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Bony, J.M. (1969). Principe du maximum, inégalité de Harnack et du problème de Cauchy pour les operateurs elliptique degeneres.Ann. Inst. Fourier, Grenoble,119(1), 277–304.
Buckley, S., Koskela, P., and Lu, G. (1996). Boman equals John.XVI-th Rolf Nevanlinna Colloquium, Laine/Martio, Eds. Walter de Gruyter, & Co., Berlin, New York, 91–99.
Caffarelli, L., Fabes, E., Mortola, S., and Salsa, S. (1981). Boundary behavior of nonnegative solutions of elliptic operators in divergence form.Indiana Univ. Math. J.,30(4), 621–640.
Capogna, L., Danielli, D., and Garofalo, N. (1993). An embedding theorem and the Harnack inequality for nonlinear subelliptic equations.Comm. P.D.E.,18(9–10), 1765–1794.
Capogna, L., Danielli, D., and Garofalo, N. (1996). Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations.Amer. J. Math.,118(6), 1153–1196.
Capogna, L., Garofalo, N., and Nhieu, D.-M. (1997). The Dirichlet problem for sub-Laplacians. Preprint.
Capogna, L. and Tang, P. (1995). Uniform domains and quasiconformal mappings in the Heisenberg group.Man. Math.,86(3), 267–282.
Citti, G. (1988). Wiener estimates at boundary points for Hörmander's operators.Boll. U.M.I.,2-B, 667–681.
Citti, G., Garofalo, N., and Lanconelli, E. (1993). Harnack inequality for sum of squares of vector fields plus a potential.Amer. J. Math.,115(3), 699–734.
Chow, W.L. (1939). Über system von linearen partiellen Differentialgleichungen erster Ordnug.Math. Ann.,117, 98–105.
Coifman, R.R. and Weiss, G. (1971). Analyse harmonique non-commutative sur certaines espaces homogènes; étude de certains intégrales singulières.Lecture Notes in Mathematics,242, springer-Verlag, Berlin, New York.
Dahlberg, B.E.J. (1977). On estimates of harmonic measure.Arch. Rat. Mech. Analysis,65, 272–288.
Danielli, D. (1995). Regularity at the boundary for solutions of nonlinear subelliptic equations.Indiana Univ. Math. J.,44(1), 269–286.
Danielli, D., Garofalo, N., and Nhieu, D.-M. (1996). Trace inequalities for Carnot-Carathéodory spaces and applications. Preprint.
Fichera, G. (1955–1956). Sulle equazioni alle derivate parziali del secondo ordine ellittico-paraboliche.Univ. Pol. Torino, Rend. Sem. Mat.,15, 27–47.
Fichera, G. (1956). Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine.Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I,18(5), 1–30.
Fichera, G. (1960). On a unified theory of boundary value problems for elliptic-parabolic equations of second order.Boundary Problems in Differential Equations, Univ. of Wisconsin Press, Madison. 97–120.
Folland, G.B. (1973). A fundamental solution for a subelliptic operator.Bull. Amer. Math. Soc.,79, 373–376.
Folland, G.B. (1975). Subelliptic estimates and function spaces on nilpotent Lie groups.Arkiv. für Math.,13, 161–207.
Folland, G.B., and Stein, E.M. (1982). Hardy spaces on homogeneous groups.Math. Notes,28, Princeton University Press, Princeton, NJ.
Garofalo, N., and Nhieu, D.M. (1996). Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces.Comm. Pure and Appl. Math.,49 1081–1144.
Garofalo, N. and Nhieu, D.M. (1997). Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces.J. d'Anal. Math.,37.
Gaveau, B. (1977). Principe de moindre action, propagation de la chaleur et estimates souselliptiques sur certaines groupe nilpotents.Acta Math.,139(1–2), 95–152.
Gilbarg, D. and Trudinger, N. (1983).Elliptic Partial Differential Equations of Second Order. Springer-Verlag.
Hansen, W. and Huber, H. (1984). The Dirichlet problem for sublaplacians on nilpotent groups—Geometric criteria for regularity.Math. Ann.,246, 537–547.
Hervé, R.M. and Hervé, M. (1972). Les functions surharmoniques dans l'axiomatique de M. Brelot associées à un operateur elliptique dégénéré.Ann. Inst. Fourier, (Grenoble),22(2), 131–145.
