Fractional Poincaré inequalities for general measures (original) (raw)

Sharp Hardy–Sobolev inequalities

Comptes Rendus Mathematique, 2004

Let Ω be a smooth bounded domain in IR N , N ≥ 3. We show that Hardy's inequality involving the distance to the boundary, with best constant (1/4), may still be improved by adding a multiple of the critical Sobolev norm. Résumé Soit Ω un ouvert borné et regulier dans IR N , N ≥ 3. On montre que l'inegalité de Hardy, liéeà la distance au bord, avec meilleure constante (1/4), péutêtre améliorée en ajoutant un multiple de la norme de Sobolev critique.

On the Hardy-Poincare inequality with

2011

Let Ω be a smooth bounded domain in R N with N ≥ 1. In this paper we study the Hardy-Poincaré inequality with weight function singular at the boundary of Ω. In particular we provide sufficient and necessary conditions on the existence of minimizers.

Poincaré inequality meets Brezis–Van Schaftingen–Yung formula on metric measure spaces

Journal of Functional Analysis

Let (X, ρ, µ) be a metric measure space of homogeneous type which supports a certain Poincaré inequality. Denote by the symbol C * c (X) the space of all continuous functions f with compact support satisfying that Lip f := lim sup r→0 sup y∈B(•,r) | f (•) − f (y)|/r is also a continuous function with compact support and Lip f = lim r→0 sup y∈B(•,r) | f (•) − f (y)|/r converges uniformly. Let p ∈ [1, ∞). In this article, the authors prove that, for any f ∈ C * c (X), sup λ∈(0,∞) λ p X µ y ∈ X : | f (x) − f (y)| > λρ(x, y)[V(x, y)] 1 p dµ(x) ∼ X [Lip f (x)] p dµ(x) with the positive equivalence constants independent of f , where V(x, y) := µ(B(x, ρ(x, y))). This generalizes a recent surprising formula of H. Brezis, J. Van Schaftingen, and P.-L. Yung from the n-dimensional Euclidean space R n to X. Applying this generalization, the authors establish new fractional Sobolev and Gagliardo-Nirenberg inequalities in X. All these results have a wide range of applications. Particularly, when applied to two concrete examples, namely, R n with weighted Lebesgue measure and the complete n-dimensional Riemannian manifold with non-negative Ricci curvature, all these results are new. The proofs of these results strongly depend on the geometrical relation of differences and derivatives in the metric measure space and the Poincaré inequality.