A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces (Q2846555)

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 NEWLINE\[NEWLINE \| f \, | B^{s,\tau}_{p,q} (\mathbb R^n)\| = \sup_{J\in \mathbb Z, M \in \mathbb Z^n} 2^{Jn\tau} \Big( \sum_{j \geq J_+} 2^{jsq} \big\| (\phi_j \hat{f})^\vee \, | L_p (Q_{J,M}) \big\|^q \Big)^{1/q} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \| f \, | F^{s,\tau}_{p,q} (\mathbb R^n) \| = \sup_{J \in \mathbb Z, M\in \mathbb Z^n} 2^{Jn\tau} \Big\| \Big( \sum_{j \geq J_+} 2^{jsq} \big| (\phi_j \hat{f} )^\vee (\cdot) \big|^q \Big)^{1/q} | L_p (Q_{J,M}) \Big\|,NEWLINE\]NEWLINE 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.