Representation formulas and weighted Poincaré inequalities for Hörmander vector fields

Bruno Franchi; Guozhen Lu; Richard L. Wheeden

Annales de l'institut Fourier (1995)

  • Volume: 45, Issue: 2, page 577-604
  • ISSN: 0373-0956

Abstract

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We derive weighted Poincaré inequalities for vector fields which satisfy the Hörmander condition, including new results in the unweighted case. We also derive a new integral representation formula for a function in terms of the vector fields applied to the function. As a corollary of the L 1 versions of Poincaré’s inequality, we obtain relative isoperimetric inequalities.

How to cite

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Franchi, Bruno, Lu, Guozhen, and Wheeden, Richard L.. "Representation formulas and weighted Poincaré inequalities for Hörmander vector fields." Annales de l'institut Fourier 45.2 (1995): 577-604. <http://eudml.org/doc/75130>.

@article{Franchi1995,
abstract = {We derive weighted Poincaré inequalities for vector fields which satisfy the Hörmander condition, including new results in the unweighted case. We also derive a new integral representation formula for a function in terms of the vector fields applied to the function. As a corollary of the $L^1$ versions of Poincaré’s inequality, we obtain relative isoperimetric inequalities.},
author = {Franchi, Bruno, Lu, Guozhen, Wheeden, Richard L.},
journal = {Annales de l'institut Fourier},
keywords = {Hörmander vector fields; weighted Poincaré inequalities; representation formulas; isoperimetric inequalities},
language = {eng},
number = {2},
pages = {577-604},
publisher = {Association des Annales de l'Institut Fourier},
title = {Representation formulas and weighted Poincaré inequalities for Hörmander vector fields},
url = {http://eudml.org/doc/75130},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Franchi, Bruno
AU - Lu, Guozhen
AU - Wheeden, Richard L.
TI - Representation formulas and weighted Poincaré inequalities for Hörmander vector fields
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 2
SP - 577
EP - 604
AB - We derive weighted Poincaré inequalities for vector fields which satisfy the Hörmander condition, including new results in the unweighted case. We also derive a new integral representation formula for a function in terms of the vector fields applied to the function. As a corollary of the $L^1$ versions of Poincaré’s inequality, we obtain relative isoperimetric inequalities.
LA - eng
KW - Hörmander vector fields; weighted Poincaré inequalities; representation formulas; isoperimetric inequalities
UR - http://eudml.org/doc/75130
ER -

