Improved weighted Poincaré inequalities in John domains and application to the divergence equation

We extend some results related to the Poincaré inequality and solvability of divergence obtained in [G. Acosta et al., Ann. Acad. Sci. Fenn. Math. 42 (2017)]. More precisely, we generalize to unbounded John domains the general theorem that provides a sufficient condition on a weight to support a weighted improved Poincaré inequality. Next, we apply this inequality to study the solvability of the divergence equation in weighted Sobolev spaces. As a consequence, we prove the solvability in weighted Sobolev spaces for weights in classes bigger than $A_p$.