On the weighted fractional Poincar\'e-type inequalities

Hurri-Syrjänen, Ritva; López-García, Fernando

doi:10.4064/cm7612-9-2018

Cited by 7 publications

(4 citation statements)

References 5 publications

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“…Notice that having vanishing mean value could also be understood as being orthogonal to the functions in C. Other applications of this type of decomposition of functions require to have orthogonality to other spaces (see [13,14]). We also refer the reader to [9] for applications to a fractional Poincaré type inequality.…”

Section: A Decomposition Of Functions and Applicationsmentioning

confidence: 99%

“…In this chapter, we prove the two corollaries stated in the Introduction about the solvability of the divergence equation in weighted Sobolev spaces for the weights ω(x) = x β 1 and ω(x) = (1 − ln(x 1 )) α . Notice that Theorem 1.1 requires p = 2, however, we analyze the general case 1 < p < ∞ since Theorem 2.5, which does not have the constraint p = 2, can be used to obtain other inequalities (such as the weighted fractional Poincaré inequality [9]) in our cuspidal domain Ω.…”

Section: The Weighted Discrete Hardy Inequalitymentioning

confidence: 99%

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An application of the weighted discrete Hardy inequality

Giao¹,

López-García²

2020

Preprint

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In a note published in 1925, G. H. Hardy stated the inequalityfor any non-negative sequence {an} n≥1 , and p > 1. This inequality is known in the literature as the classical discrete Hardy inequality. It has been widely studied and several applications and new versions have been shown.In this work, we use a characterization of a weighted version of this inequality to exhibit a sufficient condition for the existence of solutions of the differential equation div u = f in weighted Sobolev spaces over a certain plane irregular domain. The solvability of this equation is fundamental for the analysis of the Stokes equations.The proof follows from a local-to-global argument based on a certain decomposition of functions which is also of interest for its applications to other inequalities or related results in Sobolev spaces, such as the Korn inequality.

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Section: A Decomposition Of Functions and Applicationsmentioning

confidence: 99%

Section: The Weighted Discrete Hardy Inequalitymentioning

confidence: 99%

An application of the weighted discrete Hardy inequality

Giao¹,

López-García²

2020

Preprint

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“…Improvements of an inequality like (1.9) were obtained in [17] by including powers of the distance to the boundary as weights on both sides of the estimate, and also in [30], where the weights are defined by powers of the distance to a compact set of the boundary of the domain. Recently, in [9], the authors have obtained improved fractional Poincaré-Sobolev inequalities on John domains of abstract metric spaces endowed with a measure which satisfies some properties with respect to the metric.…”

Section: Introductionmentioning

confidence: 99%

“…We will stress the differences between our result and theirs in Section 4. Also, we have not been able to improve completely the results in [9], as we did not get weights defined by the distance to a compact set of the boundary instead of weights defined by the distance to the boundary, as it is done in [30].…”

Section: Introductionmentioning

confidence: 99%

A note on generalized Poincaré-type inequalities with applications to weighted improved Poincaré-type inequalities

Martínez-Perales

2021

Ann. Fenn. Math.

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The main result of this paper supports a conjecture by Pérez and Rela about the properties of the weight appearing in their recent self-improving result of generalized inequalities of Poincaré-type in the Euclidean space. The result we obtain does not need any condition on the weight, but still is not fully satisfactory, even though the result by Pérez and Rela is obtained as a corollary of ours. Also, we extend the conclusions of their theorem to the range p < 1.As an application of our result, we give a unified vision of weighted improved Poincaré-type inequalities in the Euclidean setting, which gathers both weighted improved classical and fractional Poincaré inequalities within an approach which avoids any representation formula. We obtain results related to some already existing results in the literature and furthermore we improve them in some aspects. Finally, we also explore analog inequalities in the context of metric spaces by means of the already known self-improving results in this setting. Q∈Q 1 w(Q)ˆQ M(wχ Q ) dx,

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