The best approximation to a class of functions of several variables by another class and related extremum problems

We study the relationship between several extremum problems for unbounded linear operators of convolution type in the spaces $L_\gamma = L_\gamma (\mathbb{R}^m )$ , m ≥ 1, 1 ≤ γ ≤ ∞. For the problem of calculating the modulus of continuity of the convolution operatorA on the function classQ defined by a similar operator and for the Stechkin problem on the best approximation of the operatorA on the classQ by bounded linear operators, we construct dual problems in dual spaces, which are the problems on, respectively, the best and the worst approximation to a class of functions by another class.     

Some characterizations of weighted Herz-type Hardy spaces and their applications

In this paper, the authors first establish some new real-variable characterizations of Herz-type Hardy spaces and , where ω₁,ω₃ ∈ A₁-weight, 1<q>∞,n(1−1/q)≤α<∞ and 0<p<∞. Then, using these new characterizations, they investigate the convergence of a bounded set in these spaces, and study the boundedness of some potential operators on these spaces. Supported by the NNSF of China     

Strong monotonicity of operator functions

LetA, B be bounded selfadjoint operators on a Hilbert space. We will give a formula to get the maximum subspace such that is invariant forA andB, and . We will use this to show strong monotonicity or strong convexity of operator functions. We will see that when 0≤A≤B, andB−A is of finite rank,A ^t ≤B ^t for somet>1 if and only if the null space ofB−A is invariant forA.     

Linear Liftings of Skew-Symmetric Tensor Fields to Weil Bundles

We define equivariant tensors for every non-negative integer p and every Weil algebra A and establish a one-to-one correspondence between the equivariant tensors and linear natural operators lifting skew-symmetric tensor fields of type (p, 0) on an n-dimensional manifold M to tensor fields of type (p, 0) on T ^A M if 1 ≤ p ≤ n. Moreover, we determine explicitly the equivariant tensors for the Weil algebras , where k and r are non-negative integers.     

Estimates of maximal distances between spaces whose norms are invariant under a group of operators

We consider the class A_г of n-dimensional normed spaces with unit balls of the form: , where B _n ¹ n is the unit ball of ℓ _n ¹ , Γ is a finite group of orthogonal operators acting in ℝⁿ, and U is a “random” orthogonal transformation. It is proved that this class contains spaces with a large Banach-Mazur distance between them. If the cardinality of Γ is of order n^c, it is shown that, in the power scale, the diameter of A_г in the modified Banach-Mazur distance behaves as the classical diameter and is of order n. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 33–42.     

Quasi-uniform completions of partially ordered spaces

In this paper we define partially ordered quasi-uniform spaces (X, , ≤) (PO-quasi-uniform spaces) as those space with a biconvex quasi-uniformity on the poset (X, ≤) and give a construction of a (transitive) biconvex compatible quasi-uniformity on a partially ordered topological space when its topology satisfies certain natural conditions. We also show that under certain conditions on the topology of a PO-quasi-uniform space (X, , ≤), the bicompletion of (X, ) is also a PO-quasi-uniform space ( , ⪯) with a partial order ⪯ on that extends ≤ in a natural way.      

Weighted composition operators from Bergman-type spaces into Bloch spaces

Let ϕ be an analytic self-map and u be a fixed analytic function on the open unit disk D in the complex plane ℂ. The weighted composition operator is defined by

K-functional, weighted moduli of smoothness, and best weighted polynomial approximation on a simplex

An inverse theorem for the best weighted polynomial approximation of a function in (S) is established. We also investigate Besov spaces generated by Freud weight and their connection with algebraic polynomial approximation in , wherew _α is a Jacobi-type weight onS, 0<p ≤ ∞,S is a simplex andW _λ is a Freud weight. For Ditzian-TotikK-functionals onL _p(S), 1 ≤p ≤ ∞, we obtain a new equivalence expression.     

Commutators of integral operators with variable kernels on Hardy spaces

LetT _Ω,α (0 ≤ α< n) be the singular and fractional integrals with variable kernel Ω(x, z), and [b, T_Ω,α] be the commutator generated by T_Ω,α and a Lipschitz functionb. In this paper, the authors study the boundedness of [b, T_Ω,α] on the Hardy spaces, under some assumptions such as theL ^r -Dini condition. Similar results and the weak type estimates at the end-point cases are also given for the homogeneous convolution operators . The smoothness conditions imposed on are weaker than the corresponding known results.     

