In this article, the authors study weighted anisotropic Besov and Triebel-Lizorkin spaces associated with expansive dilations and -weights. The authors show that elements of these spaces are locally integrable when the smoothness parameter~ is positive. The authors also characterize these spaces for small values of~ 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].
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Baode Li, Marcin Bownik, Dachun Yang, Wen Yuan, A Mean Characterization of Weighted Anisotropic Besov and Triebel-Lizorkin Spaces. Z. Anal. Anwend. 33 (2014), no. 2, pp. 125–147DOI 10.4171/ZAA/1503