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Anisotropic Triebel-Lizorkin spaces with doubling measures. (English) Zbl 1147.42006

A real \(n\times n\) matrix is expansive if all of its eigenvalues satisfy \(| \lambda| > 1\). A quasi-norm associated with an expansive matrix is a Borel measurable mapping \(\rho_A : \mathbb R^n \rightarrow [0,\infty)\) such that \(\rho_A(x) >0\) for \(x\not = 0\), \(\rho_A(Ax) = | \text{det}\,A| \rho_A(x)\) for \(x\in \mathbb R^n\), \(\rho_A (x+y) \leq H(\rho_A(x) + \rho_A (y))\) for \(x, y \in \mathbb R^n\), where \(H \geq 1\) is a constant. Put \(B_{\rho_A}(x,r) = \{y \in \mathbb R^n; \rho_A(x-y) < r\}\), \(x\in\mathbb R^n\), \(r>0\). A nonnegative Borel measure is called \(\rho_A\)-doubling if there exists \(\beta = \beta(\mu) > 0\) such that \[ \mu(B_{\rho_A} (x,| \text{det}\, A| r)) \leq | \text{det}\,A| ^{\beta} \mu(B_{\rho_A}(x,r)) \;\;\text{for all} \;x\in\mathbb R^n, \;r> 0. \] Let \(\varphi\) belong to the Schwarz class \(\mathcal S(\mathbb R^n)\) and satisfy: \[ \sum_{j\in\mathbb Z} \hat{\varphi} ((A^*)^j\xi) =1 \;\;\text{for all} \;\xi \in \mathbb R^n\setminus \{0\}, \] \[ \text{supp} \;\hat{\varphi} \;\text{is compact and bounded away from the origin}. \] Put \(\varphi_j(x) = | \text{det}\, A| ^j \varphi(A^jx), \;j\in\mathbb Z, \;x\in \mathbb R^n\).
Given \(\alpha \in\mathbb R\), \(0<p<\infty\), \(0<q\leq\infty\) and a \(\rho_A\)-doubling measure \(\mu\), the author introduces the anisotropic Triebel-Lizorkin space \(\dot F^{\alpha,q}_p (\mathbb R^n,A,\mu)\) norm as \[ \| f\| _{\dot F_p^{\alpha,q}} = \Big\| \Big( \sum_{j\in\mathbb Z} (| \text{det}\, A| ^{j\alpha} | f *\varphi_j| )^q \Big)^{1/q} \Big\| _{L^p(\mu)} < \infty \] and shows that this space is independent of the choice of \(\varphi\). The operators on \(\dot F^{\alpha,q}_p\) spaces are studied by transferring them with the use of wavelet transforms to the corresponding sequence spaces \(\dot f^{\alpha,q}_p(A,\mu)\). In particular, the class of almost diagonal operators is investigated. As an application, smooth atomic and molecular decompositions of spaces \(\dot F^{\alpha,q}_p (A,\mu)\) are established. The authors also develops localization techniques in the endpoint case \(p=\infty\). Furthermore, nonsmooth atomic decompositions of spaces \(\dot F^{\alpha,q}_p\) in the range \(0<p\leq 1\) are found. Finally, unweighted \(\dot F^{0,2}_p (\mathbb R^n,A)\) spaces are identified with the anisotropic (real) Hardy spaces \(H^p_A\) for \(0<p<\infty\).

MSC:

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