Two-weight Sobolev-Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators

Bruno Franchi; Cristian E. Gutiérrez; Richard L. Wheeden

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1994)

  • Volume: 5, Issue: 2, page 167-175
  • ISSN: 1120-6330

Abstract

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In this Note we prove a two-weight Sobolev-Poincaré inequality for the function spaces associated with a Grushin type operator. Conditions on the weights are formulated in terms of a strong A ∞ weight with respect to the metric associated with the operator. Roughly speaking, the strong A ∞ condition provides relationships between line and solid integrals of the weight. Then, this result is applied in order to prove Harnack's inequality for positive weak solutions of some degenerate elliptic equations.

How to cite

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Franchi, Bruno, Gutiérrez, Cristian E., and Wheeden, Richard L.. "Two-weight Sobolev-Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 5.2 (1994): 167-175. <http://eudml.org/doc/244130>.

@article{Franchi1994,
abstract = {In this Note we prove a two-weight Sobolev-Poincaré inequality for the function spaces associated with a Grushin type operator. Conditions on the weights are formulated in terms of a strong \( A\_\{\infty\} \)» weight with respect to the metric associated with the operator. Roughly speaking, the strong \( A\_\{\infty\} \)» condition provides relationships between line and solid integrals of the weight. Then, this result is applied in order to prove Harnack's inequality for positive weak solutions of some degenerate elliptic equations.},
author = {Franchi, Bruno, Gutiérrez, Cristian E., Wheeden, Richard L.},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Weighted Sobolev-Poincaré inequalities; Degenerate elliptic equations; Harnack inequality; Sobolev-Poincaré inequality; generalized Grushin operator; weight functions; reverse Hölder inequality; sub-unit curves; Harnack's inequality; positive weak solutions of some degenerate elliptic equations},
language = {eng},
month = {6},
number = {2},
pages = {167-175},
publisher = {Accademia Nazionale dei Lincei},
title = {Two-weight Sobolev-Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators},
url = {http://eudml.org/doc/244130},
volume = {5},
year = {1994},
}

TY - JOUR
AU - Franchi, Bruno
AU - Gutiérrez, Cristian E.
AU - Wheeden, Richard L.
TI - Two-weight Sobolev-Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1994/6//
PB - Accademia Nazionale dei Lincei
VL - 5
IS - 2
SP - 167
EP - 175
AB - In this Note we prove a two-weight Sobolev-Poincaré inequality for the function spaces associated with a Grushin type operator. Conditions on the weights are formulated in terms of a strong \( A_{\infty} \)» weight with respect to the metric associated with the operator. Roughly speaking, the strong \( A_{\infty} \)» condition provides relationships between line and solid integrals of the weight. Then, this result is applied in order to prove Harnack's inequality for positive weak solutions of some degenerate elliptic equations.
LA - eng
KW - Weighted Sobolev-Poincaré inequalities; Degenerate elliptic equations; Harnack inequality; Sobolev-Poincaré inequality; generalized Grushin operator; weight functions; reverse Hölder inequality; sub-unit curves; Harnack's inequality; positive weak solutions of some degenerate elliptic equations
UR - http://eudml.org/doc/244130
ER -

