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Higher order Poincaré inequalities associated with linear operators on stratified groups and applications. (English) Zbl 1023.46032

Summary: This paper considers the dual of anisotropic Sobolev spaces on any stratified groups \(\mathbb G\). For \(0\leq k<m\) and every linear bounded functional \(T\) on the anisotropic Sobolev space \(W^{m-k,p}(\Omega)\) on \(\Omega \subset \mathbb G\), we derive a projection operator \(L\) from \(W^{m,p}(\Omega)\) to the collection \(\mathbb P_{k+1}\) of polynomials of degree less than \(k+1\) such that \(T(X^I(Lu)) = T(X^Iu)\) for all \(u\in W^{m,p}(\Omega)\) and multi-index \(I\) with \(d(I)\leq k\). We then prove a general Poincaré 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 Poincaré inequalities of special interest. In particular, we obtain Poincaré 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 \(W^{\alpha,p}(\mathbb G)\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
22E25 Nilpotent and solvable Lie groups
41A10 Approximation by polynomials
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