High order representation formulas and embedding theorems on stratified groups and generalizations

Tom 142 / 2000

Guozhen Lu, Richard Wheeden Studia Mathematica 142 (2000), 101-133 DOI: 10.4064/sm-142-2-101-133


We derive various integral representation formulas for a function minus a polynomial in terms of vector field gradients of the function of appropriately high order. Our results hold in the general setting of metric spaces, including those associated with Carnot-Carathéodory vector fields, under the assumption that a suitable $L^1$ to $L^1$ Poincaré inequality holds. Of particular interest are the representation formulas in Euclidean space and stratified groups, where polynomials exist and $L^1$ to $L^1$ Poincaré inequalities involving high order derivatives are known to hold. We apply the formulas to derive embedding theorems and potential type inequalities involving high order derivatives.


  • Guozhen Lu
  • Richard Wheeden

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