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New characterizations and applications of inhomogeneous Besov and Triebel-Lizorkin spaces on homogeneous type spaces and fractals. (English) Zbl 1019.43006

The authors study spaces of homogeneous type \((X,\rho,\mu)_{d,\theta}\). In the first section they introduce the space of test functions \({\mathcal G}( \beta, \gamma)\) on \(X\), they define approximation to the identities \(\{S_k\}_{k\in\mathbb N}\), the inhomogeneous Besov spaces \(B_{pq}^s(X) \), the inhomogeneous Triebel-Lizorkin spaces \(F^s_{pq}(X)\) and they show that these spaces are the same for equivalent quasi-metrics on \(X\). Furthermore the atomic decomposition of these spaces is recalled. In the second section fractional integrals and derivatives \(I_\alpha\) \((\alpha\in{\mathbb R}\)) are introduced, where \(I_\alpha f(x)=\sum_{l=0}^\infty 2^{-{\l}\alpha}(S_l-S_{l-1})(f)(x)\). It is shown that \(I_\alpha\) maps \({\mathcal G}( \beta, \gamma)\) continuously into \({\mathcal G}( \beta+\alpha, \gamma-\max(\alpha,0))\), \(B^s_{pq}(X)\) into \( B^{s+\alpha}_{pq}(X)\) and \(F^s_{pq}(X)\) into \(F^{s+\alpha}_{pq}(X)\) (under certain conditions on \(\alpha \) and \(\beta\)). They show also the invertibility of \(I_\alpha I_{-\alpha}\) on \(B^s_{pq}(X)\) and on \(F^s_{pq}(X)\). In the third section explicit representation formulae in spaces of test functions for left and right inverses of fractional integrals and derivatives are established. The authors show that \(I-I_{\alpha}I_{-\alpha}\) is a singular integral with a standard kernel \(K(x,y)\) which goes to 0 as \(|\alpha|\) goes to \(0\) (\(I\) denoting the identity operator). In the fourth section frame decomposition characterizations of \(B_{pq}^s(X) \) and \(F^s_{pq}(X)\) by using the discrete Calderón reproducing formulae are established. In the fifth section the entropy numbers of compact embeddings between \(B_{pq}^s(X) \) or \(F^s_{pq}(X)\) (when \(\mu(X)<\infty\)) by using the frame characterizations of these spaces established in the preceding sections are estimated. Finally in the two last sections relations with Sobolev spaces on metric spaces and an application to quadratic forms on the spaces \(H^s(X):=B_{2,2}^s(X)\) are given.
Reviewer: Jean Ludwig (Metz)

MSC:

43A85 Harmonic analysis on homogeneous spaces
47B38 Linear operators on function spaces (general)
47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
35P15 Estimates of eigenvalues in context of PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B35 Function spaces arising in harmonic analysis
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47A75 Eigenvalue problems for linear operators
28A80 Fractals
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