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We give conditions on pairs of non-negative functions and which are sufficient that, for ,
Many problems in analysis are described as weighted norm inequalities that have given rise to different classes of weights, such as -weights of Muckenhoupt and -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 of weights w for which the Hardy transform is -bounded. A -weight is precisely one for which its Hardy transform is in , and also a weight whose indefinite...
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.
Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for to hold when and are N-functions with 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.
We prove weighted norm inequalities for the averaging operator Af(x) = 1/x ∫0x f of monotone functions.
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