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Fractional Sobolev norms and structure of Carnot-Carathéodory balls for Hörmander vector fields

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We study the notion of fractional $L^p$-differentiability of order $s∈(0,1)$ along vector fields satisfying the Hörmander condition on $ℝ^n$. We prove a modified version of the celebrated structure theorem for the Carnot-Carathéodory balls originally due to Nagel, Stein and Wainger. This result enables us to demonstrate that different $W^{s,p}$-norms are equivalent. We also prove a local embedding $W^{1,p} ⊂ W^{s,q}$, where q is a suitable exponent greater than p.
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  • Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40127 Bologna, Italy,
  • [1] D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J. 44 (1995), 1033-1074.
  • [2] G. Ben Arous and M. Gradinaru, Singularities of hypoelliptic Green functions, Potential Anal. 8 (1998), 217-258.
  • [3] S. Berhanu and I. Pesenson, The trace problem for vector fields satisfying Hörmander's condition, Math. Z. 231 (1999), 103-122.
  • [4] M. Biroli and U. Mosco, Sobolev inequalities on homogeneous spaces, Potential Anal. 4 (1995), 311-324.
  • [5] B J. M. Bony, Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble) 19 (1969), no. 1, 277-304.
  • [6] S. Buckley, P. Koskela and G. Lu, Subelliptic Poincaré estimates: the case p < 1, Publ. Math. 39 (1995), 313-334.
  • [7] 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.
  • [8] L. Capogna, D. Danielli and N. Garofalo, The geometric Sobolev embedding for vector fields and the isoperimetric inequality, Comm. Anal. Geom. 2 (1994), 203-215.
  • [9] L. Capogna, D. Danielli and N. Garofalo, Subelliptic mollifiers and a basic pointwise estimate of Poincaré type, Math. Z. 226 (1997), 147-154.
  • [10] L. Capogna, D. Danielli and N. Garofalo, Capacitary estimates and the local behavior of solutions to nonlinear subelliptic equations, Amer. J. Math. 118 (1996), 1153-1196.
  • [11] J. Y. Chemin et C. J. Xu, Inclusions de Sobolev en calcul de Weyl-Hörmander et champs de vecteurs sous-elliptiques, Ann. Sci. École Norm. Sup. (4) 30 (1997), 719-751.
  • [12] V. M. Chernikov and S. K. Vodop'yanov, Sobolev spaces and hypoelliptic equations I, II, Siberian Adv. Math. 6 (1996), no. 3, 27-67, and no. 4, 64-96.
  • [13] C W. L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. 117 (1940), 98-105.
  • [14] G. Citti and G. Di Fazio, Hölder continuity of the solutions for operators which are sum of squares of vector fields plus a potential, Proc. Amer. Math. Soc. 122 (1994), 741-750.
  • [15] G. Citti, N. Garofalo and E. Lanconelli, Harnack's inequality for sum of squares of vector fields plus a potential, Amer. J. Math. 115 (1993), 699-734.
  • [16] D D. Danielli, A compact embedding theorem for a class of degenerate Sobolev spaces, Rend. Sem. Mat. Univ. Politec. Torino 49 (1991), 339-420.
  • [17] C. Fefferman and D. H. Phong, Subelliptic eigenvalue problems, in: Proc. Conf. on Harmonic Analysis in honor of Antoni Zygmund, Wadsworth, Belmont, CA, 1983, 590-606.
  • [18] C. Fefferman and A. Sánchez-Calle, Fundamental solution for second order subelliptic operators, Ann. of Math. 124 (1986), 247-272.
  • [19] F G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161-207.
  • [20] B. Franchi, S. Gallot and R. Wheeden, Sobolev and isoperimetric inequalities for degenerate metrics, Math. Ann. 300 (1994), 557-571.
  • [21] B. Franchi et E. Lanconelli, Une métrique associée à une classe d'opérateurs elliptiques dégénérés, in: Conference on Linear Partial and Pseudodifferential Operators, Rend. Sem. Mat. Univ. Politec. Torino (1983) (special issue), 105-114.
  • [22] B. Franchi et E. Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa 10 (1983), 523-541.
  • [23] B. Franchi et E. Lanconelli, An embedding theorem for Sobolev spaces related to non-smooth vector fields and Harnack inequality, Comm. Partial Differential Equations 9 (1984), 1237-1264.
