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Duality and interpolation of anisotropic Triebel-Lizorkin spaces. (English) Zbl 1213.42062

Summary: We study properties of anisotropic Triebel-Lizorkin spaces associated with general expansive dilations and doubling measures on \({\mathbb{R}}^n\) using wavelet transforms. This paper is a continuation of [the author, J. Geom. Anal. 17, No. 3, 387–424 (2007; Zbl 1147.42006); Trans. Am. Math. Soc. 358, No. 4, 1469–1510 (2006; Zbl 1083.42016)], where we generalized the isotropic methods of dyadic \(\varphi\)-transforms of M. Frazier and B. Jawerth [J. Funct. Anal. 93, No. 1, 34–170 (1990; Zbl 0716.46031)] to the non-isotropic settings. By working at the level of sequence spaces, we identify the duals of anisotropic Triebel-Lizorkin spaces. We also obtain several real and complex interpolation results for these spaces.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
46B70 Interpolation between normed linear spaces
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47B38 Linear operators on function spaces (general)
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