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Register a new userOn various approaches to Besov-type spaces of variable smoothness
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Authors
A. I. Tyulenev
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The paper is concerned with Besov spaces of variable smoothness $B^{varphi_{0}}_{p,q}(mathbb{R}^{n},{t_{k}})$, in which the norms are defined in terms of convolutions with smooth functions. A relation is found between the spaces $B^{varphi_{0}}_{p,q}(mathbb{R}^{n},{t_{k}})$ and the spaces $widetilde{B}^{l}_{p,q,r}(mathbb{R}^{n},{t_{k}})$, which were introduced earlier by the author.
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The paper puts forward new Besov spaces of variable smoothness $B^{varphi_{0}}_{p,q}(G,{t_{k}})$ and $widetilde{B}^{l}_{p,q,r}(Omega,{t_{k}})$ on rough domains. A~domain~$G$ is either a~bounded Lipschitz domain in~$mathbb{R}^{n}$ or the epigraph of a~Lipschitz function, a~domain~$Omega$ is an $(varepsilon,delta)$-domain. These spaces are shown to be the traces of the spaces $B^{varphi_{0}}_{p,q}(mathbb{R}^{n},{t_{k}})$ and $widetilde{B}^{l}_{p,q,r}(mathbb{R}^{n},{t_{k}})$ on domains $G$ and~$Omega$, respectively. The extension operator $operatorname{Ext}_{1}:B^{varphi_{0}}_{p,q}(G,{t_{k}}) to B^{varphi_{0}}_{p,q}(mathbb{R}^{n},{t_{k}})$ is linear, the operator $operatorname{Ext}_{2}:widetilde{B}^{l}_{p,q,r}(Omega,{t_{k}}) to widetilde{B}^{l}_{p,q,r}(mathbb{R}^{n},{t_{k}})$ is nonlinear. As a~corollary, an exact description of the traces of 2-microlocal Besov-type spaces and weighted Besov-type spaces on rough domains is obtained.
In this article, the authors introduce Besov-type spaces with variable smoothness and integrability. The authors then establish their characterizations, respectively, in terms of $varphi$-transforms in the sense of Frazier and Jawerth, smooth atoms or Peetre maximal functions, as well as a Sobolev-type embedding. As an application of their atomic characterization, the authors obtain a trace theorem of these variable Besov-type spaces.
We continue our investigations on pointwise multipliers for Besov spaces of dominating mixed smoothness. This time we study the algebra property of the classes $S^r_{p,q}B(mathbb{R}^d)$ with respect to pointwise multiplication. In addition if $pleq q$, we are able to describe the space of all pointwise multipliers for $S^r_{p,q}B(mathbb{R}^d)$.
The present paper is concerned with new Besov-type space of variable smoothness. Nonlinear spline-approximation approach is used to give atomic decomposition of such space. Characterization of the trace space on hyperplane is also obtained.
This paper is devoted to giving definitions of Besov spaces on an arbitrary open set of $mathbb R^n$ via the spectral theorem for the Schrodinger operator with the Dirichlet boundary condition. The crucial point is to introduce some test function spaces on $Omega$. The fundamental properties of Besov spaces are also shown, such as embedding relations and duality, etc. Furthermore, the isomorphism relations are established among the Besov spaces in which regularity of functions is measured by the Dirichlet Laplacian and the Schrodinger operators.
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