Hausdorff Operators Research Articles

Hausdorff operators: problems and solutions

Hausdorff operators: problems and solutions

Generalized Product Hausdorff Operator on Two-Weighted Morrey–Herz Spaces

Generalized Product Hausdorff Operator on Two-Weighted Morrey–Herz Spaces

Estimates for Hausdorff operator associated with the Opdam–Cherednik transform and its commutator on weighted Morrey–Herz spaces

Estimates for Hausdorff operator associated with the Opdam–Cherednik transform and its commutator on weighted Morrey–Herz spaces

Characterization of the generalized Hausdorff and quasi Hausdorff operators on weighted integrable spaces

Characterization of the generalized Hausdorff and quasi Hausdorff operators on weighted integrable spaces

Hausdorff–Zhu operators on the unit ball in ℂn

We introduce and study Hausdorff–Zhu operators over the unit ball in . We give sufficient conditions for boundedness of such operators in Möbius invariant spaces, weighted Lebesgue and Bergman spaces, and Hardy spaces on the unit ball. In the framework of the Lebesgue spaces, we also provide necessity conditions for boundedness. Approximation of the identity by Hausdorff operators is also considered within Lebesgue space with a Möbius invariant Haar measure.

Boundedness and compactness of Hausdorff operators on Fock spaces

We obtain a complete characterization of the bounded Hausdorff operators acting on a Fock space F α p F^p_\alpha and taking its values into a larger one F α q , 0 > p ≤ q ≤ ∞ F^q_\alpha ,\ 0 > p \leq q \leq \infty , as well as some necessary or sufficient conditions for a Hausdorff operator to transform a Fock space into a smaller one. Some results are written in the context of mixed norm Fock spaces. Also the compactness of Hausdorff operators on a Fock space is characterized. The compactness result for Hausdorff operators on the Fock space F α ∞ F^\infty _\alpha is extended to more general Banach spaces of entire functions with weighted sup norms defined in terms of a radial weight and conditions for the Hausdorff operators to become p p -summing are also included.

Variable Herz–Morrey estimates for rough fractional Hausdorff operator

As a first attempt, we obtain the boundedness of the rough fractional Hausdorff operator on variable exponent Herz-type spaces. The method used in this paper enables us to study the operator on some other function spaces with variable exponents.

Hausdorff operators on some classical spaces of analytic functions

Abstract In this note, we start on the study of the sufficient conditions for the boundedness of Hausdorff operators $$ \begin{align*}(\mathcal{H}_{K,\mu}f)(z):=\int_{\mathbb{D}}K(w)f(\sigma_w(z))d\mu(w)\end{align*} $$ on three important function spaces (i.e., derivative Hardy spaces, weighted Dirichlet spaces, and Bloch type spaces), which is a continuation of the previous works of Mirotin et al. Here, $\mu $ is a positive Radon measure, K is a $\mu $ -measurable function on the open unit disk $\mathbb {D}$ , and $\sigma _w(z)$ is the classical Möbius transform of $\mathbb {D}$ .

On the spectra of multidimensional normal discrete Hausdorff operators

In this paper, the general case of normal discrete Hausdorff operators in is studied. The main result states that under some natural arithmetic condition, the spectrum of such an operator is rotationally invariant. Several special cases and applications are considered.

An analysis of the convergence problem of a function of hexagonal Fourier series in generalized Hölder norm using Hausdorff operator

In the present paper, we obtain the results on the degree of convergence of an ‐periodic continuous function in generalized Hölder spaces using Hausdorff operator with monotonically non‐decreasing and monotonically non‐increasing rows of its hexagonal Fourier series. Some important corollaries are also deduced from our main theorems. Applications of main theorems are also obtained in this paper.

Hardy–Littlewood maximal operators and Hausdorff operators on $ p $-adic block spaces with variable exponents

<p>In this paper, we established some sufficient conditions for the boundedness of the Hardy–Littlewood maximal operators and the Hausdorff operators on $ p $-adic Herz spaces and $ p $-adic local block spaces with variable exponents. In particular, the boundedness of the $ p $-adic maximal commutator operators, the $ p $-adic Hardy–Littlewood average operators, and the $ p $-adic Hardy-Hilbert operators on such spaces was also discussed.</p>

TWO-WEIGHT INEQUALITIES FOR HAUSDORFF OPERATOR IN LEBESGUE SPACES WITH INDICES 0  p 1

Abstract. In this abstract we establish necessary condition and sufficient condition on weight functions for the boundedness of Hausdorff operator in weighted variable Lebesgue spaces with negative exponents.

Spectral sets of generalized Hausdorff matrices on spaces of holomorphic functions on [formula omitted

Here, we study a family of bounded operators H, acting on Banach spaces of holomorphic functions X↪O(D), which are subordinated in terms of a C0-semigroup of weighted composition operators (vtCϕt), i.e., H=∫0∞vtCϕtdν(t) in the strong sense for some Borel measure ν. This family of operators extends the so-called generalized Hausdorff operators. We obtain the spectrum, point spectrum and essential spectrum of H under mild assumptions on (vtCϕt),ν and X. In particular, we obtain these spectral sets for a wide family of generalized Hausdorff operators acting on Hardy spaces, weighted Bergman spaces, weighted Dirichlet spaces and little Korenblum classes. The description for the spectra of the infinitesimal generator of (vtCϕt) is the key for our findings.

