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Register a new userGreedy bases for Besov spaces
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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 $1leq p=qle infty$.
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We study the problem of improving the greedy constant or the democracy constant of a basis of a Banach space by renorming. We prove that every Banach space with a greedy basis can be renormed, for a given $vare>0$, so that the basis becomes $(1+vare)$-democratic, and hence $(2+vare)$-greedy, with respect to the new norm. If in addition the basis is bidemocratic, then there is a renorming so that in the new norm the basis is $(1+vare)$-greedy. We also prove that in the latter result the additional assumption of the basis being bidemocratic can be removed for a large class of bases. Applications include the Haar systems in $L_p[0,1]$, $1<p<infty$, and in dyadic Hardy space $H_1$, as well as the unit vector basis of Tsirelson space.
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.
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 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.
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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