Abstract
In [118] Hardy proved the following celebrated inequality: let 1 < p < ∞ and set F(x) = ∫ o x f (t)dt, where f is a non-negative measurable function on (0, ∞). Then, if ε < 1/p′ = 1–1/p,
for some constant C > 0 independent of f. If ε > 1/p′, the inequality takes the form
where G(x) = ∫ ∞ x f (t)dt. The best possible constants C in (2.1.1) and (2.1.2) are equal and this common value was determined by Landau in [150] as
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© 2004 Springer-Verlag Berlin Heidelberg
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Edmunds, D.E., Evans, W.D. (2004). Hardy-type Operators. In: Hardy Operators, Function Spaces and Embeddings. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-07731-3_2
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DOI: https://doi.org/10.1007/978-3-662-07731-3_2
Publisher Name: Springer, Berlin, Heidelberg
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