Rough Fractional Hausdorff Operators on Morrey–Herz Spaces with Variable Exponents

Rough Fractional Hausdorff Operators on Morrey–Herz Spaces with Variable Exponents

Generalized Hausdorff operator on Bergmann spaces

Abstract In this article, we considered the generalized Hausdorff operator ℋ μ , ϕ , a {{\mathcal{ {\mathcal H} }}}_{\mu ,\phi ,a} on Bergmann space and determined the conditions on ϕ \phi and a a so that the operator is bounded. In addition, we studied the action of the Hausdorff operator on the truncated domain to estimate norm ℋ μ , ϕ , a {{\mathcal{ {\mathcal H} }}}_{\mu ,\phi ,a} and established a relation with quasi-Hausdorff operator on Bergmann space.

On Hausdorff operators in ZF$\mathsf {ZF}$

AbstractA Hausdorff space is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every with , , where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in , i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice (): is equivalent to “For every set X, the Cantor cube is effectively Hausdorff”. This enhances the result of Howard, Keremedis, Rubin and Rubin [13] that is equivalent to “Hausdorff spaces are effectively Hausdorff” in . The Boolean Prime Ideal Theorem and the statement “For every infinite set X, the Stone space of the Boolean algebra is effectively Hausdorff” are mutually independent. In particular, the latter statement is not provable in . The Axiom of Choice for non‐empty subsets of () is equivalent to each of “Separable Hausdorff spaces are effectively Hausdorff” and “The Cantor cube is effectively Hausdorff”. The Principle of Dependent Choices in conjunction with the Axiom of Choice for continuum sized families of non‐empty subsets of does not imply the axiom of choice for partitions of . The latter independence result fills the gap in information in Howard and Rubin's book “Consequences of the Axiom of Choice”. The axiom of countable choice for non‐empty subsets of is equivalent to each of “Denumerable Hausdorff spaces are effectively Hausdorff”, “Denumerable T3 spaces are completely normal” and “Denumerable Tychonoff spaces are Urysohn”.

Hausdorff operators on compact abelian groups

AbstractNecessary and sufficient conditions are given for the boundedness of Hausdorff operators on the generalized Hardy spaces , real Hardy space , , and for compact Abelian group G. Surprisingly, these conditions turned out to be the same for all groups and spaces under consideration. Applications to Dirichlet series are given. The case of the space of continuous functions on G and examples are also considered.

Sharp constants for multilinear Hausdorff operators on local Morrey-type spaces

Abstract Necessary and sufficient conditions for the boundedness of multilinear Hausdorff operators on local Morrey-type spaces are obtained and sharp constants are also given. As applications, we generalize our results to some concrete operators derived from Hausdorff operators, such as the Hardy operator and its dual operator, the weighted Hardy–Littlewood average, among others.

Hausdorff operators on Fock spaces and a coefficient multiplier problem

Let μ \mu be a positive Borel measure on the positive real axis. We study the integral operator H μ ( f ) ( z ) = ∫ ( 0 , ∞ ) 1 t f ( z t ) d μ ( t ) , z ∈ C , \begin{equation*} \mathcal {H}_{\mu }(f)(z)=\int _{(0,\infty )}\frac {1}{t}f\left (\frac {z}{t}\right ) d\mu (t),\quad z\in \mathbb {C}, \end{equation*} acting on the Fock spaces F α p F^{p}_{\alpha } , p ∈ [ 1 , ∞ ] , α > 0 p\in [1,\infty ],\alpha >0 . Its action is easily seen to be a coefficient multiplication operator by the moment sequence μ n = ∫ [ 1 , ∞ ) 1 t n + 1 d μ ( t ) . \begin{equation*} \mu _n= \int _{[1,\infty )}\frac {1}{t^{n+1}} d\mu (t). \end{equation*} We prove that ‖ H μ ‖ F α p → F α p = ∫ [ 1 , ∞ ) 1 t d μ ( t ) , 1 ≤ p ≤ ∞ . \begin{equation*} \|\mathcal {H}_{\mu }\|_{F^{p}_{\alpha }\to F^{p}_{\alpha }}=\int _{[1,\infty )}\frac {1}{t} d\mu (t),\quad 1\leq p\leq \infty . \end{equation*} It turns out that H μ \mathcal {H}_{\mu } is compact on F α p , p ∈ ( 1 , ∞ ) F^{p}_{\alpha },p\in (1,\infty ) if and only if μ ( { 1 } ) = 0 \mu (\{1\})=0 . In addition, we completely characterize the Schatten class membership of H μ \mathcal {H}_{\mu } .

TO THE SPECTRAL THEORY OF DISCRETE HAUSDORFF OPERATORS

We show that under an arithmetic condition the spectrum of a bounded multidimensional discrete Hausdorff operator in the Lebesgue space is an annulus (or a disc) centered at the origin, provided the perturbation matrices commute and are either positive or negative definite. Conditions for a point spectrum of such an operator to be empty are given and its norm is computed.