Abstract
The maps T we shall consider in this chapter act between Lebesgue spaces on an interval (a,b), where b may be infinite, and are of the form \(Tf(x)=v(x)\int_{a}^{x}u(t)f(t)dt,\) u and v being prescribed functions. They are commonly called Hardy operators, or operators of Hardy type, the operator originally studied by Hardy being that in which a=0,b= 8 and v=u= 1. Necessary and sufficient conditions for the boundedness or compactness of T are given.When u and v are both identically equal to 1 and b is finite, the exact value of the norm of T is determined; it is shown that it is attained at a function expressible in terms of generalised trigonometric functions.
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© 2011 Springer-Verlag Berlin Heidelberg
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Lang, J., Edmunds, D. (2011). Hardy Operators. In: Eigenvalues, Embeddings and Generalised Trigonometric Functions. Lecture Notes in Mathematics(), vol 2016. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18429-1_4
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DOI: https://doi.org/10.1007/978-3-642-18429-1_4
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-18267-9
Online ISBN: 978-3-642-18429-1
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