Hardy–Littlewood–Sobolev and related inequalities: Stability

Abstract

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

Sharp Stability for LSI

“…The techniques in [14,16,23,24] involve optimal transport, semigroup theory, Fourier analysis, and probability. The recent proof of (1.3) in [12] is a fundamental achievement. One interesting feature is the lack of additional assumptions for (1.3) via the Bianchi-Egnell method.…”

“…The following quantitative LSI was proven in 2022 [12]: there exists a dimensionless κ > 0 such that assuming u ∈ H 1 (e −π|x| 2 dx),…”

On the sharp constant in the Bianchi–Egnell stability inequality

“…As the proof of (1.4) in [4, 7] proceeds by compactness, it yields no explicit lower bound on the constant $$c_{BE}(s)$$. For $$s=1$$, the first constructive lower bound on $$c_{BE}(1)$$ is proved in the recent preprint [10]. Previously, in [5] a stability bound with an explicit constant, but with the $$\dot{H}^1$$‐distance replaced by a weaker notion of distance, had been obtained.…”

Stability for the logarithmic Sobolev inequality

“…The logarithmic Sobolev inequality can be viewed as a limit case of a family of the Gagliardo-Nirenberg-Sobolev (GNS) as observed in [25], in the Euclidean setting, or as a large dimension limit of the Sobolev inequality according to [9]. See [27,18] for recent developments and further references. We refer to [1,38,52,5] to reference books on the topic.…”

“…In a classical result on stability in functional inequalities, Bianchi and Egnell proved in [10] that the deficit in the Sobolev inequality measures the Ḣ1 (R d , d x) distance to the manifold of the Aubin-Talenti functions. The estimate has been made constructive in [27] where a new L 2 (R d , d x) stability result for the logarithmic Sobolev inequality is also established (also see [39] for furrther results in strong norms). Still in the Euclidean setting a first stability result in strong norms for the logarithmic Sobolev inequality appears in [33], where the authors give deficit estimates in various distances for functions inducing a Poincaré inequality.…”