We prove that for a decreasing weight w, the following inequality is sharp:where B p is the Ariño and Muckenhoupt class of weights, and p 2. The case w ≡ 1 gives a positive answer to a conjecture formulated in Kruglyak and Setterqvist (2008) [8], where this estimate is proved only when p 2 is an integer. Simple examples show that, for 1 < p < 2, or if w is not decreasing, the result is false. Finally, using a different argument, we also prove that in the case p = 1, and for arbitrary weights w ∈ B 1 , w B 1 is the best constant in the corresponding inequality.
We characterize the weights w such that ð y 0 f à ðsÞ p wðsÞ ds F ð y 0 ð f Ãà ðsÞ À f à ðsÞÞ p wðsÞ ds:Our result generalizes a result due to Bennett-De Vore-Sharpley, where the usual Lorentz L p; q norm is replaced by an equivalent expression involving the functional f Ãà À f à . Su‰cient conditions for the boundedness of maximal Calderó n-Zygmund singular integral operators between classical Lorentz spaces are also given.
The work of Coifman and Weiss concerning Hardy spaces on spaces of homogeneous type gives, as a particular case, a definition of Hp(ZN) in terms of an atomic decomposition.Other characterizations of these spaces have been studied by other authors, but it was an open question to see if they can be defined, as it happens in the classical case, in terms of a maximal function or via the discrete Riesz transforms.In this paper, we give a positive answer to this question.
We study the optimal bounds for the Hardy operator S minus the identity, as well as S and its dual operator S * , on the full range 1 ≤ p ≤ ∞, for the cases of decreasing, positive or general functions (in fact, these two kind of inequalities are equivalent for the appropriate cone of functions). For 1 < p ≤ 2, we prove that all these estimates are the same, but for 2 < p < ∞, they exhibit a completely different behavior.
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