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Representation formulas and weighted Poincaré inequalities for Hörmander vector fields. (English) Zbl 0820.46026

Summary: 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.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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[1] [BKL] , and , Boman aka John, in preparation.
[2] [BM1] and , Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur., to appear. · Zbl 0837.31006
[3] [BM2] and , Proceedings of the Conference “Potential theory and partial differential operators with nonnegative characteristic form”, Parma, February 1994, Kluwer, Amsterdam, to appear.
[4] [Bo] , Remarks on Sobolev imbedding inequalities, Lecture Notes in Math. 1351 (1989), 52-68, Springer-Verlag. · Zbl 0662.46037
[5] [Bu] , The Geometry of Geodesics, Academic Press, New York, 1955. · Zbl 0112.37002
[6] [Ca] , Inequalities for the maximal function relative to a metric, Studia Math., 57 (1976), 297-306. · Zbl 0341.44007
[7] [CDG] , , , Subelliptic mollifiers and a characterization of Rellich and Poincaré domains, Rend. Sem. Mat. Univ. Politec. Torino, 51 (1993), 361-386. · Zbl 0811.35017
[8] [Ch] , Weighted Sobolev’s inequality on domains satisfying the Boman chain condition, Proc. Amer. Math. Soc., to appear. · Zbl 0812.46020
[9] [Cou] , Espaces de Lipschitz et inegalités de Poincaré, J. Funct. Anal., to appear. · Zbl 0859.58009
[10] [CW] and , Weighted Poincaré and Sobolev inequalities and estimates for the Peano maximal function, Amer. J. Math., 107 (1985), 1191-1226. · Zbl 0575.42026
[11] [Fe] , Geometric Measure Theory, Springer, 1969. · Zbl 0176.00801
[12] [FP] and , Subelliptic eigenvalue estimates, Conference on Harmonic Analysis, Chicago, 1980, W. Beckner et al. ed., Wadsworth (1981), 590-606. · Zbl 0503.35071
[13] [F] , Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic operators, Trans. Amer. Math. Soc., 327 (1991), 125-158. · Zbl 0751.46023
[14] [FGaW1] , and , Inégalités isoperimétriques pour des métriques dégénérées, C.R. Acad. Sci. Paris, Sér. I, Math., 317 (1993), 651-654. · Zbl 0794.51011
[15] [FGaW2] , and , Sobolev and isoperimetric inequalities for degenerate metrics, Math. Ann., 300 (1994), 557-571. · Zbl 0830.46027
[16] [FGuW] , and , Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. P.D.E., 19 (1994), 523-604. · Zbl 0822.46032
[17] [11] , The Analysis of Linear Partial Differential Operators III, IV, Berlin - Heidelberg - New York, Springer ; 1985. · Zbl 0552.35032
[18] [FLW] , and , 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] and , Pointwise estimates for a class of strongly degenerate elliptic operators, Ann. Scuola Norm. Sup. Pisa, (IV) 14 (1987), 527-568. · Zbl 0685.35046
[20] [G] , Structures Métriques pour les Variétés Riemanniennes (rédigé par J. Lafontaine et P. Pansu), CEDIC Ed., Paris, 1981. · Zbl 0509.53034
[21] [GGK] , and , Criteria of general weak type inequalities for integral transforms with positive kernels, Proc. Georgian Acad. Sci. Math., 1 (1993), 11-34. · Zbl 0803.42011
[22] [H] , Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. · Zbl 0156.10701
[23] [IN] and , Hardy-Littlewood inequality for quasiregular mappings in certain domains in ℝn, Ann. Acad. Sci. Fenn. Series A.I. Math., 10 (1985), 267-282. · Zbl 0588.30023
[24] [J] , The Poincaré inequality for vector fields satisfying Hörmander’s condition, Duke Math. J., 53 (1986), 503-523. · Zbl 0614.35066
[25] [L1] , Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications, Revista Mat. Iberoamericana, 8 (1992), 367-439. · Zbl 0804.35015
[26] [L2] , The sharp Poincaré inequality for free vector fields: An endpoint result, Preprint 1992, Revista Mat. Iberoamericana, 10 (2) (1994), 453-466. · Zbl 0860.35006
[27] [L3] , Embedding theorems on Campanato-Morrey spaces for vector fields of Hörmander type and applications to subelliptic PDE, C.R. Acad. Sci. Paris, to appear. · Zbl 0916.46026
[28] [L4] , Embedding theorems into the Orlicz and Lipschitz spaces and applications to quasilinear subelliptic differential equations, Preprint, February, 1994.
[29] [L5] , A note on Poincaré type inequality for solutions to subelliptic equations, Comm. Partial Differential Equations, to appear. · Zbl 0847.35044
[30] [LW] and , (ε, δ) domains, Poincaré domains and extension theorems on weighted Sobolev spaces for degenerate vector fields, in preparation.
[31] [MS-Cos] and , Analyse sur les boules d’un opérateur sous-elliptique, preprint (1994). · Zbl 0836.35106
[32] [NSW] , and , Balls and metrics defined by vector fields I : basic properties, Acta Math., 155 (1985), 103-147. · Zbl 0578.32044
[33] [RS] and , Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247-320. · Zbl 0346.35030
[34] [S-Cal] , Fundamental solutions and geometry of the sums of squares of vector fields, Invent. Math., 78 (1984), 143-160. · Zbl 0582.58004
[35] [S-Cos] , A note on Poincaré, Sobolev and Harnack inequalities, Internat. Math. Research Notices (Duke Math. J.), 65(2) (1992), 27-38. · Zbl 0769.58054
[36] [SW] and , Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math., 114 (1992), 813-874. · Zbl 0783.42011
[37] [W] , A characterization of some weighted norm inequalities for the fractional maximal function, Studia Math., 107 (1993), 257-272. · Zbl 0809.42009
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