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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:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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