Higher order Poincaré inequalities associated with linear operators on stratified groups and applications

Abstract

This paper considers the dual of anisotropic Sobolev spaces on any stratified groups 𝔾. For 0≤k<m and every linear bounded functional T on anisotropic Sobolev space Wm−k,p(Ω) on Ω⊂𝔾, we derive a projection operator L from Wm,p(Ω) to the collection 𝒫k+1 of polynomials of degree less than k+1 such that T(XI(Lu))=T(XIu) for all uWm,p(Ω) and multi-index I with d(I)≤k. We then prove a general Poincare inequality involving this operator L and the linear functional T. As applications, we often choose a linear functional T such that the associated L is zero and consequently we can prove Poincare inequalities of special interests. In particular, we obtain Poincare inequalities for functions vanishing on tiny sets of positive Bessel capacity on stratified groups. Finally, we derive a Hedberg-Wolff type characterization of measures belonging to the dual of the fractional anisotropic Sobolev spaces Wα,p𝔾.