Hardy Operator Research Articles

Some new inequalities involving the Hardy operator

AbstractIn this paper we derive some new inequalities involving the Hardy operator, using some estimates of the Jensen functional, continuous form generalization of the Bellman inequality and a Banach space variant of it. Some results are generalized to the case of Banach lattices on .

Norm Dynamic Inequalities and Theorems of Factorization of Weighted Cesàro and Copson Spaces

In this paper, we establish some factorization theorems for weighted Cesaro and Copson spaces, obtain two sided norm dynamic inequalities, and give conditions for the boundedness of the Hardy and Copson dynamic operators on the weighted space $$L_{\lambda }^{p}({\mathbb {T}})$$ . We obtain, as special cases, the classical integral inequalities on $${\mathbb {R}}$$ and the discrete inequalities on $${\mathbb {N}}$$ .

Modified Fractional Hardy and Hardy-Littlewood Operators and Their Commutators

For modified fractional Hardy and Hardy-Littlewood operators and their commutators with symbol from a central mean oscillation space, considered as acting from a modified Herz space into another one, we find conditions of their boundedness. The sharpness of the result concerning commutators of the fractional Hardy-Littlewood operator is established.

Invariant linear functionals on [formula omitted

We consider a continuous version of the classical notion of Banach limits, i.e., normalized positive linear functionals on L∞(R+) invariant under translations f(x)↦f(x+s) of L∞(R+) for every s≥0. We give one of its characterizations in terms of the invariance under the operation of a certain linear transformation on L∞(R+). We also deal with invariant linear functionals under dilations f(x)↦f(rx), r≥1 and give a similar characterization via the Hardy operator. Applications to summability methods are presented in the last section.

New Characterizations of Weights in Hardy and Opial Type Inequalities via Solvability of Dynamic Equations

In this paper, we prove that the solvability of dynamic equations of second order is sufficient for the validity of some Hardy and Opial type inequalities with two different weights on time scales. In particular, the results give new characterizations of two different weights in inequalities containing Hardy and Opial operators. The main contribution in this paper is the characterizations of weights in discrete inequalities that will be formulated from our results as special cases.

Some properties related to trace inequalities for the multi-parameter Hardy operators on poly-trees

In this note we investigate the multi-parameter Potential Theory on the weighted d-tree (Cartesian product of several copies of uniform dyadic tree), which is connected to the discrete models of weighted Dirichlet spaces on the polydisc. We establish some basic properties of the respective potentials, capacities and equilibrium measures (in particular in the case of product polynomial weights). We explore multi-parameter Hardy inequality and its trace measures, and discuss some open problems of potential-theoretic and combinatorial nature.

Equivalence of Sobolev Norms Involving Generalized Hardy Operators

Abstract We consider the fractional Schrödinger operator with Hardy potential and critical or subcritical coupling constant. This operator generates a natural scale of homogeneous Sobolev spaces, which we compare with the ordinary homogeneous Sobolev spaces. As a byproduct, we obtain generalized and reversed Hardy inequalities for this operator. Our results extend those obtained recently for ordinary (non-fractional) Schrödinger operators and have an important application in the treatment of large relativistic atoms.

Estimate for the Entropy Numbers of the Weighted Hardy Operators that Act from Banach Space to q-Banach Space

We study weighted Hardy operators from the Banach space Lp to the q-Banach space Lq and obtain estimates for the entropy numbers of one- and two-weighted Hardy operators with nonnegative measurable weight functions.

Weighted Morrey–Herz space estimates for rough Hausdorff operator and its commutators

In this paper, we give necessary and sufficient conditions for the boundedness of rough Hausdorff operators on Herz, Morrey and Morrey–Herz spaces with absolutely homogeneous weights. Especially, the estimates for operator norms in each case are worked out. Moreover, we also establish the boundedness of the commutators of rough Hausdorff operators on the two weighted Morrey–Herz type spaces with their symbols belonging to Lipschitz space. By these, we obtain some new estimates for the high dimensional Hardy operator and adjoint Hardy operator.

Estimate for the entropy numbers of the weighted Hardy operators that act from Banach space to $q$-Banach space

Estimate for the entropy numbers of the weighted Hardy operators that act from Banach space to $q$-Banach space

Commutators of the Bilinear Hardy Operator on Herz Type Spaces with Variable Exponents

In this paper, we obtain the boundedness of bilinear commutators generated by the bilinear Hardy operator and BMO functions on products of Herz spaces and Herz-Morrey spaces with variable exponents.

Mixed Norm Inequalities for Lebesgue Spaces

We give necessary and sufficient condition for the two-dimensional mixed norm Hardy inequality to hold with functions that are non-increasing in each variable. The corresponding inequalities involving adjoint operator and the mixed Hardy operators are also discussed.

Weighted inequalities for fractional Hardy operators and commutators

In this paper, we introduce a fractional maximal operators N_{alpha } on (0,infty ) associated to the fractional Hardy operator P_{alpha } and its dual Q_{alpha }, 0leq alpha <1, and obtain some characterizations for the one-weight and two-weight inequalities for N_{alpha }. We also give some A_{p} type sufficient conditions for the two-weight inequalities for the fractional Hardy operators, the dual operators and the commutators of the fractional Hardy operators with CMO functions.

On n-dimensional fractional Hardy operators and commutators in variable Herz-type spaces

Based on the theory of variable exponents and atomic decomposition, we study the boundedness of n-dimensional fractional Hardy operators on variable Herz and Herz-type Hardy spaces, where the three main indices are variable exponents. The corresponding boundedness for the mth order commutators generated by the n-dimensional fractional Hardy operators and bounded mean oscillation (BMO) function are also considered. We note that, even in the special case of m=1, the obtained results are also new.

Дробные модифицированные операторы Харди и Харди-Литтлвуда и их коммутаторы

Дробные модифицированные операторы Харди и Харди-Литтлвуда и их коммутаторы

Two-weighted Hardy operators in $L^{p(\cdot )}$ spaces and applications

Two-weighted Hardy operators in $L^{p(\cdot )}$ spaces and applications

Sawyer's duality principle for grand Lebesgue spaces

AbstractThe aim of this paper is to extend Sawyer's duality principle from the cone of decreasing functions of the Lebesgue space to the cone of decreasing functions of the grand Lebesgue space and to prove the boundedness of classical Hardy operators in the grand Lebesgue spaces.

On Hardy Inequality in Variable Lebesgue Spaces with Mixed Norm

In this paper a two-weight boundedness of multidimensional Hardy operator and its dual operator acting from one weighted variable Lebesgue spaces with mixed norm into other weighted variable Lebesgue spaces with mixed norm spaces is proved. In particular, a new type two-weight criterion for multidimensional Hardy operator is obtained.

Hausdorff operators on Sobolev spaces Wk,1

ABSTRACTThis paper is served as a first contribution regarding the boundedness of Hausdorff operators on function spaces with smoothness. The sharp conditions are established for boundedness of Hausdorff operators on Sobolev spaces . As applications, some bounded and unbounded properties on of Hardy operator and adjoint Hardy operator are deduced.

A Characterization of Approximation of Hardy Operators in VLS

Variable exponent spaces and Hardy operator space have played an important role in recent harmonic analysis because they have an interesting norm including both local and global properties. The variable exponent Lebesgue spaces are of interest for their applications to modeling problems in physics, and to the study of variational integrals and partial differential equations with non-standard growth conditions. This studies also has been stimulated by problems of elasticity, fluid dynamics, calculus of variations, and differential equations with non-standard growth conditions. In this study, we will discuss a characterization of approximation of Hardy operators in variable Lebesgue spaces.