Weighted Hardy inequalities and Hardy transforms of weights

Joan Cerdà; Joaquim Martín

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Displaying similar documents to “Weighted Hardy inequalities and Hardy transforms of weights”

Two-weighted criteria for integral transforms with multiple kernels

Vakhtang Kokilashvili, Alexander Meskhi (2006)

Banach Center Publications

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Necessary and sufficient conditions governing two-weight $L^{p}$ norm estimates for multiple Hardy and potential operators are presented. Two-weight inequalities for potentials defined on nonhomogeneous spaces are also discussed. Sketches of the proofs for most of the results are given.

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 $ʃ_{0}^{\infty} (\prod_{i = 1}^{n} | T_{γ_{i}} {f (x) |}^{q_{i}} {|) | f (x) |}^{q_{0}} w (x) d x \leq C (ʃ_{0}^{\infty} {| f (x) |}^{p} {v (x) d x)}^{(q_{0} + \dots + q_{n}) / p}$ . 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. ...

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...

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.

ω-Calderón-Zygmund operators

Sijue Wu (1995)

Studia Mathematica

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

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 $\int_{ℝ ₊} M (ω (x) | u (x) |) e x p (- φ (x)) d x \leq C \int_{ℝ ₊} M (| u^{'} (x) |) e x p (- φ (x)) d x$ , where u belongs to some set of locally absolutely continuous functions containing $C ₀^{\infty} (ℝ ₊)$ . 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. ...

Weighted multi-parameter mixed Hardy spaces and their applications

Wei Ding, Yun Xu, Yueping Zhu (2022)

Czechoslovak Mathematical Journal

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Applying discrete Calderón’s identity, we study weighted multi-parameter mixed Hardy space $H_{mix}^{p} (ω, ℝ^{n_{1}} \times ℝ^{n_{2}})$ . Different from classical multi-parameter Hardy space, this space has characteristics of local Hardy space and Hardy space in different directions, respectively. As applications, we discuss the boundedness on $H_{mix}^{p} (ω, ℝ^{n_{1}} \times ℝ^{n_{2}})$ of operators in mixed Journé’s class.

Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces

Viktor I. Burenkov, Huseyn V. Guliyev (2004)

Studia Mathematica

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The problem of boundedness of the Hardy-Littewood maximal operator in local and global Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted $L_{p}$ -spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions are also necessary.

Weighted inequalities for the two-dimensional Hardy operator

E. Sawyer (1985)

Studia Mathematica

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The John-Nirenberg inequality for functions of bounded mean oscillation with bounded negative part

Min Hu, Dinghuai Wang (2022)

Czechoslovak Mathematical Journal

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A version of the John-Nirenberg inequality suitable for the functions $b \in BMO$ with $b^{-} \in L^{\infty}$ is established. Then, equivalent definitions of this space via the norm of weighted Lebesgue space are given. As an application, some characterizations of this function space are given by the weighted boundedness of the commutator with the Hardy-Littlewood maximal operator.

On weighted inequality of Hardy type for higher order derivatives

Raman Adegoke, James Adedayo Oguntuase (2001)

Kragujevac Journal of Mathematics

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On the best ranges for $A_{p}^{+}$ and $R H_{r}^{+}$

María Silvina Riveros, A. de la Torre (2001)

Czechoslovak Mathematical Journal

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In this paper we study the relationship between one-sided reverse Hölder classes $R H_{r}^{+}$ and the $A_{p}^{+}$ classes. We find the best possible range of $R H_{r}^{+}$ to which an $A_{1}^{+}$ weight belongs, in terms of the $A_{1}^{+}$ constant. Conversely, we also find the best range of $A_{p}^{+}$ to which a $R H_{\infty}^{+}$ weight belongs, in terms of the $R H_{\infty}^{+}$ constant. Similar problems for $A_{p}^{+}$ , $1 < p < \infty$ and $R H_{r}^{+}$ , $1 < r < \infty$ are solved using factorization.

Non-compact Littlewood-Paley theory for non-doubling measures

Michael Wilson (2007)

Studia Mathematica

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We prove weighted Littlewood-Paley inequalities for linear sums of functions satisfying mild decay, smoothness, and cancelation conditions. We prove these for general “regular” measure spaces, in which the underlying measure is not assumed to satisfy any doubling condition. Our result generalizes an earlier result of the author, proved on $ℝ^{d}$ with Lebesgue measure. Our proof makes essential use of the technique of random dyadic grids, due to Nazarov, Treil, and Volberg.

An interpolation theorem with $A_{p}$ -weighted $L^{p}$ spaces

Steven Bloom (1990)

Studia Mathematica

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Norm inequalities in weighted amalgam

Suket Kumar (2018)

Commentationes Mathematicae Universitatis Carolinae

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Hardy inequalities for the Hardy-type operators are characterized in the amalgam space which involves Banach function space and sequence space.