Complex Interpolation of Weighted Besov and Lizorkin–Triebel Spaces

We study complex interpolation of weighted Besov and Lizorkin–Triebel spaces.The used weights w0,w1 are local Muckenhoupt weights in the sense of Rychkov.As a first step we calculate the Calder′on products of associated sequence spaces.Finally,as a corollary of these investigations,we obtain results on complex interpolation of radial subspaces of Besov and Lizorkin–Triebel spaces on Rd.     

Interpolation of Functions from Besov-type Spaces on Gauß–Chebyshev Grids

In this paper, we give a unified approach to error estimates for interpolation on Gauß–Chebyshev grids for functions from certain Besov-type spaces with dominating mixed smoothness properties.     

Besov–Lipschitz and Mean Besov–Lipschitz Spaces of Holomorphic Functions on the Unit Ball

We give several characterizations of holomorphic mean Besov–Lipschitz spaces on the unit ball in ${\mathbb C^N} $ and appropriate Besov–Lipschitz spaces and prove the equivalences between them. Equivalent norms on the mean Besov–Lipschitz spaces involve different types of L ^p -moduli of continuity, while in characterizations of Hardy–Sobolev spaces we use not only the radial derivative but also the gradient. The characterization in terms of the best approximation by polynomials is also given.     

Harmonic Besov and Triebel–Lizorkin Spaces on the Ball

Harmonic Besov and Triebel–Lizorkin spaces on the unit ball in \({\mathbb R}^d\) with full range of parameters are introduced and studied. It is shown that these spaces can be identified with respective Besov and Triebel–Lizorkin spaces of distributions on the sphere. Frames consisting of harmonic functions are also developed and frame characterization of the harmonic Besov and Triebel–Lizorkin spaces is established.     

Some Applications of Besov Spaces on Fractals

Let F be a compact d-set in R^n with 0 〈 d ≤ n, which includes various kinds of fractals. The author establishes an embedding theorem for the Besov spaces Bpq^s（F） of Triebel and the Sobolev spaces W^1,P（F,d,μ） of Hajtasz when s 〉 1, 1 〈 p 〈∞ and 0 〈 q ≤ ∞. The author also gives some applications of the estimates of the entropy numbers in the estimates of the eigenvalues of some fractal pseudodifferential operators in the spaces Bpq^0（F） and Fpq^0（F）.     

Two Types of Rubio de Francia Operators on Triebel–Lizorkin and Besov Spaces

Journal of Fourier Analysis and Applications - We discuss generalizations of Rubio de Francia’s inequality for Triebel–Lizorkin and Besov spaces, continuing the research from Osipov (Sb...     

Tensor Products of Approximately Cohen–Macaulay Rings

Our aim in this article is to study a problem originally raised by Grothendieck. We show that the approximately Cohen–Macaulay property is preserved for the tensor product of algebras over a field k. We also discuss the converse problem.     

The Boundedness of Composition Operators on Triebel–Lizorkin and Besov Spaces with Different Homogeneities

In this paper, we introduce new Triebel–Lizorkin and Besov Spaces associated with the different homogeneities of two singular integral operators. We then establish the boundedness of composition of two Calder′on–Zygmund singular integral operators with different homogeneities on these Triebel–Lizorkin and Besov spaces.     

Tensor Product of Massey Products

In this paper, we interpret Massey products in terms of realizations （twitsting cochains） of certain differential graded coalgebras with values in differential graded algebras. In the case where the target algebra is the cobar construction of a differential graded commutative Hopf algebra, we construct the tensor product of realizations and show that the tensor product is strictly associative, and commutative up to homotopy.     

On the Dirac–Frenkel Variational Principle on Tensor Banach Spaces

Foundations of Computational Mathematics - The main goal of this paper is to extend the so-called Dirac–Frenkel variational principle in the framework of tensor Banach spaces. To this end we...     

Some Extensions of Optimal Interpolation in Spaces of Lorentz–Zygmund Type

The results on optimal interpolation from [7] are extended to quasinormed spaces with p<1, to spaces with varying secondary parameters , E and to spaces of functions defined on the interval (1,). As a tool for doing this, we construct special mappings which transform these cases into the basic one, considered in [7].     

Interpolation of Ideal Measures by Abstract K and J Spaces

We work with the abstract K and J interpolation method generated by a sequence lattice Г. We investigate the deviation of an interpolated operator from a given operator ideal by establishing formulae for the ideal measure of the interpolated operator in terms of the ideal measures of restrictions of the operator. Formulae are given in terms of the norms of the shift operators on Г.     

