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

Guozhen Lu; Richard Wheeden

Studia Mathematica (2000)

  • Volume: 142, Issue: 2, page 101-133
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Lu, Guozhen, and Wheeden, Richard. "High order representation formulas and embedding theorems on stratified groups and generalizations." Studia Mathematica 142.2 (2000): 101-133. <http://eudml.org/doc/216792>.

@article{Lu2000,
abstract = {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.},
author = {Lu, Guozhen, Wheeden, Richard},
journal = {Studia Mathematica},
keywords = {Poincaré inequalities; doubling measures; stratified groups; polynomials; representation formulas; vector fields; embedding theorems; representation formula; Poincaré inequality; embedding theorem; polynomial; stratified group; vector field; doubling measure; Carnot-Carathéodory vector fields},
language = {eng},
number = {2},
pages = {101-133},
title = {High order representation formulas and embedding theorems on stratified groups and generalizations},
url = {http://eudml.org/doc/216792},
volume = {142},
year = {2000},
}

TY - JOUR
AU - Lu, Guozhen
AU - Wheeden, Richard
TI - High order representation formulas and embedding theorems on stratified groups and generalizations
JO - Studia Mathematica
PY - 2000
VL - 142
IS - 2
SP - 101
EP - 133
AB - 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.
LA - eng
KW - Poincaré inequalities; doubling measures; stratified groups; polynomials; representation formulas; vector fields; embedding theorems; representation formula; Poincaré inequality; embedding theorem; polynomial; stratified group; vector field; doubling measure; Carnot-Carathéodory vector fields
UR - http://eudml.org/doc/216792
ER -

References

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