On improved fractional Sobolev–Poincaré inequalities

Abstract

We prove a certain improved fractional Sobolev-Poincaré inequality on John domains; the proof is based on the equivalence of the corresponding weak and strong type inequalities. We also give necessary conditions for the validity of an improved fractional Sobolev-Poincaré inequality, in particular, we show that a domain having a finite measure and satisfying this inequality, and a 'separation property', is a John domain.2010 Mathematics Subject Classification. 46E35 (26D10).

Improved Poincaré inequalities in fractional Sobolev spaces

“…For the case p = 1 we will make use of the following "weak implies strong" result. It is proved in [13,Theorem 4.1] in the case µ = ν, but the reader can easily check that the same proof holds for two different measures.…”

“…More recently, some authors have turned their attention to fractional generalizations of Poincaré and Sobolev-Poincaré inequalities, where a fractional seminorm appears instead of the norm in W 1,p (Ω). Indeed, in [13,18] the following inequalities were introduced for John domains:…”

“…with 1 ≤ p ≤ q ≤ np n−sp and s, τ ∈ (0, 1). The seminorm appearing on the RHS of (1.2) can be seen to be equivalent on Lipschitz domains to the usual seminorm in W s,p (Ω), that is, integrating over Ω × Ω (see [12, equation (13)]), but it can be strictly smaller than the usual seminorm for general John domains (see [13,Proposition 3.4]). Moreover, it is easy to see that, unlike the classical Poincaré inequality, the inequality holds for the class of Ahlfors n-regular domains, which is larger than that of the John domains, but if we turn to the inequality with the stronger seminorm, there are Ahlfors n-regular domains for which the inequality fails (see [13,Theorem 3.1]).…”

Weighted generalized Korn inequalities on John domains

“…Examples of domains with the separation property are simply connected plane domains. Buckley and Koskela have also been used to prove the equivalence of the John condition with other inequalities in Sobolev spaces such as the one known as the inf‐sup condition and an improved version of the fractional Sobolev‐Poincaré inequality …”

“…Buckley and Koskela 9 have also been used to prove the equivalence of the John condition with other inequalities in Sobolev spaces such as the one known as the inf-sup condition 3 and an improved version of the fractional Sobolev-Poincaré inequality. 10 In this work, we deal with a generalized version of (1) known simply as the generalized Korn inequality or the conformal Korn inequality, where the linearized strain vector (v) is replaced by its trace-free part l(v) defined by…”