Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations

Franchi, Bruno

doi:10.1090/S0002-9947-1991-1040042-8

Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations
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Trans. Amer. Math. Soc. 327 (1991), 125-158 Request permission

Abstract:

In this paper we prove a Sobolev-Poincaré inequality for a class of function spaces associated with some degenerate elliptic equations. These estimates provide us with the basic tool to prove an invariant Harnack inequality for weak positive solutions. In addition, Hölder regularity of the weak solutions follows in a standard way.

References

A.-P. Calderón, Inequalities for the maximal function relative to a metric, Studia Math. 57 (1976), no. 3, 297–306. MR 442579, DOI 10.4064/sm-57-3-297-306
Filippo Chiarenza, Aldo Rustichini, and Raul Serapioni, De Giorgi-Moser theorem for a class of degenerate non-uniformly elliptic equations, Comm. Partial Differential Equations 14 (1989), no. 5, 635–662. MR 993823, DOI 10.1080/03605308908820623
Sagun Chanillo and Richard L. Wheeden, Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions, Amer. J. Math. 107 (1985), no. 5, 1191–1226. MR 805809, DOI 10.2307/2374351
Sagun Chanillo and Richard L. Wheeden, Harnack’s inequality and mean-value inequalities for solutions of degenerate elliptic equations, Comm. Partial Differential Equations 11 (1986), no. 10, 1111–1134. MR 847996, DOI 10.1080/03605308608820458
Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116. MR 643158, DOI 10.1080/03605308208820218
Bruno Franchi and Ermanno 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), no. 4, 523–541. MR 753153
Bruno Franchi and Ermanno Lanconelli, An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality, Comm. Partial Differential Equations 9 (1984), no. 13, 1237–1264. MR 764663, DOI 10.1080/03605308408820362
Bruno Franchi and Ermanno Lanconelli, Une condition géométrique pour l’inégalité de Harnack, J. Math. Pures Appl. (9) 64 (1985), no. 3, 237–256 (French). MR 823405
C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 590–606. MR 730094
B. Franchi and R. Serapioni, Pointwise estimates for a class of strongly degenerate elliptic operators: a geometrical approach, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 4, 527–568 (1988). MR 963489
José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
David Jerison, The Poincaré inequality for vector fields satisfying Hörmander’s condition, Duke Math. J. 53 (1986), no. 2, 503–523. MR 850547, DOI 10.1215/S0012-7094-86-05329-9
Robert V. Kohn, New integral estimates for deformations in terms of their nonlinear strains, Arch. Rational Mech. Anal. 78 (1982), no. 2, 131–172. MR 648942, DOI 10.1007/BF00250837
David Kinderlehrer and Guido Stampacchia, An introduction to variational inequalities and their applications, Pure and Applied Mathematics, vol. 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 567696
S. Kusuoka and D. Stroock, Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator, Ann. of Math. (2) 127 (1988), no. 1, 165–189. MR 924675, DOI 10.2307/1971418
Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539
Antonio Sánchez-Calle, Fundamental solutions and geometry of the sum of squares of vector fields, Invent. Math. 78 (1984), no. 1, 143–160. MR 762360, DOI 10.1007/BF01388721
N. Th. Varopoulos, Analysis on Lie groups, J. Funct. Anal. 76 (1988), no. 2, 346–410. MR 924464, DOI 10.1016/0022-1236(88)90041-9

The Harnack’s inequality for second order degenerate elliptic operators

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