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

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• 作者：William S. Cohn ; Guozhen Lu and Shanzhen Lu
• 刊名：Mathematische Zeitschrift
• 出版年：2003
• 出版时间：June 2003
• 年：2003
• 卷：244
• 期：2
• 页码：309-335
• 全文大小：247 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1823

文摘

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 W mk,p (Ω) on Ω𝔾, we derive a projection operator L from W m,p (Ω) to the collection 𝒫 _k+1 of polynomials of degree less than k+1 such that T(X I (Lu))=T(X I u) for all uW m,p (Ω) and multi-index I with d(I)≤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 interests. 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 α,p 𝔾.