A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces. (English) Zbl 1283.46027
Let \(\{\phi_j \}^\infty_{j=0}\) be the, nowadays classical, inhomogeneous dyadic resolution of unity in \(\mathbb R^n\), underlying the Fourier-analytical definition of the spaces \(B^s_{p,q} (\mathbb R^n)\) and \(F^s_{p,q} (\mathbb R^n)\), where \(s\in \mathbb R\), \(0<p,q \leq \infty\) (\(p<\infty\) for \(F\)-spaces). This has been complemented in recent times by localized (or Morreyfied) spaces \(B^{s,\tau}_{p,q} (\mathbb R^n)\) and \(F^{s,\tau}_{p,q} (\mathbb R^n)\), \(\tau \geq 0\), quasi-normed by
\[
\| f \, | B^{s,\tau}_{p,q} (\mathbb R^n)\| = \sup_{J\in \mathbb Z, M \in \mathbb Z^n} 2^{Jn\tau} \Big( \sum_{j \geq J_+} 2^{jsq} \big\| (\phi_j \hat{f})^\vee \, | L_p (Q_{J,M}) \big\|^q \Big)^{1/q}
\]
and
\[
\| f \, | F^{s,\tau}_{p,q} (\mathbb R^n) \| = \sup_{J \in \mathbb Z, M\in \mathbb Z^n} 2^{Jn\tau} \Big\| \Big( \sum_{j \geq J_+} 2^{jsq} \big| (\phi_j \hat{f} )^\vee (\cdot) \big|^q \Big)^{1/q} | L_p (Q_{J,M}) \Big\|,
\]
where \(Q_{J,M} = 2^{-J} (0,1)^n + 2^{-J}M\) are dyadic cubes and \(J_+ =\max (J,0)\). The paper under review extends this set-up in several directions. In particular, \((\phi_j \hat{f} )^\vee\) is replaced by Peetre’s maximal function and, instead of the Lebesgue spaces \(L_p\), the authors deal with a class of general basic spaces, including many other spaces of interest. In this general framework, the authors deal with atomic and wavelet decompositions, pointwise and Fourier multipliers, embeddings, equivalent characterizations by differences and spaces on domains. Specifications cover, for example, weighted Lebesgue spaces, Morrey spaces and Orlicz spaces.
Reviewer: Hans Triebel (Jena)
MSC:
46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
42B35 | Function spaces arising in harmonic analysis |
42B25 | Maximal functions, Littlewood-Paley theory |
42B15 | Multipliers for harmonic analysis in several variables |
42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |