The compactness and the concentration compactness via p-capacity

Anoop, T. V.; Das, Ujjal

doi:10.1007/s10231-021-01098-2

Cited by 7 publications

(15 citation statements)

References 19 publications

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“…Such weighted Hardy-Sobolev inequality has been discussed in [3,5,6,44]. A similar study has been done for the Hardy-Rellich inequalities too, see [7,39].…”

Section: Introductionmentioning

confidence: 72%

“…Since (2.4), (2.5) hold and g ∈ H p,p (R N ), it follows from Theorem 31 of [6] that g ∈ F p,p (R N ). Therefore,…”

Section: 5mentioning

confidence: 97%

“…Then part (i) implies g ∈ H p,r (R N ). For r = p, it follows from Proposition 17 of [6] that g ∈ F p,p (R N ). For r > p we will use Theorem 31 of [6] to show that g ∈ F p,p (R N ).…”

Section: 5mentioning

confidence: 99%

“…For r = p, it follows from Proposition 17 of [6] that g ∈ F p,p (R N ). For r > p we will use Theorem 31 of [6] to show that g ∈ F p,p (R N ). Since 1 < N p < p * p * −r , we have g ∈ L N p (R N ) and hence, g ∈ H p,p (R N ).…”

Section: 5mentioning

confidence: 99%

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On weighted logarithmic-Sobolev & logarithmic-Hardy inequalities

Das¹

2020

Preprint

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For N ≥ 3 and p ∈ (1, N ), we look for g ∈ L 1 loc (R N ) that satisfies the following weighted logarithmic Sobolev inequality:we identify a Banach function space H p,r (R N ) such that the above inequality holds for g ∈ H p,r (R N ). For γ > r r−p , we also find a class of g for which the best constant C γ in the above inequality is attained in D 1,p 0 (R N ). Further, for a closed set E with Assouad dimension = d < N and a ∈ (−), we establish the following logarithmic Hardy inequalityfor some C > 0, where δ E (x) is the distance between x and E. The second order extension of the logarithmic Hardy inequality is also obtained.

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“…Such weighted Hardy-Sobolev inequality has been discussed in [3,5,6,44]. A similar study has been done for the Hardy-Rellich inequalities too, see [7,39].…”

Section: Introductionmentioning

confidence: 72%

“…Since (2.4), (2.5) hold and g ∈ H p,p (R N ), it follows from Theorem 31 of [6] that g ∈ F p,p (R N ). Therefore,…”

Section: 5mentioning

confidence: 97%

Section: 5mentioning

confidence: 99%

Section: 5mentioning

confidence: 99%

See 2 more Smart Citations

On weighted logarithmic-Sobolev & logarithmic-Hardy inequalities

Das¹

2020

Preprint

Self Cite

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“…( ) Some of the non-compact cases for = are discussed in [6] using the variant of concentration compactness lemma and Maz'ya's capacity conditions.…”

Section: Formentioning

confidence: 99%

Admissible function spaces for weighted Sobolev inequalities

Anoop¹,

Biswas²,

Das³

2021

CPAA

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Let , ∈ N with 1 ≤ ≤ and let Ω = Ω 1 × Ω 2 be an open set in R × R − . For ∈ (1, ∞) and ∈ (0, ∞), we consider the following weighted Sobolev type inequality:for some > 0. Depending on the values of , , , we have identified various pairs of Lorentz spaces, Lorentz-Zygmund spaces and weighted Lebesgue spaces for ( 1 , 2 ) so that (0.1) holds. Furthermore, we give a sufficient condition on 1 , 2 so that the best constant in (0.1) is attained in the Beppo-Levi space 1, 0 (Ω)-the completion of 1 (Ω) with respect to ‖∇ ‖ (Ω) .

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On the generalised Brézis–Nirenberg problem

Anoop

Das

2022

Nonlinear Differ. Equ. Appl.

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The compactness and the concentration compactness via p-capacity

Cited by 7 publications

References 19 publications

On weighted logarithmic-Sobolev & logarithmic-Hardy inequalities

On weighted logarithmic-Sobolev & logarithmic-Hardy inequalities

Admissible function spaces for weighted Sobolev inequalities

On the generalised Brézis–Nirenberg problem

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