Reverse conformally invariant Sobolev inequalities on the sphere

Frank, Rupert L.; König, Tobias; Tang, Hanli

doi:10.1016/j.jfa.2021.109339

Cited by 5 publications

(2 citation statements)

References 38 publications

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“…Constants are anyway non-constructive as they rely on [7]. Starting with [85], another approach (which will not be reviewed here) has been developed for studying the stability of the critical points of (S 𝛼 ) in 𝐻 −1 : see [3,25,30,48,56,88]. Results are again non-constructive and we are not aware of any counterpart in the case of (HLS 𝜆 ).…”

Section: Quantitative Stability Resultsmentioning

confidence: 99%

Hardy–Littlewood–Sobolev and related inequalities: Stability

Dolbeault

Esteban

2022

The Physics and Mathematics of Elliott Lieb

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The purpose of this chapter 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).It is with great pleasure that we dedicate this paper to Elliott Lieb on the occasion of his 90th birthday. A short review of some functional inequalitiesFunctional inequalities play a very important role in various fields of mathematics, ranging from geometry, analysis, and probability theory to mathematical physics. For many problems, the precise value of the best constants matters and was actively studied, often in relation with the explicit knowledge of the optimizers. A standard scheme goes as follows: by rearrangement and symmetrisation, optimality is reduced to a smaller class of functions, for instance, to radial functions. After proving that the equality case is achieved, the Euler-Lagrange equations are solved by ODE techniques, which allows to classify the optimal functions and compute the best constants. This is the strategy of E.H. Lieb in [72] for the Hardy-Littlewood-Sobolev inequality which can be written as

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Section: Quantitative Stability Resultsmentioning

confidence: 99%

Hardy–Littlewood–Sobolev and related inequalities: Stability

Dolbeault

Esteban

2022

The Physics and Mathematics of Elliott Lieb

View full text Add to dashboard Cite

show abstract

“…Constants are anyway non-constructive as they rely on [7]. Starting with [85], another approach (which will not be reviewed here) has been developed for studying the stability of the critical points of (S ) in −1 : see [3,25,30,48,56,88]. Results are again non-constructive and we are not aware of any counterpart in the case of (HLS ).…”

Section: Quantitative Stability Resultsmentioning

confidence: 99%

Hardy-Littlewood-Sobolev and related inequalities: stability

Dolbeault¹,

Esteban²

2022

Preprint

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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).It is with great pleasure that we dedicate this paper to Elliott Lieb on the occasion of his 90th birthday. A short review of some functional inequalitiesFunctional inequalities play a very important role in various fields of mathematics, ranging from geometry, analysis, and probability theory to mathematical physics. For many problems, the precise value of the best constants matters and was actively studied, often in relation with the explicit knowledge of the optimizers. A standard scheme goes as follows: by rearrangement and symmetrisation, optimality is reduced to a smaller class of functions, for instance, to radial functions. After proving that the equality case is achieved, the Euler-Lagrange equations are solved by ODE techniques, which allows to classify the optimal functions and compute the best constants. This is the strategy of E.H. Lieb in [72] for the Hardy-Littlewood-Sobolev inequality which can be written as

show abstract

On the Hang–Yang conjecture for GJMS equations on $${\mathbb {S}}^n$$

Hyder,

Ngô

2023

Math. Ann.

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Reverse conformally invariant Sobolev inequalities on the sphere

Cited by 5 publications

References 38 publications

Hardy–Littlewood–Sobolev and related inequalities: Stability

Hardy–Littlewood–Sobolev and related inequalities: Stability

Hardy-Littlewood-Sobolev and related inequalities: stability

On the Hang–Yang conjecture for GJMS equations on $${\mathbb {S}}^n$$

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