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Continuous primitives for higher degree differential forms in Euclidean spaces, Heisenberg groups and applications

Annalisa Baldi, Bruno Franchi, Pierre Pansu

Abstract

It is shown that higher degree exact differential forms on compact Riemannian $n$-manifolds possess continuous primitives whose uniform norm is controlled by their $L^n$ norm. A contact sub-Riemannian analogue is proven, with differential forms replaced with Rumin differential forms.

Continuous primitives for higher degree differential forms in Euclidean spaces, Heisenberg groups and applications

Abstract

It is shown that higher degree exact differential forms on compact Riemannian -manifolds possess continuous primitives whose uniform norm is controlled by their norm. A contact sub-Riemannian analogue is proven, with differential forms replaced with Rumin differential forms.
Paper Structure (26 sections, 48 theorems, 257 equations)

This paper contains 26 sections, 48 theorems, 257 equations.

Table of Contents

  1. Introduction
  2. The problem
  3. The method
  4. Organization of the paper
  5. Heisenberg groups
  6. Definitions and preliminary results
  7. Convolution in Heisenberg groups
  8. Rumin's complex of differential forms
  9. Rumin's Laplacian
  10. Poincaré inequalities in Heisenberg groups
  11. Bourgain-Brezis' duality argument for the global Poincaré inequality
  12. Currents and measures
  13. Poincaré and Sobolev inequalities for currents via an approximation result
  14. Proof of Theorem \ref{['main global']}--i): continuous primitives of forms in $E_0^h$, $h\neq n+1$
  15. The main global result in degree $n+1$: Poincaré inequalities with Beppo Levi-Sobolev spaces
  16. ...and 11 more sections

Key Result

Theorem 1

Theorems & Definitions (102)

  • Theorem 1
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Proposition 2.7
  • Theorem 2.8
  • Theorem 2.9: CvS2009, Theorem 1
  • ...and 92 more