Weighted Inequalities of Hardy Type

“…It can be said that by this time quite a complete weight theory has been established for the Hardy operator. An attractive chronology of the well-known results can be found, for example, in [21,22,32,39]. In [4,26,27] it is shown that the The first author was supported by INTAS grant of South-Caucasian Republics, No.…”

Section: Introductionmentioning

confidence: 97%

On a Hardy Type General Weighted Inequality in Spaces L p(·)

Mamedov

Harman

2010

Integr. Equ. Oper. Theory

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A Hardy type two-weighted inequality is investigated for the multidimensional Hardy operator in the norms of generalized Lebesgue spaces L p(·) . Equivalent necessary and sufficient conditions are found for the L p(·) −→ L q(·) boundedness of the Hardy operator when exponents q(0) < p(0), q(∞) < p(∞). It is proved that the condition for such an inequality to hold coincides with the condition for the validity of two-weighted Hardy inequalities with constant exponents if we require of the exponents to be regular near zero and at infinity. Mathematics Subject Classification (2000). 26D10, 45P05.

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“…This area has been developed recently very extensively and along with these investigations, the theory of Lorentz spaces, including the studies of their geometric structure, has been evolved independently (e.g. [6,9,11,10,14,19,18,27]). For instance, in the papers [10,11] the authors characterized the embeddings between the spaces Γ p,w for different parameters p and w. As a consequence, they characterized the Köthe dual of this space, and gave new conditions for the boundedness of several operators (identity, Hardy-Littlewood maximal function, Hilbert transform, etc.)…”

Section: Introductionmentioning

confidence: 99%

Gâteaux derivatives and their applications to approximation in Lorentz spaces Γ_p,w

Ciesielski

Kamińska

Płuciennik

2009

Mathematische Nachrichten

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We establish the formulas of the left‐ and right‐hand Gâteaux derivatives in the Lorentz spaces Γp,w = {f: ∫0α (f **)p w < ∞}, where 1 ≤ p < ∞, w is a nonnegative locally integrable weight function and f ** is a maximal function of the decreasing rearrangement f * of a measurable function f on (0, α), 0 < α ≤ ∞. We also find a general form of any supporting functional for each function from Γp,w , and the necessary and sufficient conditions for which a spherical element of Γp,w is a smooth point of the unit ball in Γp,w . We show that strict convexity of the Lorentz spaces Γp,w is equivalent to 1 < p < ∞ and to the condition ∫0∞ w = ∞. Finally we apply the obtained characterizations to studies the best approximation elements for each function f ∈ Γp,w from any convex set K ⊂ Γp,w (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Weighted Inequalities of Hardy Type

Cited by 429 publications

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Some new refined Hardy type inequalities with general kernels and measures

Some new refined Hardy type inequalities with general kernels and measures

On a Hardy Type General Weighted Inequality in Spaces L p(·)

Gâteaux derivatives and their applications to approximation in Lorentz spaces Γ_p,w

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Weighted Inequalities of Hardy Type

Cited by 429 publications

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Some new refined Hardy type inequalities with general kernels and measures

Some new refined Hardy type inequalities with general kernels and measures

On a Hardy Type General Weighted Inequality in Spaces L p(·)

Gâteaux derivatives and their applications to approximation in Lorentz spaces Γp,w

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Gâteaux derivatives and their applications to approximation in Lorentz spaces Γ_p,w