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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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We prove thatthe Banach space $(oplus_{n=1}^infty ell_p^n)_{ell_q}$, which is isomorphic to certain Besov spaces, has a greedy basis whenever $1leq p leqinfty$ and $1<q<infty$. Furthermore, the Banach spaces $(oplus_{n=1}^infty ell_p^n)_{ell_1}$, with $1<ple infty$, and $(oplus_{n=1}^infty ell_p^n)_{c_0}$, with $1le p<infty$ do not have a greedy bases. We prove as well that the space $(oplus_{n=1}^infty ell_p^n)_{ell_q}$ has a 1-greedy basis if and only if $1leq p=qle infty$.
We study the embeddings of (homogeneous and inhomogeneous) anisotropic Besov spaces associated to an expansive matrix $A$ into Sobolev spaces, with focus on the influence of $A$ on the embedding behaviour. For a large range of parameters, we derive sharp characterizations of embeddings.
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.
The purpose of this paper is to give a definition and prove the fundamental properties of Besov spaces generated by the Neumann Laplacian. As a by-product of these results, the fractional Leibniz rule in these Besov spaces is obtained.
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