Maximal Functions on Classical Lorentz Spaces and Hardy's Inequality with Weights for Nonincreasing Functions

Ariño, Miguel Á.; Muckenhoupt, Benjamin

doi:10.2307/2001699

Cited by 100 publications

(144 citation statements)

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“…Hence, the function CT, E (pJ is well defined and it is an increasing convex function. As was proved in [3], we have that M is bounded on Ap(wo) and so, Therefore F E T, (A'(wP,)) and by hypothesis and Theorem 3.13 we can find a decomposition F = x j A j b j where, llFllTm(A1(wE)) ~l l (~f ) p l l " l ( w~) = cll~fl15Pp(wo) 5 cllfl15Pp(wo) < 02. (ii) Another interesting example is given by the following choice of the function parameters: Suppose 0 < 8 < 1 and 0 I y < min(28, 2 (1 -8)) and set Since we need to assume the condition cp$((t)/tP-l EB, we easily find that this is equivalent to also assuming that p < 2/(2 -28, + yo).…”

Section: Weighted Inequalities For Maximal Functionssupporting

confidence: 52%

Weighted Tent Spaces

Soria

1992

Mathematische Nachrichten

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We generalize the theory of tent spaces introduced in [9] and [lo], to consider weighted norms related to some function parameters (see [ll]). We study their atomic decomposition, from which we obtain a weighted inequality for a certain fractional maximal operator. We also find the dual spaces, and get a new class of CARLESON measures and we identify the intermediate spaces when using several methods of interpolation. 0 1.

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Section: Weighted Inequalities For Maximal Functionssupporting

confidence: 52%

Weighted Tent Spaces

Soria

1992

Mathematische Nachrichten

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“…The first implication depends on his duality principle [50] (Theorem 1). The second implication uses the argument in Lemma 2.1 of [1], which itself follows from a result (Lemma 21) of Stromberg and Torchinsky [53]. The third implication uses (3.5), and the fourth is elementary.…”

Section: Theorem Cmentioning

confidence: 87%

Weighted Fourier Inequalities: New Proofs and Generalizations

Benedetto

Heinig

2003

Journal of Fourier Analysis and Applications

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Fourier transform inequalities in weighted Lebesgue spaces are proved. The inequalities are generalizations of the Plancherel theorem, they are characterized in terms of uncertainty principle relations between pairs of weights, and they are put in the context of existing weighted Fourier transform inequalities. The proofs are new and relatively elementary, and they give rise to good and explicit constants controlling the continuity of the Fourier transform operator. The smaller the constant is, the more applicable the inequality will be in establishing weighted uncertainty principle or entropy inequalities. There are two essentially different proofs, one depending on operator theory and one depending on Lorentz spaces. The results from these approaches are quantitatively compared, leading to classical questions concerning multipliers and to new questions concerning wavelets.

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“…The following example shows that the assumption that w is strictly decreasing cannot be omitted. In general there are more smooth points in Λ p,w than in Γ p,w , for given 1 ≤ p < ∞ and a decreasing weight w. 1) . Assume also that f = w. By [22, Theorem 2.14], the left-and right-hand Gâteaux derivatives of the norm in Λ p,w are given by…”

Section: Theorem 53mentioning

confidence: 99%

“…Γ p,w is an interpolation space between L 1 and L ∞ yielded by the Lions-Peetre K-method [4,17]. Obviously Γ p,w ⊂ Λ p,w , and it is wellknown that they coincide as sets with equivalent (quasi) norms if and only if the Hardy operator H 1 is bounded on Λ p,w , which is equivalent to the so-called condition B p , an integral condition satisfied by the weight w [1,27,30]. In particular, the Hardy operator is bounded on L q,p := Λ p,w with w(t) = t p/q−1 , 1 < p, q < ∞ [12], and in that case Λ p,w = Γ p,w as sets with equivalent (quasi) norms.…”

Section: Introductionmentioning

confidence: 99%

Gâteaux derivatives and their applications to approximation in Lorentz spaces Γ_p,w

Ciesielski

Kamińska

Płuciennik

2009

Mathematische Nachrichten

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We establish the formulas of the left‐ and right‐hand Gâteaux derivatives in the Lorentz spaces Γp,w = {f: ∫0α (f **)p w < ∞}, where 1 ≤ p < ∞, w is a nonnegative locally integrable weight function and f ** is a maximal function of the decreasing rearrangement f * of a measurable function f on (0, α), 0 < α ≤ ∞. We also find a general form of any supporting functional for each function from Γp,w , and the necessary and sufficient conditions for which a spherical element of Γp,w is a smooth point of the unit ball in Γp,w . We show that strict convexity of the Lorentz spaces Γp,w is equivalent to 1 < p < ∞ and to the condition ∫0∞ w = ∞. Finally we apply the obtained characterizations to studies the best approximation elements for each function f ∈ Γp,w from any convex set K ⊂ Γp,w (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Maximal Functions on Classical Lorentz Spaces and Hardy's Inequality with Weights for Nonincreasing Functions

Cited by 100 publications

References 0 publications

Weighted Tent Spaces

Weighted Tent Spaces

Weighted Fourier Inequalities: New Proofs and Generalizations

Gâteaux derivatives and their applications to approximation in Lorentz spaces Γ_p,w

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Maximal Functions on Classical Lorentz Spaces and Hardy's Inequality with Weights for Nonincreasing Functions

Cited by 100 publications

References 0 publications

Weighted Tent Spaces

Weighted Tent Spaces

Weighted Fourier Inequalities: New Proofs and Generalizations

Gâteaux derivatives and their applications to approximation in Lorentz spaces Γp,w

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Gâteaux derivatives and their applications to approximation in Lorentz spaces Γ_p,w