Hörmander, L. (1967). Hypoelliptic second order differential equations.Acta Math.,119, 147–171.
Hunt, R.R., and Wheeden, R.L. (1968). On the boundary values of harmonic functions.Trans. Amer. Math. Soc.,132, 307–322.
Hunt, R.R., and Wheeden, R.L. (1970). Positive harmonic functions of Lipschitz domains.Trans. Amer. Math. Soc.,147, 507–527.
Huber, H. (1986). Examples of irregular domains for some hypoelliptic differential operators.Expo Math.,4, 189–192.
Jerison, D.S. (1981). The Dirichlet problem for the Kohn Laplacian on the Heisenberg group, Parts I and II.J. Funct. Analysis,43, 97–142.
Jerison, D.S. (1983). Boundary regularity in the Dirichlet problem for □ b on CR manifolds.Comm. Pure and Appl. Math.,36, 143–181.
Jerison, D.S., and Kenig, C.E. (1982). Boundary behavior of harmonic functions in non-tangentially accessible domains.Adv. in Math.,46, 80–147.
Jones, P.W. (1982). A geometric localization theorem.Adv. in Math.,46, 71–79.
Kaplan, A. (1980). Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms.Trans. Amer. Math. Soc.,258, 147–153.
Kemper, J. (1972). Temperatures in several variables; kernel functions, representations, and parabolic boundary values.Trans. Amer. Math. Soc.,167, 243–262.
Kenig, C.E. (1994). Harmonic analysis techniques for second order elliptic boundary value problems.Regional Conference Series in Mathematics,83, AMS.
Kohn, J.J., and Nirenberg, L. (1965). Non-coercive boundary value problems.Comm. Pure and Appl. Math.,18, 443–492.
Korányi, A. (1983). Geometric aspects of analysis on the Heisenberg group.Topics in Modern Harmonic Analysis,1, 209–258. 1st. Naz. Alta Mat. F. Severi. Rome.
Korányi, A., and Riemann, H.M. (1995). Foundations for the theory of quasiconformal mappings on the Heisenberg group.Adv. in Math.,111(1), 1–87.
Lewis, J. Private communication.
Littman, W., Stampacchia, G., and Weinberger, H.F. (1963). Regular points for elliptic equations with discontinuous coefficients.Ann. S. N. S. Pisa,3(17), 43–79.
Lu, G. (1994). On Harnack's inequality for a class of strongly degenerate Schrödinger operators formed by vector fields.Diff. Int. Eq.,7(1), 73–100.
Nagel, A., Stein, E.M., and Wainger, S. (1985). Balls and metrics defined by vector fields I: basic properties.Acta Math.,155, 103–147.
Negrini, P., and Scornazzani, V. (1987). Wiener criterion for a class of degenerate elliptic operators.J. Diff. Eq.,66, 151–167.
Nhieu, D.M. (1994). Extension theorems for Sobolev spaces in stratified nilpotent Lie groups. Preprint.
Oleinik, O.A., and Radkevich, E.V. (1973). Second order equations with non-negative characteristic form. (Mathematical Analysis 1969), Moscow: Itogi Nauki (1971) [Russian], English translation:Amer. Math. Soc., Providence, RI.
Pansu, P. (1989). Métriques de Carnot-Carathéodory et quasisométries des espaces symmetriques de rang un.Ann. Math.,129, 1–60.
Sanchez-Calle, A. (1984). Fundamental solutions and geometry of sum of squares of vector fields.Inv. Math.,78, 143–160.
Varadarajan, V.S. (1974).Lie Groups, Lie Algebras, and Their Representation. Prentice Hall, Englewood Cliffs, NJ
Vodop'yanov, S.K. (1995). Weighted Sobolev spaces and boundary behavior of solutions to degenerate hypoelliptic equations.Sib. Math. J.,36(2), 246–264.
Wittmann, R. (1987). A nontangential limit theorem.Osaka J. Math.,24, 61–72.
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Acknowledgments and Notes. Nicola Garofalo-Supported by the NSF, grant DMS-9706892.
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Capogna, L., Garofalo, N. Boundary behavior of nonnegative solutions of subelliptic equations in NTA domains for carnot-carathéodory metrics. The Journal of Fourier Analysis and Applications 4, 403–432 (1998). https://doi.org/10.1007/BF02498217
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DOI: https://doi.org/10.1007/BF02498217