New applications of Besov-type and Triebel-Lizorkin-type spaces
Given a bump function \(\psi\in \mathcal S(\mathbb R^n)\) with \(\chi_{B(0,2)}\leq \psi\leq \chi_{B(0,4)}\) and denoting \(\phi_j(x)=\psi(2^{-j}x)-\psi(2^{-j-1}), j\in \mathbb Z\), the Besov-Morrey spaces \(\mathcal N^s_{u,p,q}(\mathbb R^n)\) , for \(0<p\leq u<\infty, 0<q\leq\infty, s\in \mathbb R\), are defined as the set of elements in \(\mathcal S'_\infty\), where \(\mathcal S_\infty\) stands for the Schwarz functions with \(\int \phi(x)x^\gamma dx=0\) for \(\gamma\in (\mathbb N\cup \{0\})^n\), such that \[ \sum_{j=-\infty}^\infty \sup_P |P|^{q/u-q/p}\left(\int_P |2^{js}\mathcal F^{-1}\phi_j\mathcal F f(x)|^pdx\right)^{q/p}<\infty \] where \(P\) stands for a dyadic cube. Similarly the Triebel-Lizorkin-Morrey spaces \(\mathcal E^s_{u,p,q}(\mathbb R^n)\) are given by the condition \[ \sup_P |P|^{1/u-1/p} \int_P \left(\sum_{j=-\infty}^\infty|2^{js}\mathcal F^{-1}\phi_j\mathcal F f(x)|^q\right)^{p/q} dx <\infty. \] These spaces were considered by \textit{Y. Sawano} and \textit{H. Tanaka} [Math. Z. 257, No. 4, 871--905 (2007; Zbl 1133.42041)]. Also \textit{D. Yang} and \textit{W. Yuan} [J. Funct. Anal. 255, No. 10, 2760--2809 (2008; Zbl 1169.46016)] introduced, for \(0\leq \tau< \infty, 0<q\leq \infty, s\in \mathbb R\), the Besov-type spaces \(\mathcal B^{s,\tau}_{p,q}(\mathbb R^n)\) and the Triebel-Lizorkin spaces \(\mathcal F^{s,\tau}_{p,q}(\mathbb R^n)\), closely related to the previous ones, and given by the conditions \[ \sup_P |P|^{-\tau q} \sum_{j=-\log_2 \ell(P)}^\infty\left(\int_P |2^{js}\mathcal F^{-1}\phi_j\mathcal F f(x)|^p dx\right)^{q/p} <\infty \] and \[ \sup_P |P|^{-\tau q} \int_P \left(\sum_{j=-\log_2 \ell(P)}^\infty|2^{js}\mathcal F^{-1}\phi_j\mathcal F f(x)|^q\right)^{p/q} dx < \infty \] respectively. The authors show that for \(0<p<u<\infty\) and \(0<q<\infty\) then \(\mathcal N^s_{u,p,q}(\mathbb R^n)\) is a proper subspace of \(\mathcal B^{s,1/p-1/u}_{p,q}(\mathbb R^n)\) and that \(\mathcal N^s_{u,p,\infty}(\mathbb R^n)=\mathcal B^{s,1/p-1/u}_{p,\infty}(\mathbb R^n)\) while for \(0<p\leq u<\infty\) and \(0<q\leq \infty\) the spaces \(\mathcal E^s_{u,p,q}(\mathbb R^n)\) and \( \mathcal F^{s,1/p-1/u}_{p,q}(\mathbb R^n)\) coincide. This is done using the wavelet characterization of the spaces \(\mathcal N^s_{u,p,q}(\mathbb R^n)\) and \(\mathcal E^s_{u,p,q}(\mathbb R^n)\). As a corollary they see that the family \(\mathcal F^{s,\tau}_{p,q}(\mathbb R^n)\) covers the Morrey as well as the Morrey-Hardy spaces previously considered by the authors. Also the atomic decompositions of all the spaces and some trace results on the spaces \(\mathcal F^{s,\tau}_{p,q}(\mathbb R^n)\) are established.
scientific article; zbMATH DE number 5644906
scientific article