References

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  1. [BKL] S. BUCKLEY, P. KOSKELA and G. LU, Boman aka John, in preparation. 
  2. [BM1] M. BIROLI and U. MOSCO, Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur., to appear. Zbl0837.31006
  3. [BM2] M. BIROLI and U. MOSCO, Proceedings of the Conference “Potential theory and partial differential operators with nonnegative characteristic form”, Parma, February 1994, Kluwer, Amsterdam, to appear. 
  4. [Bo] B. BOJARSKI, Remarks on Sobolev imbedding inequalities, Lecture Notes in Math. 1351 (1989), 52-68, Springer-Verlag. Zbl0662.46037MR90b:46068
  5. [Bu] H. BUSEMANN, The Geometry of Geodesics, Academic Press, New York, 1955. Zbl0112.37002MR17,779a
  6. [Ca] A.P. CALDERÓN, Inequalities for the maximal function relative to a metric, Studia Math., 57 (1976), 297-306. Zbl0341.44007MR56 #960
  7. [CDG] L. CAPOGNA, D. DANIELLI, N. GAROFALO, Subelliptic mollifiers and a characterization of Rellich and Poincaré domains, Rend. Sem. Mat. Univ. Politec. Torino, 51 (1993), 361-386. Zbl0811.35017MR96b:35030
  8. [Ch] S.-K. CHUA, Weighted Sobolev's inequality on domains satisfying the Boman chain condition, Proc. Amer. Math. Soc., to appear. Zbl0812.46020
  9. [Cou] T. COULHON, Espaces de Lipschitz et inegalités de Poincaré, J. Funct. Anal., to appear. Zbl0859.58009
  10. [CW] S. CHANILLO and R.L. WHEEDEN, Weighted Poincaré and Sobolev inequalities and estimates for the Peano maximal function, Amer. J. Math., 107 (1985), 1191-1226. Zbl0575.42026MR87f:42045
  11. [Fe] H. FEDERER, Geometric Measure Theory, Springer, 1969. Zbl0176.00801MR41 #1976
  12. [FP] C. FEFFERMAN and D.H. PHONG, Subelliptic eigenvalue estimates, Conference on Harmonic Analysis, Chicago, 1980, W. Beckner et al. ed., Wadsworth (1981), 590-606. Zbl0503.35071
  13. [F] B. FRANCHI, Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic operators, Trans. Amer. Math. Soc., 327 (1991), 125-158. Zbl0751.46023MR91m:35095
  14. [FGaW1] B. FRANCHI, S. GALLOT and R.L. WHEEDEN, Inégalités isoperimétriques pour des métriques dégénérées, C.R. Acad. Sci. Paris, Sér. I, Math., 317 (1993), 651-654. Zbl0794.51011MR95e:46039
  15. [FGaW2] B. FRANCHI, S. GALLOT and R.L. WHEEDEN, Sobolev and isoperimetric inequalities for degenerate metrics, Math. Ann., 300 (1994), 557-571. Zbl0830.46027MR96a:46066
  16. [FGuW] B. FRANCHI, C.E. GUTIERREZ and R.L. WHEEDEN, Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. P.D.E., 19 (1994), 523-604. Zbl0822.46032MR96h:26019
  17. [FL] B. FRANCHI and E. LANCONELLI, Hölder regularity for a class of linear non uniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa, (IV) 10 (1983), 523-541. Zbl0552.35032MR85k:35094
  18. [FLW] B. FRANCHI, G. LU and R.L. WHEEDEN, Weighted Poincaré inequalities for Hörmander vector fields and local regularity for a class of degenerate elliptic equations, Proceedings of the Conference “Potential theory and partial differential operators with nonnegative characteristic form”, Parma, February 1994, Kluwer, Amsterdam, to appear. 
  19. [FS] B. FRANCHI and R. SERAPIONI, Pointwise estimates for a class of strongly degenerate elliptic operators, Ann. Scuola Norm. Sup. Pisa, (IV) 14 (1987), 527-568. Zbl0685.35046MR90e:35076
  20. [G] M. GROMOV, Structures Métriques pour les Variétés Riemanniennes (rédigé par J. Lafontaine et P. Pansu), CEDIC Ed., Paris, 1981. Zbl0509.53034MR85e:53051
  21. [GGK] I. GENEBASHVILI, A. GOGATISHVILI and V. KOKILASHVILI, Criteria of general weak type inequalities for integral transforms with positive kernels, Proc. Georgian Acad. Sci. Math., 1 (1993), 11-34. Zbl0803.42011MR94j:42030
  22. [H] L. HÖRMANDER, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. Zbl0156.10701MR36 #5526
  23. [IN] T. IWANIEC and C.A. NOLDER, Hardy-Littlewood inequality for quasiregular mappings in certain domains in ℝn, Ann. Acad. Sci. Fenn. Series A.I. Math., 10 (1985), 267-282. Zbl0588.30023MR87d:30022
  24. [J] D. JERISON, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J., 53 (1986), 503-523. Zbl0614.35066MR87i:35027
  25. [L1] G. LU, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications, Revista Mat. Iberoamericana, 8 (1992), 367-439. Zbl0804.35015MR94c:35061
  26. [L2] G. LU, The sharp Poincaré inequality for free vector fields: An endpoint result, Preprint 1992, Revista Mat. Iberoamericana, 10 (2) (1994), 453-466. Zbl0860.35006MR96g:26023
  27. [L3] G. LU, Embedding theorems on Campanato-Morrey spaces for vector fields of Hörmander type and applications to subelliptic PDE, C.R. Acad. Sci. Paris, to appear. Zbl0916.46026
  28. [L4] G. LU, Embedding theorems into the Orlicz and Lipschitz spaces and applications to quasilinear subelliptic differential equations, Preprint, February, 1994. 
  29. [L5] G. LU, A note on Poincaré type inequality for solutions to subelliptic equations, Comm. Partial Differential Equations, to appear. Zbl0847.35044
  30. [LW] G. LU and R.L. WHEEDEN, (ε, δ) domains, Poincaré domains and extension theorems on weighted Sobolev spaces for degenerate vector fields, in preparation. 
  31. [MS-Cos] P. MAHEUX and L. SALOFF-COSTE, Analyse sur les boules d'un opérateur sous-elliptique, preprint (1994). Zbl0836.35106
  32. [NSW] A. NAGEL, E.M. STEIN and S. WAINGER, Balls and metrics defined by vector fields I : basic properties, Acta Math., 155 (1985), 103-147. Zbl0578.32044MR86k:46049
  33. [RS] L.P. ROTHSCHILD and E.M. STEIN, Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247-320. Zbl0346.35030MR55 #9171
  34. [S-Cal] A. SÁNCHEZ-CALLE, Fundamental solutions and geometry of the sums of squares of vector fields, Invent. Math., 78 (1984), 143-160. Zbl0582.58004MR86e:58078
  35. [S-Cos] L. SALOFF-COSTE, A note on Poincaré, Sobolev and Harnack inequalities, Internat. Math. Research Notices (Duke Math. J.), 65(2) (1992), 27-38. Zbl0769.58054
  36. [SW] E.T. SAWYER and R.L. WHEEDEN, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math., 114 (1992), 813-874. Zbl0783.42011MR94i:42024
  37. [W] R.L. WHEEDEN, A characterization of some weighted norm inequalities for the fractional maximal function, Studia Math., 107 (1993), 257-272. Zbl0809.42009MR94m:42044

Citations in EuDML Documents

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  1. Francescopaolo Montefalcone, Sets of finite perimeter associated with vector fields and polyhedral approximation
  2. Bruno Franchi, Richard Wheeden, Compensation couples and isoperimetric estimates for vector fields
  3. Bruno Franchi, Francesco Serra Cassano, Gehring's lemma for metrics and higher integrability of the gradient for minimizers of noncoercive variational functionals
  4. Bruno Franchi, Piotr Hajłasz, How to get rid of one of the weights in a two-weight Poincaré inequality?
  5. Daniele Morbidelli, Fractional Sobolev norms and structure of Carnot-Carathéodory balls for Hörmander vector fields
  6. Guozhen Lu, Richard Wheeden, High order representation formulas and embedding theorems on stratified groups and generalizations
  7. Francescopaolo Montefalcone, Some relations among volume, intrinsic perimeter and one-dimensional restrictions of B V functions in Carnot groups
  8. Franchi, Bruno, B V spaces and rectifiability for Carnot-Carathéodory metrics: an introduction
  9. Donatella Danielli, Nicola Garofalo, Duy-Minh Nhieu, Trace inequalities for Carnot-Carathéodory spaces and applications

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