On the coincidence of certain approaches to smoothness spaces related to Morrey spaces (Q2862567)

Let \(\{ \varphi_j \}^\infty_{j=0}\) be the nowadays well-known dyadic resolution of unity in \(\mathbb R^n\). Let NEWLINE\[NEWLINE Q_{k,m} = 2^{-k} m + 2^{-k} (0,1)^n, \qquad k\in \mathbb N_0, \quad m\in \mathbb Z^n, NEWLINE\]NEWLINE be dyadic cubes in \(\mathbb R^n\). Let \(0<p,q \leq \infty\), \(s\in \mathbb R^n\), \(0\leq \tau <\infty\). Then \({\mathcal B}^{s,\tau}_{p,q} (\mathbb R^n)\) is the collection of all \(f \in S'(\mathbb R^n)\) such that NEWLINE\[NEWLINE \| f \, | {\mathcal B}^{s,\tau}_{p,q} (\mathbb R^n) \| = \sup_{k\in \mathbb N_0, m \in \mathbb Z^n} 2^{kn\tau} \Big( \sum^\infty_{j=k} 2^{jsq} \Big( \int_{Q_{k,m}} \big| (\varphi_j \hat{f} )^\vee (x) \big|^p dx \Big)^{q/p} \Big)^{1/q} NEWLINE\]NEWLINE is finite. Similarly, for \({\mathcal F}^{s,\tau}_{p,q} (\mathbb R^n)\). Both together are abbreviated as \({\mathcal A}^{s,\tau}_{p,q} (\mathbb R^n)\), \({\mathcal A}\in \{ {\mathcal B}, {\mathcal F} \}\). It is the main aim of this paper to characterize these spaces in terms of atoms and wavelets. This is expressed by the coincidence NEWLINE\[NEWLINE {\mathcal L}^r A^s_{p,q} (\mathbb R^n) = {\mathcal A}^{s,\tau}_{p,q} (\mathbb R^n), \qquad \tau = \frac{1}{p} + \frac{r}{n}, NEWLINE\]NEWLINE where \({\mathcal L}^r A^s_{p,q} (\mathbb R^n)\) are the so-called local spaces introduced by \textit{H. Triebel} [Local function spaces, heat and Navier-Stokes equations. Zürich: European Mathematical Society (EMS) (2013; Zbl 1280.46002)] in terms of wavelet decompositions.