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Characterisations of function spaces of generalised smoothness. (English) Zbl 1116.46024
This rather extensive paper presents a theory of function spaces of generalized smoothness which extends the theory of the well-known Besov spaces \(B^s_{p,q}\) and Triebel-Lizorkin spaces \(F^s_{p,q}\). The spaces are defined on the basis of a refinement of the standard decomposition method in which the following two types of sequences play a role. A sequence \(N=(N_j)_{j\in\mathbb N_0}\) is strongly increasing if there is a positive constant \(d_0\) and a natural number \(\kappa_0\) such that \(d_0N_j\leq N_k\) for all \(j\), \(k\) with \(0\leq j\leq k\), and \(2N_j\leq N_k\) for all \(j\), \(k\) with \(j+\kappa_0\leq k\). The sequence \(N\) is of bounded growth if there exists a constant \(d_1>0\) and \(J\in\mathbb N_0\) such that \(N_{j+1}\leq d_1N_j\) for any \(j\geq J_0\). A sequence \(\sigma=(\sigma_j)\) of positive real numbers is admissible if both \((\sigma_j)\) and \((\sigma_j^{-1})\) are of bounded growth, i.e. \(d_0\sigma_j\leq\sigma_{j+1}\leq d_1\sigma_j\) for all \(j\in\mathbb N\).
To define the corresponding decomposition set \(\Omega^{N,J}_j=\{\xi\in\mathbb R^n:|\xi|\leq N_{j+J\kappa_0}\}\) if \(j=0,1,\dots,J\kappa_0-1\) and \(\Omega^{N,J}_j=\{\xi\in\mathbb R^n:N_{j-J\kappa_0}\leq|\xi|\leq N_{j+J\kappa_0}\}\) if \(j\geq J\kappa_0\) and let \(\Phi^{N,J}\) be a class of all function systems \(\varphi^{N,J}=(\varphi^{N,J}_j)\) be a sequence of non-negative smooth functions with a support in \(\Omega^{N,J}_j\) such that \(| D^\gamma\varphi^{N,J}_j(\xi)|\leq c_\gamma\langle\xi\rangle^{-\gamma}\) for any \(\gamma\in\mathbb N^n_0\), \(j\in\mathbb N_0\), and \(0<\sum_{j=0}^\infty\varphi^{N,J}_j(\xi)=c_\varphi<\infty\) for any \(\xi\in\mathbb R^n\). Let \(1<p<\infty\). For \(1\leq q\leq\infty\) the Besov space of generalized smoothness \(B^{\sigma,N}_{p,q}\) is the class of tempered distributions \(f\) with the norm \(\| f\mid B^{\sigma,N}_{p,q}\|=\|(\sigma_j\varphi^{N,J}_j(D)f)_{j\in\mathbb N_0}\mid l_q(L_p)\|<\infty\). For \(1<q<\infty\) the Triebel-Lizorkin space of generalized smoothness \(F^{\sigma,N}_{p,q}\) is the class of tempered distributions \(f\) with the norm \[ \| f\mid F^{\sigma,N}_{p,q}\|=\|(\sigma_j\varphi^{N,J}_j(D)f(\cdot))_{j\in\mathbb N_0}\mid L_p(l_q)\|<\infty. \] If \(N_j=2^j\) and \(\sigma_j=2^{js}\) then the spaces \(B^{\sigma,N}_{p,q}\) and \(F^{\sigma,N}_{p,q}\) consider with the usual Besov spaces \(B^s_{p,q}\) and Triebel-Lizorkin spaces \(F^s_{p,q}\). The authors investigate many properties of the spaces of generalized smoothness. They use the Michlin-Hörmander-type theorem on Fourier multipliers to show the consistency of the definition and prove the theorem of Littlewood-Paley type, \(F^{1,N}_{p,2}=L_p\). They prove the analogues of the usual embeddings, \[ L_p\hookrightarrow B^{1,N}_{p,\infty}\hookrightarrow B^0_{p,\infty},\quad B^0_{p,1}\hookrightarrow B^{1,N}_{p,1}\hookrightarrow L_p, \] characterize the dual spaces etc.
The main results of the paper concerns the characterization with local means and the atomic decomposition under the assumption that \(N\) satisfies \(\lambda_0N_j\leq N_{j+1}\leq \lambda_1N_{N+1}\) with \(1<\lambda_0\leq\lambda_1\).

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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