On a variant of the Hardy inequality between weighted Orlicz spaces

Kałamajska, Agnieszka; Pietruska-Pałuba, Katarzyna

doi:10.4064/sm193-1-1

Cited by 17 publications

(13 citation statements)

References 38 publications

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“…This work is a continuation of our previous papers [16] and [18], where we have found sufficient conditions for the validity of (1) and (2) for general weights. While in the aforementioned papers we have developed abstract theorems, now we use these abstract tools to construct inequalities primarily for weights being power functions, but also for power-logarithmic and power-exponential weights.…”

Section: Introductionsupporting

confidence: 63%

“…Now we describe theorems concerning inequalities (5) and (6), considering separately the case with both constants C 1 C 2 positive (subsection 3.1, based on [18]), and the case where C 1 = C 1 = 0 (subsection 3.2, based on [16]). …”

Section: Summary Of Results For General Weightsmentioning

confidence: 99%

“…In the current work, we focus on issues (ii), (iii) and (iv), mentioning that the sets of functions admissible in our inequalities can be proper supersets, as well as proper subsets of sets of Hardy transforms [16].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Kałamajska

Pietruska-Pałuba

2012

Open Mathematics

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Section: Introductionsupporting

confidence: 63%

Section: Summary Of Results For General Weightsmentioning

confidence: 99%

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

Kałamajska

Pietruska-Pałuba

2012

Open Mathematics

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“…Remark It follows from that the assumptions (H2) and (H3) imply the following Poincaré inequality. (H1')For every

v \in W_{ρ, 0}^{1, 2} false(normalΩ false)

\int_{normalΩ} | v (x) {false|}^{2} ρ (x) d x \leq C_{A} \sum_{i, j = 1}^{n} \int_{normalΩ} a_{i j} (x) \frac{\partial v}{\partial x_{i}} \frac{\partial v}{\partial x_{j}} d x,

where

C_{A} \leq \frac{C_{P}}{c_{1}}

is given constant independent on v . Constructions of Poincaré inequality can be found for examples in the literature …”

Section: Resultsmentioning

confidence: 99%

Existence and uniqueness of the solution of linear degenerate PDEs in weighted Sobolev space with nonhomogeneous boundary condition

Dhara

2017

Math Methods in App Sciences

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We prove the existence and uniqueness of solution to the nonhomogeneous degenerate elliptic PDE of second order with boundary data in weighted Orlicz-Slobodetskii space. Our goal is to consider the possibly general assumptions on the involved constraints: the class of weights, the boundary data, and the admitted coefficients. We also provide some estimates on the spectrum of our degenerate elliptic operator. KEYWORDS boundary value problems, degenerate elliptic PDEs, upper and lower bounds of eigenvalues, weighted Slobodetskii spaces, weighted Sobolev spaces 544

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“…Many authors consider generalized versions of the inequalities with remainder terms [1,4,28] as well as those expressed in Orlicz setting [15,17,44,46]. Recently, Hardy-type inequalities are investigated also on Riemannian manifolds [26].…”

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confidence: 99%

Hardy inequalities resulted from nonlinear problems dealing with A-Laplacian

Skrzypczak

2014

Nonlinear Differ. Equ. Appl.

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Abstract. We derive Hardy inequalities in weighted Sobolev spaces via anticoercive partial differential inequalities of elliptic type involving ALaplacian −ΔAu = −divA(∇u) ≥ Φ, where Φ is a given locally integrable function and u is defined on an open subset Ω ⊆ R n . Knowing solutions we derive Caccioppoli inequalities for u. As a consequence we obtain Hardy inequalities for compactly supported Lipschitz functions involving certain measures, having the formwhereĀ(t) is a Young function related to A and satisfying Δ -condition, while FĀ(t) = 1/(Ā(1/t)). Examples involvingĀ(t) = t p log α (2+t), p ≥ 1, α ≥ 0 are given. The work extends our previous work (Skrzypczaki, in Nonlinear Anal TMA 93:30-50, 2013), where we dealt with inequality −Δpu ≥ Φ, leading to Hardy and Hardy-Poincaré inequalities with the best constants. Mathematics Subject Classification (2010). Primary 26D10; Secondary 31B05, 35D30, 35J60, 35R45.

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On a variant of the Hardy inequality between weighted Orlicz spaces

Cited by 17 publications

References 38 publications

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

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

Existence and uniqueness of the solution of linear degenerate PDEs in weighted Sobolev space with nonhomogeneous boundary condition

Hardy inequalities resulted from nonlinear problems dealing with A-Laplacian

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