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Weighted Hardy inequalities and Hardy transforms of weights

Joan Cerdà, Joaquim Martín (2000)

Studia Mathematica

Many problems in analysis are described as weighted norm inequalities that have given rise to different classes of weights, such as A p -weights of Muckenhoupt and B p -weights of Ariño and Muckenhoupt. Our purpose is to show that different classes of weights are related by means of composition with classical transforms. A typical example is the family M p of weights w for which the Hardy transform is L p ( w ) -bounded. A B p -weight is precisely one for which its Hardy transform is in M p , and also a weight whose indefinite...

Weighted Hardy's inequalities for negative indices.

Dmitryi V. Prokhorov (2004)

Publicacions Matemàtiques

In the paper we obtain a precise characterization of Hardy type inequalities with weights for the negative indices and the indices between 0 and 1 and establish a duality between these cases.

Weighted L Φ integral inequalities for operators of Hardy type

Steven Bloom, Ron Kerman (1994)

Studia Mathematica

Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for Φ 2 - 1 ( ʃ Φ 2 ( w ( x ) | T f ( x ) | ) t ( x ) d x ) Φ 1 - 1 ( ʃ Φ 1 ( C u ( x ) | f ( x ) | ) v ( x ) d x ) to hold when Φ 1 and Φ 2 are N-functions with Φ 2 Φ 1 - 1 convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored.

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