Improved Poincaré inequalities and solutions of the divergence in weighted norms

Abstract

The improved Poincaré inequality ∣∣φ-φΩ∣∣ Lp(Ω)≤C ∣∣d∇φ Lp(Ω) Where Ω ⊂ Rn is a bounded domain and d(x) is the distance from x to the boundary of Ω, has many applications. In particular, it can be used to obtain a decomposition of functions with vanishing integral into a sum of locally supported functions with the same property. Consequently, it can be used to go from local to global results, i.e., to extend to very general bounded domains results which are known for cubes. For example, this methodology can be used to prove the existence of solutions of the divergence in Sobolev spaces. The goal of this paper is to analyze the generalization of these results to the case of weighted norms. When the weight is in Ap the arguments used in the un-weighted case can be extended without great difficulty. However, we will show that the improved Poincaré inequality, as well as its above mentioned applications, can be extended to a more general class of weights.