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On some nonuniform cases of the weighted Sobolev and Poincaré inequalities


Authors: F. I. Mamedov and R. A. Amanov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 20 (2008), nomer 3.
Journal: St. Petersburg Math. J. 20 (2009), 447-463
MSC (2000): Primary 46E35
DOI: https://doi.org/10.1090/S1061-0022-09-01055-3
Published electronically: April 7, 2009
MathSciNet review: 2454455
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Abstract | References | Similar Articles | Additional Information

Abstract: Weighted inequalities $ \Vert f\Vert _{q,\nu,B_0}\le C\sum^{n}_{j=1}\Vert f_{xj}\Vert _{p,\omega_j,B_0}$ of Sobolev type $ (\supp f\subset B_0)$ and of Poincaré type $ (\bar f_{\nu,B_0}=0)$ are studied, with different weight functions for each partial derivative $ f_{x_j}$, for parallelepipeds $ B_0\subset E_n, n\ge 1$. Also, weighted inequalities $ \Vert f\Vert _{q,\nu}\le C\Vert Xf\Vert _{p,\omega}$ of the same type are considered for vector fields $ X=\{X_j\}$, $ j=1, \ldots, m$, with infinitely differentiable coefficients satisfying the Hörmander condition.


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  • 1. V. G. Maz'ya, Sobolev spaces, Leningrad. Gos. Univ., Leningrad, 1985; English transl., Springer-Verlag, Berlin, 1985. MR 0807364 (87g:46055); MR 0817985 (87g:46056)
  • 2. J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577-591. MR 0159138 (28:2356)
  • 3. B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), 523-541. MR 0753153 (85k:35094)
  • 4. F. Chiarenza and R. Serapioni, A Harnack inequality for degenerate parabolic equations, Comm. Partial Differential Equations 9 (1984), 719-749. MR 0748366 (86c:35082)
  • 5. E. T. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), 813-874. MR 1175693 (94i:42024)
  • 6. P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), 1-60. MR 0979599 (90e:53058)
  • 7. M. Gromov, Carnot-Carathéodory spaces seen from within, Sub-Riemannian Geometry, Progr. Math., vol. 144, Birkhäuser, Basel, 1996, pp. 79-323. MR 1421823 (2000f:53034)
  • 8. B. Franchi, Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations, Trans. Amer. Math. Soc. 327 (1991), 125-158. MR 1040042 (91m:35095)
  • 9. B. Franchi, C. E. Guttiérez, and R. L. Wheeden, Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. Partial Differential Equations 19 (1994), 523-604. MR 1265808 (96h:26019)
  • 10. L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. MR 0222474 (36:5526)
  • 11. L. Capogna, D. Danielli, and N. Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations 18 (1993), 1765-1794. MR 1239930 (94j:35038)
  • 12. -, The geometric Sobolev embedding for vector fields and the isoperimetric inequality, Comm. Anal. Geom. 2 (1994), 203-215. MR 1312686 (96d:46032)
  • 13. D. Danielli, Regularity at the boundary for solutions of nonlinear subelliptic equations, Indiana Univ. Math. J. 44 (1995), 269-286. MR 1336442 (97b:35028)
  • 14. M.  de Guzmán, Differentiation of integrals in $ R^n$, Lecture Notes in Math., vol. 481, Springer-Verlag, Berlin-New York, 1975. MR 0457661 (56:15866)
  • 15. C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vols. I, II (Chicago, Ill., 1981), Wadsworth, Belmont, CA, 1983, pp. 590-606. MR 0730094 (86c:35112)
  • 16. P. Hajlasz and P. Strzelecki, Subelliptic $ p$-harmonic maps into spheres and the ghost of Hardy spaces, Max-Planck-Inst. Mat. Naturwiss., Leipzig, Preprint no. 36, 1997, pp. 1-22. Math. Ann. 312 (1998), 341-362. MR 1671796 (2000b:35033)
  • 17. A. Nagel, E. M. Stein, and S. Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), 103-147. MR 0793239 (86k:46049)
  • 18. N. Garofalo and D. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), 1081-1144. MR 1404326 (97i:58032)
  • 19. G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207. MR 0494315 (58:13215)
  • 20. D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), 503-523. MR 0850547 (87i:35027)
  • 21. G. Lu, The sharp Poincaré inequality for free vector fields: An endpoint result, Preprint, 1992; Rev. Math. Iberoamericana 10 (1994), 453-466. MR 1286482 (96g:26023)
  • 22. B. Franchi, G. Lu, and R. L. Wheeden, Representation formulas and weighted Poincaré inequalities for Hörmander vector fields, Ann. Inst. Fourier (Grenoble) 45 (1995), 577-604. MR 1343563 (96i:46037)
  • 23. L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices 1992, no. 2, 27-38. MR 1150597 (93d:58158)
  • 24. D. Danielli, Formules de représentation et théorèmes d'inclusion pour des opérateurs sous elliptiques, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), 987-990. MR 1168522 (93e:35020)
  • 25. C. Pérez, Two weighted norm inequalities for Riesz potentials and uniform $ L^p$-weighted Sobolev inequalities, Indiana Univ. Math. J. 39 (1990), 31-44. MR 1052009 (92a:42024)
  • 26. H. Federer, Geometric measure theory, Grundlehren Math. Wiss., Bd. 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325 (41:1976)
  • 27. W. Fleming and R. Rishel, An integral formula for the total gradient variation, Arch. Math. 11 (1960), 218-222. MR 0114892 (22:5710)

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Additional Information

F. I. Mamedov
Affiliation: Institute of Mathematics and Mechanics, National Academy of Sciences, Azerbaidzhan, and Dichle University, Diyarbakir, Turkey
Email: farman-m@mail.ru

R. A. Amanov
Affiliation: Institute of Mathematics and Mechanics, National Academy of Sciences, Azerbaidzhan
Email: rabilamanov@hotmail.com

DOI: https://doi.org/10.1090/S1061-0022-09-01055-3
Keywords: Sobolev and Poincar\'e inequalities, Carnot-Carath\'eodory metric, Besicovitch property
Received by editor(s): June 14, 2006
Published electronically: April 7, 2009
Additional Notes: The work of the first author was supported in part by INTAS (grant no. 8792)
Article copyright: © Copyright 2009 American Mathematical Society

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