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

Kenneth Andersen, Benjamin Muckenhoupt (1982)

Studia Mathematica

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Kenneth Andersen, Benjamin Muckenhoupt (1982)

Studia Mathematica

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Néstor Aguilera, Carlos Segovia (1977)

Studia Mathematica

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Shuichi Sato (1989)

Studia Mathematica

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Christoph J. Neugebauer (1991)

Publicacions Matemàtiques

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We prove weighted norm inequalities for the averaging operator Af(x) = 1/x ∫ f of monotone functions.

E. Sawyer (1985)

Studia Mathematica

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Michelangelo Franciosi (1989)

Studia Mathematica

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Steven Bloom (1997)

Studia Mathematica

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Let ${T}_{\gamma}f\left(x\right)={\u0283}_{0}^{x}k{(x,y)}^{\gamma}f\left(y\right)dy$, where k is a nonnegative kernel increasing in x, decreasing in y, and satisfying a triangle inequality. An nth-order Opial inequality has the form ${\u0283}_{0}^{\infty}({\prod}_{i=1}^{n}|{T}_{{\gamma}_{i}}{f\left(x\right)|}^{{q}_{i}}{\left|\right)\left|f\left(x\right)\right|}^{{q}_{0}}w\left(x\right)dx\le C({\u0283}_{0}^{\infty}{\left|f\left(x\right)\right|}^{p}{v\left(x\right)dx)}^{({q}_{0}+\dots +{q}_{n})/p}$. Such inequalities can always be simplified to nth-order reduced inequalities, where the exponent ${q}_{0}=0$. When n = 1, the reduced inequality is a standard weighted norm inequality, and characterizing the weights is easy. We also find necessary and sufficient conditions on the weights for second-order reduced Opial inequalities to hold. ...

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.

José García-Cuerva (1991)

Publicacions Matemàtiques

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I am going to discuss the work José Luis Rubio did on weighted norm inequalities. Most of it is in the book we wrote together on the subject [12].

David Cruz-Uribe, SFO, C. Neugebauer, V. Olesen (1995)

Studia Mathematica

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We introduce the one-sided minimal operator, ${m}^{+}f$, which is analogous to the one-sided maximal operator. We determine the weight classes which govern its two-weight, strong and weak-type norm inequalities, and show that these two classes are the same. Then in the one-weight case we use this class to introduce a new one-sided reverse Hölder inequality which has several applications to one-sided $\left({A}_{p}^{+}\right)$ weights.