Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces

Bownik, Marcin; Ho, Kwok-Pun

doi:10.1090/S0002-9947-05-03660-3

Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces
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by Marcin Bownik and Kwok-Pun Ho PDF

Trans. Amer. Math. Soc. 358 (2006), 1469-1510 Request permission

Abstract:

Weighted anisotropic Triebel-Lizorkin spaces are introduced and studied with the use of discrete wavelet transforms. This study extends the isotropic methods of dyadic $\varphi$-transforms of Frazier and Jawerth (1985, 1989) to non-isotropic settings associated with general expansive matrix dilations and $A_\infty$ weights. In close analogy with the isotropic theory, we show that weighted anisotropic Triebel-Lizorkin spaces are characterized by the magnitude of the $\varphi$-transforms in appropriate sequence spaces. We also introduce non-isotropic analogues of the class of almost diagonal operators and we obtain atomic and molecular decompositions of these spaces, thus extending isotropic results of Frazier and Jawerth.

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