Related papers: Stability results for Sobolev, logarithmic Sobolev…
In this paper, we review recent results on stability and instability in logarithmic Sobolev inequalities, with a particular emphasis on strong norms. We consider several versions of these inequalities on the Euclidean space, for the…
This paper is devoted to stability results for the Gaussian logarithmic Sobolev inequality, with explicit stability constants.
These notes are an extended version of a series of lectures given at the CIME Summer School in Cetraro in June 2022. The goal is to explain questions about optimal functional inequalities on the example of the sharp Sobolev inequality and…
We develop a general framework for using duality to "transfer" stability results for a functional inequality to its dual inequality. As an application, we prove a stability bound for the Hardy-Littlewood-Sobolev inequality, which is related…
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…
This paper extends a stability estimate of the Sobolev Inequality established by G. Bianchi and H. Egnell in their paper "A note on the Sobolev Inequality." Bianchi and Egnell's Stability Estimate answers the question raised by H. Brezis…
The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard…
In this paper, we present recent stability results with explicit and dimensionally sharp constants and optimal norms for the Sobolev inequality and for the Gaussian logarithmic Sobolev inequality obtained by the authors in [24]. The…
In this paper, we establish several improved Caffarelli-Kohn-Nirenberg and Hardy-type inequalities. Our main results are divided into two parts. In the first part, we consider the following Caffarelli-Kohn-Nirenberg inequality:…
This paper is devoted to improvements of functional inequalities based on scalings and written in terms of relative entropies. When scales are taken into account and second moments fixed accordingly, deficit functionals provide explicit…
This paper investigates sharp stability estimates for the fractional Hardy-Sobolev inequality: $$\mu_{s,t}\left(\mathbb{R}^N\right) \left(\int_{\mathbb{R}^N} \frac{|u|^{2^*_s(t)}}{|x|^t} \,{\rm d}x \right)^{\frac{2}{2^*_s(t)}} \leq…
We prove a quantitative Sobolev inequality in cones of Bianchi-Egnell type, which implies a stability property. Our result holds for any cone as long as the minimizers of the Sobolev quotient are nondegenerate, which is the case of most…
We prove a sharp quantitative version for the stability of the Sobolev inequality with explicit constants. Moreover, the constants have the correct behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative…
The goal of this short note is to prove qualitative stability for a family of trace Sobolev inequalities first proven by Carlen \& Loss for $p=2$ and by Maggi and the author for $p\in (1,n)$. This answers an open problem raised in a recent…
The study of what we now call Sobolev inequalities has been studied for almost a century in various forms, while it has been eighty years since Sobolev's seminal mathematical contributions. Yet there are still things we don't understand…
Our main goal is to explicitly compute the best constant for the Sobolev-type inequality involving the polyharmonic operator obtained in (Analysis and Applications 22, pp. 1417-1446, 2024). To achieve this goal, we also establish both…
A symmetrization techique, introduced by Cianchi, Fusco, Maggi and Pratelli concerning the Sobolev inequality, is adapted to the Gagliardo-Nirenberg-Sobolev inequality (GNS) to obtain a reduction step of the problem of showing its…
We review recent results regarding the problem of the stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds. We shall describe techniques and methods from smooth and non-smooth geometry, the fruitful…
In this paper, we establish the stability for the Hardy-Littlewood-Sobolev (HLS) inequalities with explicit lower bounds. By establishing the relation between the stability of HLS inequalities and the stability of fractional Sobolev…
The logarithmic Sobolev inequality is fundamental in mathematical physics. Associated stability estimates are equivalent to uncertainty principles. Via a second moment bound, $W^{1,1}$ estimates are obtained in one dimension and similar…