589–626] and isotropic Triebel–Lizorkin spaces in [Publ. Mat. 57 (2013), 57–82].