Some weighted inequalities for general one-sided maximal operators

F. Martín-Reyes; A. de la Torre

Displaying similar documents to “Some weighted inequalities for general one-sided maximal operators”

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

Kenneth Andersen, Benjamin Muckenhoupt (1982)

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Two-weight weak type maximal inequalities in Orlicz classes

Luboš Pick (1991)

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Necessary and sufficient conditions are shown in order that the inequalities of the form $ϱ (M_{μ} f > λ) Φ (λ) \leq C ʃ_{X} Ψ (C | f (x) |) σ (x) d μ$ , or $ϱ (M_{μ} f > λ) \leq C ʃ_{X} Φ (C λ^{- 1} | f (x) |) σ (x) d μ$ hold with some positive C independent of λ > 0 and a μ-measurable function f, where (X,μ) is a space with a complete doubling measure μ, $M_{μ}$ is the maximal operator with respect to μ, Φ, Ψ are arbitrary Young functions, and ϱ, σ are weights, not necessarily doubling.

Two weighted inequalities for convolution maximal operators.

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

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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 < ∞.

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.

Weighted Hardy inequalities and Hardy transforms of weights

Joan Cerdà, Joaquim Martín (2000)

Studia Mathematica

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Many problems in analysis are described as weighted norm inequalities that have given rise to different classes of weights, such as $A_{p}$ -weights of Muckenhoupt and $B_{p}$ -weights of Ariño and Muckenhoupt. Our purpose is to show that different classes of weights are related by means of composition with classical transforms. A typical example is the family $M_{p}$ of weights w for which the Hardy transform is $L_{p} (w)$ -bounded. A $B_{p}$ -weight is precisely one for which its Hardy transform is in $M_{p}$ , and also a weight...

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

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 $(A_{p}^{+})$ weights.

Weighted generalized weak type inequalities for modified Hardy operators.

Pedro Ortega Salvador (2000)

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Two-weight mixed ф-inequalities for the one-sided maximal function

Qinsheng Lai (1995)

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Weighted $L_{Φ}$ integral inequalities for operators of Hardy type

Steven Bloom, Ron Kerman (1994)

Studia Mathematica

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Necessary and sufficient conditions are given on the weights t, u, v, and w, in order for $Φ_{2}^{- 1} (ʃ Φ_{2} (w (x) | T f (x) |) t (x) d x) \leq Φ_{1}^{- 1} (ʃ Φ_{1} (C u (x) | f (x) |) v (x) d x)$ to hold when $Φ_{1}$ and $Φ_{2}$ are N-functions with $Φ_{2} \circ Φ_{1}^{- 1}$ convex, and T is the Hardy operator or a generalized Hardy operator. Weak-type characterizations are given for monotone operators and the connection between weak-type and strong-type inequalities is explored.

Weighted inequalities for one-sided maximal functions in Orlicz spaces

Pedro Ortega Salvador (1998)

Studia Mathematica

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Let $M_{g}^{+}$ be the maximal operator defined by $M_{g}^{+} ⨍ (x) = s u p_{h > 0} (ʃ_{x}^{x + h} | ⨍ | g) / (ʃ_{x}^{x + h} g)$ , where g is a positive locally integrable function on ℝ. Let Φ be an N-function such that both Φ and its complementary N-function satisfy $Δ_{2}$ . We characterize the pairs of positive functions (u,ω) such that the weak type inequality $u (x \in ℝ | M_{g}^{+} ⨍ (x) > λ) \leq C / (Φ (λ)) ʃ_{ℝ} Φ (| ⨍ |) ω$ holds for every ⨍ in the Orlicz space $L_{Φ} (ω)$ . We also characterize the positive functions ω such that the integral inequality $ʃ_{ℝ} Φ (| M_{g}^{+} ⨍ |) ω \leq ʃ_{ℝ} Φ (| ⨍ |) ω$ holds for every $⨍ \in L_{Φ} (ω)$ . Our results include some already obtained for functions in $L^{p}$ and yield...

On boundedness properties of certain maximal operators

M. Menárguez (1995)

Colloquium Mathematicae

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It is known that the weak type (1,1) for the Hardy-Littlewood maximal operator can be obtained from the weak type (1,1) over Dirac deltas. This theorem is due to M. de Guzmán. In this paper, we develop a technique that allows us to prove such a theorem for operators and measure spaces in which Guzmán's technique cannot be used.

ω-Calderón-Zygmund operators

Sijue Wu (1995)

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We prove a T1 theorem and develop a version of Calderón-Zygmund theory for ω-CZO when $ω \in A_{\infty}$ .

Restrictions from $ℝ^{n}$ to $ℤ^{n}$ of weak type (1,1) multipliers

Nakhlé Asmar, Earl Berkson, Jean Bourgain (1994)

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