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Hardy-Littlewood-Sobolev and related inequalities: stability
Analysis of PDEs · Submitted 7 Feb 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We also contribute to the topic with some observations on constructive stability estimates for (HLS).
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The abstract, sentence by sentence
The purpose of this text is twofold. We present a review of the existing stability results for Sobolev, Hardy-Littlewood-Sobolev (HLS) and related inequalities. We also contribute to the topic with some observations on constructive stability estimates for (HLS).
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The purpose of this text is twofold. | background 0.55 | 0.26 | 0.02 | asserted 0.88 |
| s1 | We present a review of the existing stability results for Sobolev, Hardy-Littlewood-Sobolev (HLS) and related inequalities. | contribution 0.64 | 0.24 | 0.09 | asserted 0.95 |
| s2 | We also contribute to the topic with some observations on constructive stability estimates for (HLS). | contribution 1.00 | 0.92 | 0.24 | asserted 0.97 |
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The paper
Original abstract
The purpose of this text is twofold. We present a review of the existing stability results for Sobolev, Hardy-Littlewood-Sobolev (HLS) and related inequalities. We also contribute to the topic with some observations on constructive stability estimates for (HLS).
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How the literature uses it
1 arXiv paper cite this in 1 sentence
Papers citing it
Cites it as background 1 paper
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We refer to the beautiful treatises [ref] which covers, for example, the stability of isoperimetric inequality, $1$-Sobolev inequality, Caffarelli-Kohn-Nirenberg inequality, log-Sobolev inequality, Gagliardo-Nirenberg-Sobolev inequality, Hardy-Littlewood-Sobolev, logarithmic Hardy-Littlewood-Sobolev inequality on the Euclidean space.
Sharp quantitative stability of Struwe's decomposition of the Poincar\'e-Sobolev inequalities on the hyperbolic space: Part I · The quantitative stability problem · 1.00
What it cites · 13 sentences without a confident label
Builds on 1 paper
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A stability result similar to Theorem [ref] can be established for [ref] with the particular choice $λ=n-2α$ and $p=t=2 n/(n-2α)$ of the parameters, $α∈(0,n/2)$, by adapting the computations of [ref] .
Fractional Sobolev and Hardy-Littlewood-Sobolev inequalities · A local stability result in the norm of relative uniform convergence · 0.85
Uses the method or tool of 2 papers
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By a standard trick used for instance in [ref] , [ref] and [ref] , the expansion of the square in [equation] with $g^r=f$ so that $fq^2=gp^2r$ if $q=2 n/(n-2α)$, shows that [equation] Slightly better results are obtained using flow methods as in [ref] which have not been detailed in Section [ref].
Fractional Sobolev and Hardy-Littlewood-Sobolev inequalities · Constructive stability results · 0.96
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Bakry, that can be found in [ref] (see a detailed computation in [ref] ): the [ref] inequality with $q=2 (p-1)$, whose optimal functions are generated by fp 𝗎_q(x):=(1+|x|^2)^-1/(q-1) ∀ x∈^d , is rewritten as a Sobolev inequality in a higher, artificial, dimension such that the optimal functions appear as Aubin-Talenti functions.
Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method · Quantitative stability results · 0.95
Cites as background 2 papers
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Notice that stability results in logarithmic Sobolev inequalities is a widely studied question, with various results in $L^1$ and in Wasserstein distances: see [ref] and references therein.
Instability results for the logarithmic Sobolev inequality and its application to related inequalities · Constructive stability results · 0.97
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Optimality results for the logarithmic Sobolev (log-Sobolev) inequality, the logarithmic Hardy-Littlewood-Sobolev (log-HLS) inequality, the Onofri inequality or the Caffarelli-Kohn-Nirenberg inequalities, among many other functional inequalities, have attracted a lot of attention over the years and we may refer for instance to [ref] for some key papers and to [ref] for a short review.
Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method · A short review of some functional inequalities · 0.99
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