  • [24] B. Franchi, G. Lu and R. Wheeden, Representation formulas and weighted Poincaré inequalities for Hörmander vector fields, Ann. Inst. Fourier (Grenoble) 45 (1995), 577-604.
  • [25] B. Franchi, G. Lu and R. Wheeden, A relationship between Poincaré type inequalities and representation formulas in spaces of homogeneous type, Internat. Math. Res. Notices 1996, no. 1, 1-14.
  • [26] B. Franchi, R. Serapioni and F. Serra Cassano, Approximation and embedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields, Boll. Un. Mat. Ital. B (7) 11 (1997), 83-117.
  • [27] N. Garofalo and D. M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), 1081-1144.
  • [28] N. Garofalo and D. M. Nhieu, Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces, J. Anal. Math. 74 (1998), 67-97.
  • [29] P. Hajłasz and P. Koskela, Sobolev meets Poincaré, C. R. Acad. Sci. Paris 320 (1995), 1211-1215.
  • [30] P. Hajłasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. (to appear).
  • [31] P. Hajłasz and P. Strzelecki, Subelliptic p-harmonic maps into spheres and the ghost of Hardy spaces, Math. Ann. 312 (1998), 341-362.
  • [32] G. Hochschild, La structure des groupes de Lie, Dunod, Paris, 1968.
  • [33] H L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171.
  • [34] L. Hörmander and A. Melin, Free systems of vector fields, Ark. Mat. 16 (1978), 83-88.
  • [35] J D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), 503-523.
  • [36] D. Jerison and A. Sánchez-Calle, Estimates for the heat kernel for a sum of squares of vector fields, Indiana Univ. Math. J. 35 (1986), 835-854.
  • [37] D. Jerison and A. Sánchez-Calle, Subelliptic second order differential operators, in: Lecture Notes in Math. 1277, Springer, 1987, 46-77.
  • [38] K N. V. Krylov, Hölder continuity and $L^p$ estimates for elliptic equations under general Hörmander's condition, Topol. Methods Nonlinear Anal. 9 (1997), 249-258.
  • [39] S. Kusuoka and D. W. Strook, Applications of the Malliavin calculus, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 34 (1987), 392-442.
  • [40] S. Kusuoka and D. W. Strook, Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator, Ann. of Math. 127 (1988), 165-189.
  • [41] E. Lanconelli, Stime subellittiche e metriche Riemanniane singolari, Seminario di Analisi Matematica, Dipartimento di Matematica, Università di Bologna, A. A. 1982-83.
  • [42] E. Lanconelli and D. Morbidelli, On the Poincaré inequality for vector fields, Ark. Mat. (to appear).
  • [43] G. Lu, Existence and size estimates for the Green's functions of differential operators constructed from degenerate vector fields, Comm. Partial Differential Equations 17 (1992), 1213-1251.
  • [44] G. Lu, Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations, Publ. Math. 40 (1996), 301-329.
  • [45] G. Lu, A note on a Poincaré type inequality for solutions to subelliptic equations, Comm. Partial Differential Equations 21 (1996), 235-254.
  • [46] P. Maheux et L. Saloff-Coste, Analyse sur les boules d'un opérateur sous-elliptique, Math. Ann. 303 (1995), 713-746.
  • [47] A. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields I: basic properties, Acta Math. 155 (1985), 103-147.
  • [48] R W. C. Rheinboldt, Local mapping relations and global implicit function theorems, Trans. Amer. Math. Soc. 138 (1969), 183-198.
  • [49] L. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247-320.
  • [50] S K. Saka, Besov spaces and Sobolev spaces on a nilpotent Lie group, Tôhoku Math. J. 31 (1979), 383-437.
  • [51] L. Saloff-Coste, A note on Poincaré, Sobolev and Harnack inequalities, Internat. Math. Res. Notices 1992, no. 2, 27-38.
  • [52] A. Sánchez-Calle, Fundamental solutions and geometry of sum of squares of vector fields, Invent. Math. 78 (1984), 143-160.
  • [53] J.-P. Serre, Lie Algebras and Lie Groups, Lecture Notes in Math. 1500, Springer, 1992.
  • [54] T. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge Univ. Press, 1992.
  • [55] S. K. Vodop'yanov and I. G. Markina, Exceptional sets for solutions of subelliptic equations, Siberian Math. J. 36 (1995), 694-706.
  • [56] X C. J. Xu, Regularity for quasi linear second order subelliptic equations, Comm. Pure Appl. Math. 45 (1992), 77-96.
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