Displaying similar documents to “Weighted L Φ integral inequalities for operators of Hardy type”

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 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...

First and second order Opial inequalities

Steven Bloom (1997)

Studia Mathematica

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Let T γ f ( x ) = ʃ 0 x k ( x , y ) γ f ( y ) d y , 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 ʃ 0 ∞ ( ∏ i = 1 n | T γ i f ( x ) | q i | ) | f ( x ) | q 0 w ( x ) d x ≤ C ( ʃ 0 ∞ | f ( x ) | p v ( x ) d x ) ( q 0 + … + 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. ...

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.

New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights

Agnieszka Kałamajska, Katarzyna Pietruska-Pałuba (2012)

Open Mathematics

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We obtain Hardy type inequalities ∫ 0 ∞ M ω r u r ρ r d r ⩽ C 1 ∫ 0 ∞ M u r ρ r d r + C 2 ∫ 0 ∞ M u ' r ρ r d r , and their Orlicz-norm counterparts ω u L M ( ℝ + , ρ ) ⩽ C ˜ 1 u L M ( ℝ + , ρ ) + C ˜ 2 u ' L M ( ℝ + , ρ ) , with an N-function M, power, power-logarithmic and power-exponential weights ω, ρ, holding on suitable dilation invariant supersets of C 0∞(ℝ+). Maximal sets of admissible functions u are described. This paper is based on authors’ earlier abstract results and applies them to particular classes of weights.

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).

Multilinear Calderón-Zygmund operators on weighted Hardy spaces

Wenjuan Li, Qingying Xue, Kôzô Yabuta (2010)

Studia Mathematica

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Grafakos-Kalton [Collect. Math. 52 (2001)] discussed the boundedness of multilinear Calderón-Zygmund operators on the product of Hardy spaces. Then Lerner et al. [Adv. Math. 220 (2009)] defined A p ⃗ weights and built a theory of weights adapted to multilinear Calderón-Zygmund operators. In this paper, we combine the above results and obtain some estimates for multilinear Calderón-Zygmund operators on weighted Hardy spaces and also obtain a weighted multilinear version of an inequality for...

On a variant of the Hardy inequality between weighted Orlicz spaces

Agnieszka Kałamajska, Katarzyna Pietruska-Pałuba (2009)

Studia Mathematica

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Let M be an N-function satisfying the Δ₂-condition, and let ω, φ be two other functions, with ω ≥ 0. We study Hardy-type inequalities ∫ ℝ ₊ M ( ω ( x ) | u ( x ) | ) e x p ( - φ ( x ) ) d x ≤ C ∫ ℝ ₊ M ( | u ' ( x ) | ) e x p ( - φ ( x ) ) d x , where u belongs to some set of locally absolutely continuous functions containing C ₀ ∞ ( ℝ ₊ ) . We give sufficient conditions on the triple (ω,φ,M) for such inequalities to be valid for all u from a given set . The set may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants. ...