References

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  1. BOJARSKE, B., Remarks on Sobolev imbedding inequalities. Complex Analysis, Lecture Notes in Mathematics1351, Springer, 1989, 52-68. Zbl0662.46037MR982072DOI10.1007/BFb0081242
  2. CALDERON, A. P., Inequalities for the maximal function relative to a metric. Studia Math., 57, 1976, 297-306. Zbl0341.44007MR442579
  3. CAPOGNA, L. - DANIELLI, D. - GAROFALO, N., Embedding theorems and the Harnack inequality for solutions of nonlinear subelliptic equations. C.R. Acad. Sci. Paris, 316, 1993, 809-814. Zbl0797.35063MR1218266
  4. CHANILLO, S. - WHEEDEN, R. L., Weighted Poincaré and Sobolev inequalities and estimates for the Peano maximal function. Amer. J. Math., 107, 1985, 1191-1226. Zbl0575.42026MR805809DOI10.2307/2374351
  5. CHANILLO, S. - WHEEDEN, R. L., Harnack's inequality and mean-value inequalities for solutions of degenerate elliptic equations. Comm. Partial Differential Equations, 11, 1986, 1111-1134. Zbl0634.35035MR847996DOI10.1080/03605308608820458
  6. CHUA, S., Weighted Sobolev inequality on domains satisfying chain conditions. Proc. Amer. Math. Soc., to appear. Zbl0812.46020
  7. DAVID, G. - SEMMES, S., Strong A ∞ weights, Sobolev inequalities and quasiconformal mappings In: C. SADOSKY (éd.), Analysis and Partial Differential Equations. Marcel Dekker, 1990, 101-111. Zbl0752.46014MR1044784
  8. FABES, E. - KENIG, C. - SERAPIONI, R., The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations, 7, 1982, 77-116. Zbl0498.35042MR643158DOI10.1080/03605308208820218
  9. FERNANDES, J. C., Mean value and Harnack inequalities for a certain class of degenerate parabolic equations. Revista Matematica Iberoamericana, 7, 1991, 247-286. Zbl0761.35050MR1165071DOI10.4171/RMI/112
  10. FEFFERMAN, C. - PHONG, D. H., Subelliptic eigenvalue problems. In: W. BECKNER et al (eds.), Conference on Harmonic Analysis (Chicago 1980). Wadsworth1981, 590-606. Zbl0503.35071MR730094
  11. FRANCHI, B., Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations. Trans. Amer. Math. Soc., 327, 1991, 125-158. Zbl0751.46023MR1040042DOI10.2307/2001837
  12. FRANCHI, B., Inégalités de Sobolev pour des champs de vecteurs lipschitziens. C.R. Acad. Sci. Paris, 311, 1990, 329-332. Zbl0734.46023MR1071637
  13. FRANCHI, B. - LANCONELLI, E., Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa, 10, (4), 1983, 523-541. Zbl0552.35032MR753153
  14. FRANCHI, B. - SERAPIONI, R., Pointwise estimates for a class of strongly degenerate elliptic operators: a geometrical approach. Ann. Scuola Norm. Sup. Pisa, 14, (4), 1987, 527-568. Zbl0685.35046MR963489
  15. FRANCHI, B. - GUTIERREZ, C. - WHEEDEN, R. L., Weighted Sobolev-Poincaré inequalities for Grushin type operators. Comm. Partial Differential Equations, to appear. Zbl0822.46032MR1265808DOI10.1080/03605309408821025
  16. FRANCHI, B. - GALLOT, S. - WHEEDEN, R. L., Sobolev and isoperimetric inequalities for degenerate metrics. Math. Ann., to appear. Zbl0830.46027MR1314734DOI10.1007/BF01450501
  17. FRANCHI, B. - GALLOT, S. - WHEEDEN, R. L., Inégalités isopérimetriques pour des métriques dégénérées. C.R. Acad. Sci. Paris, Ser. A, 317, 1993, 651-654. Zbl0794.51011MR1245092
  18. GEHRING, F. W., The L p -integrability of the partial derivatives of a quasi conformai mapping. Acta Math., 130, 1973, 265-277. Zbl0258.30021MR402038
  19. GUTIERREZ, C. - WHEEDEN, R. L., Mean-value and Harnack inequalities for degenerate parabolic equations. Colloq. Math., 60/61, 1990, 157-194. Zbl0785.35057MR1096367
  20. IWANIEK, T. - NOLDER, C. A., Hardy-Littlewood inequality for quasiregular mappings in certain domains in R n . Ann. Acad. Sci. Fenn., Series A I Math.10, 1985, 267-282. Zbl0588.30023MR802488
  21. JERISON, D., The Poincaré inequality for vector fields satisfying Hörmander condition. Duke Math. J., 53, 1986, 503-523. Zbl0614.35066MR850547DOI10.1215/S0012-7094-86-05329-9
  22. LU, G., The sharp Poincaré inequality for free vector fields: an endpoint result. Revista Matématica Iberoamericana, to appear. Zbl0860.35006MR1286482DOI10.4171/RMI/158
  23. NAGEL, A. - STEIN, E. M. - WAINGER, S., Balls and metrics defined by vector fields I: basic properties. Acta Math., 155, 1985, 103-147. Zbl0578.32044MR793239DOI10.1007/BF02392539
  24. SALINAS, O., Harnack inequality and Green function for a certain class of degenerate elliptic differential operators. Revista Matematica Iberoamericana, 7, 1991, 313-349. Zbl0770.35023MR1165073DOI10.4171/RMI/114
  25. SAWYER, E. - WHEEDEN, R. L., Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Amer. J. Math., 114, 1992, 813-874. Zbl0783.42011MR1175693DOI10.2307/2374799

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