The one-sided minimal operator and the one-sided reverse Holder inequality

David Cruz-Uribe; SFO; C. Neugebauer; V. Olesen

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Norm inequalities for the minimal and maximal operator, and differentiation of the integral.

David Cruz-Uribe, Christoph J. Neugebauer, Victor Olesen (1997)

Publicacions Matemàtiques

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We study the weighted norm inequalities for the minimal operator, a new operator analogous to the Hardy-Littlewood maximal operator which arose in the study of reverse Hölder inequalities. We characterize the classes of weights which govern the strong and weak-type norm inequalities for the minimal operator in the two weight case, and show that these classes are the same. We also show that a generalization of the minimal operator can be used to obtain information about the differentiability...

Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions

Kenneth Andersen, Benjamin Muckenhoupt (1982)

Studia Mathematica

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Some weighted inequalities for general one-sided maximal operators

F. Martín-Reyes, A. de la Torre (1997)

Studia Mathematica

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We characterize the pairs of weights on ℝ for which the operators $M_{h, k}^{+} f (x) = s u p_{c > x} h (x, c) ʃ_{x}^{c} f (s) k (x, s, c) d s$ are of weak type (p,q), or of restricted weak type (p,q), 1 ≤ p < q < ∞, between the Lebesgue spaces with the coresponding weights. The functions h and k are positive, h is defined on $(x, c) : x < c$ , while k is defined on $(x, s, c) : x < s < c$ . If $h (x, c) = {(c - x)}^{- β}$ , $k (x, s, c) = {(c - s)}^{α - 1}$ , 0 ≤ β ≤ α ≤ 1, we obtain the operator $M_{α, β}^{+} f = s u p_{c > x} 1 / {(c - x)}^{β} ʃ_{x}^{c} f (s) / {(c - s)}^{1 - α} d s$ . For this operator, under the assumption 1/p - 1/q = α - β, we extend the weak type characterization to the case p = q and prove that in the case of equal...

Weighted norm inequalities relating the $g *_{λ}$ and the area functions

Néstor Aguilera, Carlos Segovia (1977)

Studia Mathematica

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Equivalence of norms in one-sided Hp spaces.

Liliana de Rosa, Carlos Segovia (2002)

Collectanea Mathematica

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One-sided versions of maximal functions for suitable defined distributions are considered. Weighted norm equivalences of these maximal functions for weights in the Sawyer's Aq+ classes are obtained.

Weighted weak type inequalities for certain maximal functions

Hugo Aimar, Liliana Forzani (1991)

Studia Mathematica

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We give an A_p type characterization for the pairs of weights (w,v) for which the maximal operator Mf(y) = sup 1/(b-a) ʃ_a^b |f(x)|dx, where the supremum is taken over all intervals [a,b] such that 0 ≤ a ≤ y ≤ b/ψ(b-a), is of weak type (p,p) with weights (w,v). Here ψ is a nonincreasing function such that ψ(0) = 1 and ψ(∞) = 0.

Two weighted inequalities for convolution maximal operators.

Ana Lucía Bernardis, Francisco Javier Martín-Reyes (2002)

Publicacions Matemàtiques

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Let φ: R → [0,∞) an integrable function such that φχ = 0 and φ is decreasing in (0,∞). Let τf(x) = f(x-h), with h ∈ R {0} and f(x) = 1/R f(x/R), with R > 0. In this paper we characterize the pair of weights (u, v) such that the operators Mf(x) = sup|f| * [τφ](x) are of weak type (p, p) with respect to (u, v), 1 < p < ∞.

A stability result on Muckenhoupt's weights.

Juha Kinnunen (1998)

Publicacions Matemàtiques

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We prove that Muckenhoupt's A-weights satisfy a reverse Hölder inequality with an explicit and asymptotically sharp estimate for the exponent. As a by-product we get a new characterization of A-weights.

Weighted norm inequalities for generalized Hankel conjugate transformations

K. Andersen, R. Kerman (1981)

Studia Mathematica

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The Muckenhoupt class A₁(R)

B. Bojarski, C. Sbordone, I. Wik (1992)

Studia Mathematica

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It is shown that the Muckenhoupt structure constants for f and f* on the real line are the same.

The equivalence of two conditions for weight functions

Benjamin Muckenhoupt (1974)

Studia Mathematica

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