Improved Poincaré inequalities in fractional Sobolev spaces

Abstract

We obtain improved fractional Poincaré and Sobolev Poincaré inequalities including powers of the distance to the boundary in John, s-John domains and Hölder-α domains, and discuss their optimality.2010 Mathematics Subject Classification. Primary: 26D10; Secondary: 46E35.

Improved fractional Poincaré type inequalities on John domains

“…At the right hand side of the inequality, we will obtain a weight of the form v F Φ,γ (x, y)=min z∈{x,y} d(z) γ Φ(d F (z)), where Φ will be an appropriate power of φ. Our results extend and improve results in [10], [24], [34] in several ways. On one hand, the obtained class of weights is larger than the ones previously considered, even in the Euclidean setting.…”

“…At this point, a "weak implies strong" argument, which also holds in our setting (see the comments preceding [10,Lemma 3.2. ] and also [14,Proposition 5], [18,Theorem 4], [13]) gives us the extremal case p=1 with weight w F φ at the left-hand side and v F Φ,γ at the right hand side.…”

“…This section is devoted to the study of improved fractional (q, p)-Poincaré inequalities on bounded John domains, where 1≤p≤q<∞. Particular cases of these inequalities include some already known results in the Euclidean case, such as the unweighted inequalities considered in [24] and the inequalities where the weights are powers of the distance to the boundary considered in [10], [34]. We will come back to these special examples in Section 4.…”

“…The proof makes use of some of the arguments in [24] and [10]. The fundamental idea is the classical one to obtain ordinary Poincaré inequalities: to bound the oscillation of the function u by the fractional integral of its derivative by using regularity properties of the function and the domain and then use the boundedness properties of the fractional integral operator.…”

“…However, throughout this work, we will use this terminology to refer to inequalities including powers of the distance to the boundary as weights, as in the classical case. Improvements of inequality (1.2) in this sense were obtained in [10] by including powers of the distance to the boundary to appropriate powers as weights on both sides of the inequality, and in [34], where the weights are defined by powers of the distance to a compact set of the boundary of the domain.…”