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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 inequalities for monotone and concave functions

Hans Heinig, Lech Maligranda (1995)

Studia Mathematica

Characterizations of weight functions are given for which integral inequalities of monotone and concave functions are satisfied. The constants in these inequalities are sharp and in the case of concave functions, constitute weighted forms of Favard-Berwald inequalities on finite and infinite intervals. Related inequalities, some of Hardy type, are also given.

Weighted inequalities for monotone functions.

L. Maligranda (1997)

Collectanea Mathematica

We give characterizations of weights for which reverse inequalities of the Hölder type for monotone functions are satisfied. Our inequalities with general weights and with sharp constants complement previous results.

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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