On the Chaoticity of Some Tensor Product Weighted Backward Shift Operators Acting on Some Tensor Product Fock–Bargmann Spaces

In Advances in Mathematical Physics (2011) we showed that the weighted shift \(z^{p}\frac{d^{p+1}}{dz^{p+1}} (p=0, 1, 2,\ldots )\) acting on classical Bargmann space \(\mathbb {B}_{p}\) is chaotic operator. In Journal of Mathematical physics (2014), we constructed an chaotic weighted shift \(\mathbb {M}^{*^{p}}\mathbb {M}^{p+1} (p=0, 1, 2,\ldots )\) on some lattice Fock–Bargmann \(\mathbb {E}_{p}^{\alpha }\) generated by the orthonormal basis \( {e_{m}^{(\alpha ,p)}(z) = e_{m}^{\alpha } ; m=p, p+1,\ldots }\) where \( {e_{m}^{\alpha }(z) = (\frac{2\nu }{\pi })^{1/4}e^{\frac{\nu }{2}z^{2}}e^{-\frac{\pi ^{2}}{\nu }(m +\alpha )^{2} +2i\pi (m +\alpha )z}; m \in \mathbb {N}}\) with \(\nu , \alpha \) are real numbers; \(\nu > 0\), \(\mathbb {M}\) is an weighted shift and \(\mathbb {M^{*}}\) is the adjoint of the \(\mathbb {M}\). In this paper we study the chaoticity of tensor product \(\mathbb {M}^{*^{p}}\mathbb {M}^{p+1}\otimes z^{p}\frac{d^{p}}{dz^{p+1}} (p=0, 1, 2, \ldots )\) acting on \(\mathbb {E}_{p}^{\alpha }\otimes \mathbb {B}_{p}\).     

Tensor Products of Type III Factor Representations of Cuntz–Krieger Algebras

We introduced a non-symmetric tensor product of any two states or any two representations of Cuntz–Krieger algebras associated with a certain non-cocommutative comultiplication in our previous work. In this paper, we show that a certain set of KMS states is closed with respect to the tensor product. From this, we obtain formulae of tensor products of type III factor representations of Cuntz–Krieger algebras which are different from results of the tensor product of factors of type III.     

Variable 2-Microlocal Besov–Triebel–Lizorkin-Type Spaces

This article is devoted to the study of variable 2-microlocal Besov-type and Triebel–Lizorkin-type spaces. These variable function spaces are defined via a Fourier-analytical approach. The authors then characterize these spaces by means of φ-transforms, Peetre maximal functions, smooth atoms, ball means of differences and approximations by analytic functions. As applications, some related Sobolev-type embeddings and trace theorems of these spaces are also established. Moreover, some obtained results, such as characterizations via approximations by analytic functions, are new even for the classical variable Besov and Triebel–Lizorkin spaces.     

N-Widths and Average Widths of Besov Classes in Sobolev Spaces

In this paper, we consider the n-widths and average widths of Besov classes in the usual Sobolev spaces. The weak asymptotic results concerning the Kolmogorov n-widths, the linear n-widths, the Gel＇fand n-widths, in the Sobolev spaces on T^d, and the infinite-dimensional widths and the average widths in the Sobolev spaces on Ra are obtained, respectively.     

The Radon–Nikodym Property for Tensor Products of Banach Lattices

Let 1 ≤ p < ∞. We show that , the Fremlin projective tensor product of ℓ_p with a Banach lattice X, has the Radon–Nikodym property if and only if X has the Radon–Nikodym property; and that , the Wittstock injective tensor product of ℓ_p with a Banach lattice X, has the Radon–Nikodym property if and only if X has the Radon–Nikodym property and each positive operator from ℓ_p' to X is compact, where 1/p +1/p'= 1 and let ℓ_p' = c₀ if p = 1. The author gratefully acknowledges support from the Office of Naval Research Grant # N00014-03-1-0621     

Addendum to “Besov–Lipschitz and Mean Besov–Lipschitz Spaces”: The Ahern–Schneider Inequality

Let f be a function holomorphic in the unit ball of \(\mathbb C^N\) , and \(\mathcal Rf\) the radial derivative of f. It is proved that the Ahern–Schneider inequality \(\|\nabla f\|_{H^p}\le C_p\|\mathcal Rf\|_{H^p}\) holds for 0?<?p?<?1. This fills a gap in the proof of the main result in the paper “Besov–Lipschitz and mean Besov–Lipschitz spaces of holomorphic functions on the unit ball” [Potential Analysis] by Jevti? and Pavlovi?.     

Multilinear Calderón-Zygmund Operator on Products of Hardy Spaces

In this paper, the authors establish the boundedness of the multilinear Calderón-Zygmund operator from products of Hardy spaces into Hardy spaces.     

Toeplitz Products on Bergman–Sobolev Spaces over the Unit Polydisk

In this paper, we characterize the compactness and Fredholmness of Toeplitz operators and Toeplitz products on Bergman–Sobolev spaces over the unit polydisk. We also calculate the essential norm of finite sums of finite Toeplitz products on these spaces.