Displaying similar documents to “Necessary and sufficient conditions for the two-weight weak type maximal inequality in Orlicz class”

Two-weight weak type maximal inequalities in Orlicz classes

Luboš Pick (1991)

Studia Mathematica

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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 ) ≤ C ʃ X Ψ ( C | f ( x ) | ) σ ( x ) d μ , or ϱ ( M μ f > λ ) ≤ 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.

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.

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 ∈ ℝ | M g + ⨍ ( x ) > λ ) ≤ 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 ≤ ʃ ℝ Φ ( | ⨍ | ) ω holds for every ⨍ ∈ 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 ) ≤ 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 ∘ 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.

Maximal function in Beurling-Orlicz and central Morrey-Orlicz spaces

Lech Maligranda, Katsuo Matsuoka (2015)

Colloquium Mathematicae

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We define Beurling-Orlicz spaces, weak Beurling-Orlicz spaces, Herz-Orlicz spaces, weak Herz-Orlicz spaces, central Morrey-Orlicz spaces and weak central Morrey-Orlicz spaces. Moreover, the strong-type and weak-type estimates of the Hardy-Littlewood maximal function on these spaces are investigated.

Orlicz spaces for which the Hardy-Littlewood maximal operators is bounded.

Diego Gallardo (1988)

Publicacions Matemàtiques

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Let M be the Hardy-Littlewood maximal operator defined by: Mf(x) = supx ∈ Q 1/|Q| ∫Q |f| dx, (f ∈ Lloc(Rn)), where the supreme is taken over all cubes Q containing x and |Q| is the Lebesgue measure of Q. In this paper we characterize the Orlicz spaces Lφ*, associated to N-functions φ, such that M is bounded in Lφ*....

Orlicz-Morrey spaces and the Hardy-Littlewood maximal function

Eiichi Nakai (2008)

Studia Mathematica

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We prove basic properties of Orlicz-Morrey spaces and give a necessary and sufficient condition for boundedness of the Hardy-Littlewood maximal operator M from one Orlicz-Morrey space to another. For example, if f ∈ L(log L)(ℝⁿ), then Mf is in a (generalized) Morrey space (Example 5.1). As an application of boundedness of M, we prove the boundedness of generalized fractional integral operators, improving earlier results of the author.

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 ⊂ A ∞ = ⋃ q > 1 R H q although the intersection of all these R H ϕ gives a proper subset of ⋂ q > 1 R H q .

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).

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.

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...