WEIGHTED ESTIMATES FOR MULTIVARIATE HAUSDORFF OPERATORS

In this paper,some weighted estimates for the multivariate Hausdorff operators are obtained.It is proved that the multivariate Hausdorff operators are bounded on L p spaces with power weights,which is based on the boundedness of multivariate Hausdorff operators on Herz spaces,and are bounded on weighted L p spaces with the weights satisfying the homogeneity of degree zero.     

Hausdorff operators on Euclidean spaces

Hausdorff operator is an important operator raised from the dilation on Euclidean space and rooted in the classical summability of number series and Fourier series. It is also connected to many well known operators in real and complex analysis. This article is a survey of some recent developments and extensions on the Hausdorff operator. Particularly, various boundedness properties of the Hausdorff operators, studied recently by our research group, are addressed.     

Hausdorff measure of noncompactness of certain matrix operators on the sequence spaces of generalized means

In the present paper, we derive some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the sequence spaces of generalized means. Further, by applying the Hausdorff measure of noncompactness, we obtain the necessary and sufficient conditions for such operators to be compact.     

Defining Hausdorff operators on Euclidean spaces

After 2000, an interest in the Hausdorff operators grew, first of all in the sense of a diversity of spaces on which these operators were considered. We try to introduce a ‘correct’ definition of the Hausdorff operator on Euclidean spaces. This is supplemented by a variety of known and possible applications.     

Hausdorff Operators on Real Hardy Spaces H1 Over Homogeneous Spaces with Local Doubling Property

We give conditions for boundedness of Hausdorff operators on real Hardy spaces H¹ over homogeneous spaces of locally compact groups with local doubling property. Special cases of the hyperbolic plane and 2-sphere are considered. In this cases Hausdorff operators look as orbital like integrals.

     

Spectral Packing Dimensions through Power-Law Subordinacy

We offer a method of classification of spectral measures of discrete one-dimensional Schrödinger operators with respect to packing measures, which can be seen as dual to results for Hausdorff measures in subordinacy theory. We apply this method to classes of sparse operators, and give an example whose spectral measure has different Hausdorff and packing dimensions, and others for which such dimensions coincide. Some dynamical motivations are also mentioned.     

The question of equivalence for generalized Hausdorff matrices

In the present paper we investigate when Hausdorff matrices and generalized Hausdorff matrices, with the same mass function, are equivalent, as bounded operators on c and ?p.     

Fredholm operator families -I

Integral Equations and Operator Theory - First, we give a characterization of semi-Fredholm operators (i.e. those which are left or right invertible modulo compact operators) on Hausdorff tvs which...     

(Lp,Lq)-Boundedness of Hausdorff Operators with Power Weight on Euclidean Spaces

In this paper, we prove the (Lp,Lq)-boundedness of (fractional) Hausdorff operators with power weight on Euclidean spaces.As special cases, we can obtain some well known results about Hardy operators.     

加权Morrey-Herz空间上的广义Hausdorff算子

引入了一种新的广义Hausdorff算子.它包括了许多著名的算子作为其特殊情形.我们得到了这些算子在加权Morrey-Herz空间上有界的充分必要条件,还得到了相应的新的算子范数不等式.它们是许多著名结果有意义的改进和推广.     

The point spectra for certain classes of operators in B(ℓ p )

In a recent paper [5] Maddox determined the point spectrum of the Cesaro matrices of order >0, considered as series-to-series operators. In this paper we obtain the point spectrum for the Endl generalized Hausdorff matrices, the Hausdorff matrices, and for a wide class of generalized Hausdorff matrices, which include the Endl and ordinary Hausdorff matrices as special cases. We also obtain point spectrum results for a wide class of weighted mean matrices. Since the Cesaro matrices are examples of Hausdorff matrices, our results contain the corresponding theorems of Maddox as special cases.     

Hausdorff Operators on Function Spaces

The authors mainly study the Hausdorff operators on Euclidean space Rn.They establish boundedness of the Hausdorff operators in various function spaces,such as Lebesgue spaces,Hardy spaces,local Hardy ...     

Hausdorff operators on the Heisenberg group

This paper is devoted to the high-dimensional and multilinear Hausdorff operators on the Heisenberg group H n. The sharp bounds for the strong type(p, p)(1 ≤ p ≤∞) estimates of n-dimensional Hausdorff operators on H n are obtained. The sharp bounds for strong(p, p) estimates are further extended to multilinear cases. As an application, we derive the sharp constant for the multilinear Hardy operator on H n. The weak type(p, p)(1 ≤ p ≤∞) estimates are also obtained.     

Sparse potentials with fractional Hausdorff dimension

We construct non-random bounded discrete half-line Schrödinger operators which have purely singular continuous spectral measures with fractional Hausdorff dimension (in some interval of energies). To do this we use suitable sparse potentials. Our results also apply to whole line operators, as well as to certain random operators. In the latter case we prove and compute an exact dimension of the spectral measures.     

Fractal entropies and dimensions for microstates spaces

Using Voiculescu's notion of a matricial microstate we introduce fractal dimensions and entropies for finite sets of selfadjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of free Hausdorff dimension is an algebraic invariant. We compute the free Hausdorff dimension in the cases where the set generates a finite-dimensional algebra or where the set consists of a single selfadjoint. We show that the Hausdorff dimension becomes additive for such sets in the presence of freeness.     

Weighted Lebesgue and Central Morrey Estimates for p-Adic Multilinear Hausdorff Operators and Their Commutators

Ukrainian Mathematical Journal - We establish the sharp boundedness of p-adic multilinear Hausdorff operators on the product of Lebesgue and central Morrey spaces associated with both power weights...     

On measure-compact operators

We obtain new sufficient solvability conditions for equations with measure-compact operators comparable with partially additive ones. We also prove new conditions under which these operators are locally condensing with respect to the Hausdorff measure of noncompactness. As an application of the obtained results we prove one property of bifurcation points which occur, in particular, in nonlinear mechanics.     

Sull'interpolazione di operatori nonlineari noncompatti in spazi di funzioni integrabili

Usando la misura di non compattezza di Hausdorff, si dimostra un teorema elementare di interpolazione per gli operatori nonlineari noncompatti tra retticoli di Banach di funzioni misurabili. Il teorema astratto viene illustrato tramite un'equazione integrale nonlineare del tipo di Hammerstein. An elementary interpolation theorem involving the Hausdorff measure of noncompactness for nonlinear operators in Banach lattices of measurable functions is proved. The abstract theorem is illustrated by a nonlinear noncompact integral operator equation of Hammerstein type.

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(Conferenza tenuta l'11 ottobre 1988)     

A semigroup proof of a non-ergodic Markov process central limit theorem with third order terms

A semigroup product for linear operators is used to get a central limit theorem with third order terms which applies to all ergodic and non-ergodic discrete time Markov processes with continuous transition probability densities on a compact Hausdorff space.     

Dilation operators and integral operators on amalgam space $$(L_{p},l_{q})$$(Lp,lq)

This paper establishes the Hardy–Littlewood–Pólya inequalities, the Hardy inequalities and the Hilbert inequalities on amalgam spaces. Moreover, it also gives the mapping properties of the Mellin convolutions, the Hadamard fractional integrals and the Hausdorff operators on amalgam spaces. We establish these properties by some estimates for the operator norms of the dilation operators on amalgam spaces.