Two-weight weak type maximal inequalities in Orlicz classes

Luboš Pick

Displaying similar documents to “Two-weight weak type maximal inequalities in Orlicz classes”

Necessary and sufficient conditions for the two-weight weak type maximal inequality in Orlicz class

Yanbo Ren, Shuang Ding (2022)

Czechoslovak Mathematical Journal

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We collect known and prove new necessary and sufficient conditions for the weighted weak type maximal inequality of the form $Φ_{1} (λ) ϱ ({x \in X : M_{μ} f (x) > λ}) \leq c \int_{X} Φ_{2} (c | f (x) |) σ (x) d μ (x),$ which extends some known results.

Weak bounds for the maximal function in weighted Orlicz spaces

Richard Bagby (1990)

Studia Mathematica

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Weighted Orlicz space integral inequalities for the Hardy-Littlewood maximal operator

S. Bloom, R. Kerman (1994)

Studia Mathematica

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Necessary and sufficient conditions are given for the Hardy-Littlewood maximal operator to be bounded on a weighted Orlicz space when the complementary Young function satisfies $Δ_{2}$ . Such a growth condition is shown to be necessary for any weighted integral inequality to occur. Weak-type conditions are also investigated.

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

Qinsheng Lai (1995)

Studia Mathematica

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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...

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 norm inequalities on spaces of homogeneous type

Qiyu Sun (1992)

Studia Mathematica

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We give a characterization of the weights (u,w) for which the Hardy-Littlewood maximal operator is bounded from the Orlicz space L_Φ(u) to L_Φ(w). We give a characterization of the weight functions w (respectively u) for which there exists a nontrivial u (respectively w > 0 almost everywhere) such that the Hardy-Littlewood maximal operator is bounded from the Orlicz space L_Φ(u) to L_Φ(w).

Reverse-Holder classes in the Orlicz spaces setting

E. Harboure, O. Salinas, B. Viviani (1998)

Studia Mathematica

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In connection with the $A_{ϕ}$ classes of weights (see [K-T] and [B-K]), we study, in the context of Orlicz spaces, the corresponding reverse-Hölder classes $R H_{ϕ}$ . We prove that when ϕ is $Δ_{2}$ and has lower index greater than one, the class $R H_{ϕ}$ coincides with some reverse-Hölder class $R H_{q}, q > 1$ . For more general ϕ we still get $R H_{ϕ} \subset A_{\infty} = ⋃_{q > 1} R H_{q}$ although the intersection of all these $R H_{ϕ}$ gives a proper subset of $⋂_{q > 1} R H_{q}$ .

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...

Gagliardo-Nirenberg inequalities in weighted Orlicz spaces

Agnieszka Kałamajska, Katarzyna Pietruska-Pałuba (2006)

Studia Mathematica

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We derive inequalities of Gagliardo-Nirenberg type in weighted Orlicz spaces on ℝⁿ, for maximal functions of derivatives and for the derivatives themselves. This is done by an application of pointwise interpolation inequalities obtained previously by the first author and of Muckenhoupt-Bloom-Kerman-type theorems for maximal functions.

On the two-weight problem for singular integral operators

David Cruz-Uribe, Carlos Pérez (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We give $A_{p}$ type conditions which are sufficient for two-weight, strong $(p, p)$ inequalities for Calderón-Zygmund operators, commutators, and the Littlewood-Paley square function $g_{λ}^{*}$ . Our results extend earlier work on weak $(p, p)$ inequalities in [13].