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,

Problems similar to the additive divisor problem

For multiplicative functions ƒ(n), let the following conditions be satisfied: ƒ(n)≥0 ƒ(p ^r)≤A ^r,A>0, and for anyε>0 there exist constants ,α>0 such that and Σ _p≤x ƒ(p) lnp≥αx. For such functions, the following relation is proved:

Convolution operators onH p (H k n )

LetG=H ^p (H _k ⁿ ) be the (2n+1)-dimensional Heisenberg group over local fieldK. In this paper we prove some theorems about convolution operators onH ^p (G) and vector-valued Hardy spaces. As an example, the distribution for some φ∈S(G), ξ φ=0 is a ramified 0-type kernel. These results can be applied to characterizeH ^p (G) spaces by square functions.     

On the Fourier transformation of positive,positive definite measures on commutative hypergroups,and dual convolution structures

We show that the support of the Fourier transform of a positive, positive definite measure on a commutative hypergroupK contains a positive character. This generalizes the known fact that the support of the Plancherel measure π contains a positive character (which in general is not the identity character1). It follows that contains a positive character for whenever a dual convolution exists. In particular, if1∈supp π, then1 is this character. We also give some further general results about the support of dual convolution products in terms ofsupp π. Some examples associated with Gelfand pairs and, in particular, non-compact Riemannian symmetric spaces of rank 1 are discussed.     

Inversion of matrices over a pseudocomplemented lattice

We compute the greatest solutions of systems of linear equations over a lattice (P, ≤). We also present some applications of the results obtained to lattice matrix theory. Let (P, ≤) be a pseudocomplemented lattice with and and let A = ‖a _ij ‖ _n×n , where a _ij ∈ P for i, j = 1,..., n. Let A* = ‖a _ij ^′ ‖ _n×n and for i, j = 1,..., n, where a* is the pseudocomplement of a ∈ P in (P, ≤). A matrix A has a right inverse over (P, ≤) if and only if A · A* = E over (P, ≤). If A has a right inverse over (P, ≤), then A* is the greatest right inverse of A over (P, ≤). The matrix A has a right inverse over (P, ≤) if and only if A is a column orthogonal over (P, ≤). The matrix D = A · A* is the greatest diagonal such that A is a left divisor of D over (P, ≤). Invertible matrices over a distributive lattice (P, ≤) form the general linear group GL _n (P, ≤) under multiplication. Let (P, ≤) be a finite distributive lattice and let k be the number of components of the covering graph Γ(join(P,≤) − , ≤), where join(P, ≤) is the set of join irreducible elements of (P, ≤). Then GL _a (P, ≤) ≅ = S _n ^k . We give some further results concerning inversion of matrices over a pseudocomplemented lattice. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 139–154, 2005.     

Integral transforms of functions with the derivative in a halfplane

LetA be the class of normalized analytic functions in the unit disk Δ and define the class

Structure of bounded solutions for a class of abstract differential equations

Summary The paper is concerned with bounded solutions of an equationu′(t)=Bu(t) in Hilbert spaces, . A representation formula is obtained depending on the zeros of Rez(θ).

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Riassunto è studiata la struttura delle soluzioni limitate di una equazione differenzialeu′(t)=Bu(t) in uno spazio di Hilbert, ove , in funzione dei zeri di Rez(θ).

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This research is supported through a grant of the National Research Council Canada.     

Non-commutative Clarkson Inequalities for n-Tuples of Operators

Let A ₀, ... , A _n−1 be operators on a separable complex Hilbert space , and let α₀,..., α _n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,

具有粗糙核的多线性算子在Herz型空间的有界性

§ 1　Introduction and main resultsLet b∈BMO(Rn) and T be a standard Calderon-Zygmund singular integral operator.Define the commutator[b,T] as follows.[b,T] f(x) =b(x) Tf(x) -T(bf) (x) .In [3 ] ,the boundedness ofthe commutator[b,T] wasestablished on Lp(Rn) .There are thesimilar results in [1 ,2 ] when the commutator was substituted with the multilinearoperators generated by the singular integral operator T and a Taylor series A(see thedefinition below) .Recently,many mathematicians h…     

Hardy spaces estimates for a class of multilinear homogeneous operators

It is proved that a class of multilinear singular integral operators with homogeneous kernels are bounded operators from the product spaces to the Hardy spacesH ^r , (ℝ ⁿ ) and the weak Hardy spaceH ^r,∞ (ℝ ⁿ . As an application of this result, the L ^p ,(ℝ ⁿ ) boundedness of a class of commutator for the singular integral with homogeneous kernels is obtained. Project supparted in part by the National Natural Science Foundation of Chind (Grant No. 19131080) of China and Doctoral Programme Foundation of Institution of Higher Education (Grant No. 98002703) of China.     

Entire solutions of differential equations with finite order transcendental entire coefficients

In this paper, we investigate the complex oscillation of the differential equation