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Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces. (English) Zbl 1083.42016

The main purpose of this paper is to extend the isotropic methods of dyadic $\phi$-transforms of M. Frazier and B. Jawerth in [Indiana Univ. Math. J. 34, 777–799 (1985; Zbl 0551.46018); J. Funct. Anal. 93, No. 1, 34–170 (1990; Zbl 0716.46031)] to non-isotropic settings associated with ${A}_{\infty }$ weights and general expansive matrix dilations which are $n×n$ real matrices all of whose eigenvalues $\lambda$ satisfy $|\lambda |>1$. Such expansive matrix dilations have fair generality and wide applications in the multidimensional theory of wavelets, which are the main advantage and novelty of this approach.

Let $A$ be a such dilation and ${A}^{*}$ the adjoint (transpose) of $A$. Let $\phi$ be a Schwartz function on ${ℝ}^{n}$ satisfying $\text{supp}\phantom{\rule{0.166667em}{0ex}}\stackrel{^}{\phi }=\overline{\left\{\xi \in {ℝ}^{n}:\phantom{\rule{4pt}{0ex}}\stackrel{^}{\phi }\left(\xi \right)\ne 0\right\}}\subset {\left[-\pi ,\pi \right]}^{n}\setminus \left\{0\right\}$ and ${sup}_{j\in ℤ}|\stackrel{^}{\phi }\left({\left({A}^{*}\right)}^{j}\xi \right)|>0$ for all $\xi \in {ℝ}^{n}\setminus \left\{0\right\}$, where $\stackrel{^}{\phi }$ denotes the Fourier transform of $\phi$. For $\alpha \in {ℝ}^{n}$, $0, $0 and $w\in {A}_{\infty }$ (the Muckenhoupt class associated to $A$), the authors define the weighted anisotropic Triebel-Lizorkin space ${\stackrel{˙}{F}}_{p}^{\alpha ,q}$ as the collection of all tempered distributions (modulo polynomials) such that

${\parallel F\parallel }_{{\stackrel{˙}{F}}_{p}^{\alpha ,q}}=\parallel {\left\{\sum _{j\in ℤ}{\left(|detA|}^{j\alpha }|f*{\phi }_{j}{|\right)}^{q}\right\}}^{1/q}{\parallel }_{{L}^{p}\left(w\phantom{\rule{0.166667em}{0ex}}dx\right)}<\infty ·$

In the standard dyadic case $A=2I$, $detA={2}^{n}$ and then factor ${|detA|}^{j\alpha }$ in the above definition would be ${2}^{jn\alpha }$ instead of the classical ${2}^{j\alpha }$. Using the Calderón reproducing formula, the authors prove that the definition of ${\stackrel{˙}{F}}_{p}^{\alpha ,q}$ is independent of the choice of $\phi$.

Moreover, in close analogy with the isotropic theory, the authors show that these weighted anisotropic Triebel-Lizorkin spaces are characterized by the magnitude of the $\phi$-transforms in appropriate sequence spaces. The authors also introduce non-isotropic analogues of the class of almost diagonal operators and obtain atomic and molecular decompositions of these spaces, thus extending the isotropic result of Frazier and Jawerth mentioned above. Finally, the authors also study inhomogeneous weighted anisotropic Triebel-Lizorkin spaces and outline analogous decomposition results for these spaces.

##### MSC:
 42B35 Function spaces arising in harmonic analysis 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 42B25 Maximal functions, Littlewood-Paley theory 42C40 Wavelets and other special systems 47B37 Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47B38 Operators on function spaces (general)