Science China Mathematics

, Volume 62, Issue 6, pp 1029–1040 | Cite as

L1-Poincaré and Sobolev inequalities for differential forms in Euclidean spaces

  • Annalisa Baldi
  • Bruno FranchiEmail author
  • Pierre Pansu


In this paper, we prove Poincaré and Sobolev inequalities for differential forms in L1(ℝn). The singular integral estimates that it is possible to use for Lp, p > 1, are replaced here with inequalities which go back to Bourgain and Brezis (2007).


differential forms Sobolev-Poincaré inequalities homotopy formula 


58A10 26D15 46E35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The first author and the second author were supported by Funds for Selected Research Topics from the University of Bologna, MAnET Marie Curie Initial Training Network, GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica “F. Severi”), Italy, and PRIN (Progetti di ricerca di Rilevante Interesse Nazionale) of the MIUR (Ministero dell’Istruzione dell’Università e della Ricerca), Italy. The third author was supported by MAnET Marie Curie Initial Training Network, Agence Nationale de la Recherche (Grant Nos. ANR-10-BLAN 116-01 GGAA and ANR-15-CE40-0018 SRGI), and thanks the hospitality of Isaac Newton Institute, of EPSRC (Engineering and Physical Sciences Research Council) (Grant No. EP/K032208/1) and Simons Foundation.


  1. 1.
    Baldi A, Franchi B, Pansu P. Poincaré and Sobolev inequalities for differential forms in Heisenberg groups. ArXiv: 1711.09786, 2017zbMATHGoogle Scholar
  2. 2.
    Baldi A, Franchi B, Pansu P. L 1-Poincaré inequalities for differential forms on Euclidean spaces and Heisenberg groups. ArXiv:1902.04819, 2019Google Scholar
  3. 3.
    Benilan P, Brezis H, Crandall M G. A semilinear equation in L 1(ℝN). Ann Sc Norm Super Pisa Cl Sci (5), 1975, 2: 523–555zbMATHGoogle Scholar
  4. 4.
    Bourgain J, Brezis H. New estimates for elliptic equations and Hodge type systems. J Eur Math Soc (JEMS), 2007, 9: 277–315MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Federer H, Fleming W H. Normal and integral currents. Ann of Math (2), 1960, 72: 458–520MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Folland G B. Lectures on Partial Differential Equations. Tata Institute of Fundamental Research. Lectures on Mathematics and Physics, vol. 70. Berlin: Springer-Verlag, 1983Google Scholar
  7. 7.
    Folland G B, Stein E M. Hardy Spaces on Homogeneous Groups. Mathematical Notes, vol. 28. Princeton: Princeton University Press, 1982Google Scholar
  8. 8.
    Franchi B, Gallot S, Wheeden R L. Sobolev and isoperimetric inequalities for degenerate metrics. Math Ann, 1994, 300: 557–571MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Franchi B, Lu G, Wheeden R L. Representation formulas and weighted Poincaré inequalities for Hörmander vector fields. Ann Inst Fourier (Grenoble), 1995, 45: 577–604MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Iwaniec T, Lutoborski A. Integral estimates for null Lagrangians. Arch Ration Mech Anal, 1993, 145: 25–79MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jost J. Riemannian Geometry and Geometric Analysis, 5th ed. Berlin: Springer-Verlag, 2008zbMATHGoogle Scholar
  12. 12.
    Lanzani L, Stein E M. A note on div curl inequalities. Math Res Lett, 2005, 12: 57–61MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Long R L, Nie F S. Weighted Sobolev inequality and eigenvalue estimates of Schrödinger operators. In: Lecture Notes in Mathematics, vol. 1494. Berlin: Springer, 1991, 131–141CrossRefzbMATHGoogle Scholar
  14. 14.
    Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43. Princeton: Princeton University Press, 1993Google Scholar
  15. 15.
    Van Schaftingen J. Limiting Bourgain-Brezis estimates for systems of linear differential equations: Theme and variations. J Fixed Point Theory Appl, 2014, 15: 273–297MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di BolognaBolognaItaly
  2. 2.Laboratoire de Mathématiques d’OrsayUniversité Paris-Sud, CNRS, Université Paris-SaclayOrsayFrance

Personalised recommendations