A Hardy-type inequality in two dimensions (original) (raw)

Weighted Norm Inequalities for Integral Operators

1996

We consider a large class of positive integral operators acting on functions which are dened on a space of homogeneous type with a group struc- ture. We show that any such operator has a discrete (dyadic) version which is always essentially equivalent in norm to the original operator. As an appli- cation, we study conditions of \testing type," like those

On weighted norm integral inequality of G. H. Hardy’s type

In this paper, we give a necessary and sufficient condition on Hardy's integral inequality: X [T f ] p wdµ ≤ C X f p vdµ ∀f ≥ 0 (1) where w, v are non-negative measurable functions on X, a non-negative function f defined on (0, ∞), K(x, y) is a non-negative and measurable on X ×X, (T f)(x) = ∞ 0 K(x, y)f (y)dy and C is a constant depending on K, p but independent of f. This work is a continuation of our recent result in [9].

On Some Integral Inequalities of Hardy-Type Operators

Advances in Pure Mathematics, 2013

In recent time, hardy integral inequalities have received attentions of many researchers. The aim of this paper is to obtain new integral inequalities of hardy-type which complement some recent results.

A Characterization of a Two-Weight Inequality for Discrete Two-Dimensional Hardy Operators

Zeitschrift für Analysis und ihre Anwendungen, 1999

We establish necessary and sufficient conditions on a weight pair (v, w) governing the boundedness of the Riesz potential operator I α defined on a homogeneous group G from L p dec,r (w, G) to L q (v, G), where L p dec,r (w, G) is the Lebesgue space defined for non-negative radially decreasing functions on G. The same problem is also studied for the potential operator with product kernels I α 1 ,α 2 defined on a product of two homogeneous groups G 1 × G 2. In the latter case weights, in general, are not of product type. The derived results are new even for Euclidean spaces. To get the main results we use Sawyer-type duality theorems (which are also discussed in this paper) and two-weight Hardy-type inequalities on G and G 1 × G 2 , respectively. MSC: 42B20; 42B25

On a Hardy Type General Weighted Inequality in Spaces L p(·)

Integral Equations and Operator Theory, 2010

A Hardy type two-weighted inequality is investigated for the multidimensional Hardy operator in the norms of generalized Lebesgue spaces L p(·) . Equivalent necessary and sufficient conditions are found for the L p(·) −→ L q(·) boundedness of the Hardy operator when exponents q(0) < p(0), q(∞) < p(∞). It is proved that the condition for such an inequality to hold coincides with the condition for the validity of two-weighted Hardy inequalities with constant exponents if we require of the exponents to be regular near zero and at infinity.

Weighted weak-type inequalities for generalized Hardy operators

Journal of Inequalities and Applications, 2006

We characterize the pairs of weights (v,w) for which the Hardy-Steklov-type operator T f (x) = g(x) h(x) s(x) K(x, y) f (y)dy applies L p (v) into weak-L q (w), q < p, assuming certain monotonicity conditions on g, s, h, and K.

On Hardy Inequality in Variable Lebesgue Spaces with Mixed Norm

Indian Journal of Pure and Applied Mathematics, 2018

In this talk, 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.

An integral operator inequality with applications

Journal of Inequalities and Applications, 1999

Linear integral operators are defined acting in the Lebesgue integration spaces on intervals of the real line. A necessary and sufficient condition is given for these operators to be bounded, and a characterisation is given for the operator bounds. There are applications of the results to integral inequalities; also to properties of the domains of self-adjoint unbounded operators, in Hilbert function spaces, associated with the classical orthogonal polynomials and their generalisations.