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Register a new userSome new function spaces of variable smoothness
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Authors
A. I. Tyulenev
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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.
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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.
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.
In this paper we give exact values of the best $n$-term approximation widths of diagonal operators between $ell_p(mathbb{N})$ and $ell_q(mathbb{N})$ with $0<p,qleq infty$. The result will be applied to obtain the asymptotic constants of best $n$-term approximation widths of embeddings of function spaces with mixed smoothness by trigonometric system.
We show that a Banach space with numerical index one cannot enjoy good convexity or smoothness properties unless it is one-dimensional. For instance, it has no WLUR points in its unit ball, its norm is not Frechet smooth and its dual norm is neither smooth nor strictly convex. Actually, these results also hold if the space has the (strictly weaker) alternative Daugavet property. We construct a (non-complete) strictly convex predual of an infinite-dimensional $L_1$ space (which satisfies a property called lushness which implies numerical index~1). On the other hand, we show that a lush real Banach space is neither strictly convex nor smooth, unless it is one-dimensional. In particular, if a subspace $X$ of the real space $C[0,1]$ is smooth or strictly convex, then $C[0,1]/X$ contains a copy of $C[0,1]$. Finally, we prove that the dual of any lush infinite-dimensional real space contains a copy of $ell_1$.
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 $S^r_{p,q}B(mathbb{R}^d)$.
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