Degree theory for VMO maps on metric spaces

Uguzzoni, Francesco; Lanconelli, Ermanno

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 3, pp. 569-601.

We construct a degree theory for Vanishing Mean Oscillation functions in metric spaces, following some ideas of Brezis & Nirenberg. The underlying sets of our metric spaces are bounded open subsets of ${ℝ}^N$ and their boundaries. Then, we apply our results in order to analyze the surjectivity properties of the $L$-harmonic extensions of VMO vector-valued functions. The operators $L$ we are dealing with are second order linear differential operators sum of squares of vector fields satisfying the hypoellipticity condition of Hörmander.

[1] H. Bahouri - J. Y. Chemin - C. J. Xu, Trace and trace lifting theorems in weight Sobolev spaces, preprint. | MR | Zbl

[2] S. Berhanu - I. Pesenson, The trace problem for vector fields satisfying Hörmander's condition, Math. Z. 231 (1999), 103-122. | MR | Zbl

[3] F. Bethuel - H. Brezis - F. Hélein, Ginzburg-Landau vortices, In: “Progress in Nonlinear Differential Equations and their Applications” 13 Birkhäuser, Boston, 1994. | MR | Zbl

[4] A. Bonfiglioli - E. Lanconelli - F. Uguzzoni, Uniform Gaussian estimates of the fundamental solutions for heat operators on Carnot groups, to appear in Adv. Differential Equations. | MR | Zbl

[5] A. Bonfiglioli - F. Uguzzoni, Families of diffeomorphic sub-Laplacians and free Carnot groups, preprint. | MR | Zbl

[6] 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), 277-304. | Numdam | MR | Zbl

[7] H. Brezis - J. M. Coron, Large solutions for harmonic maps in two dimensions, Comm. Math. Phys. 92 (1983), 203-215. | MR | Zbl

[8] H. Brezis - L. Nirenberg, Degree theory and BMO; Part I: compact manifolds without boundaries, Selecta Math. 1 (1995), 197-263. | MR | Zbl

[9] H. Brezis - L. Nirenberg, Degree theory and BMO; Part II: compact manifolds with boundaries, Selecta Math. 2 (1996), 309-368. | MR | Zbl

[10] S. M. Buckley, Inequalities of John-Nirenberg type in doubling spaces, J. Anal. Math. 79 (1999), 215-240. | MR | Zbl

[11] L. Capogna - N. Garofalo - D. M. Nhieu, A version of a theorem of Dahlberg for the subelliptic Dirichlet problem, Math. Res. Lett. 5 (1998), 541-549. | MR | Zbl

[12] C. Castaing - M. Valadier, “Convex analysis and measurable multifunctions”, Lecture Notes in Math. 580, Springer-Verlag, New York, 1977. | MR | Zbl

[13] D. Christodoulou, On the geometry and dynamics of crystalline continua, Ann. Inst. H. Poincaré, Phys. Théor. 69 (1998), 335-358. | Numdam | MR | Zbl

[14] R. Coifman - G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer-Verlag, Berlin-New York, 1971. | MR | Zbl

[15] D. Danielli - N. Garofalo - D. M. Nhieu, Trace inequalities for Carnot-Carathéodory spaces and applications, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), 195-252. | Numdam | MR | Zbl

[16] D. Danielli - N. Garofalo - D. M. Nhieu, Sub-elliptic Besov spaces and the characterization of traces on lower dimensional manifolds, preprint. | MR | Zbl

[17] D. Danielli - N. Garofalo - D. M. Nhieu, Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carathéodory spaces, preprint. | MR | Zbl

[18] H. A. De Kleine - J. E. Girolo, A degree theory for almost continuous functions, Fund. Math. 101 (1978), 39-52. | MR | Zbl

[19] M. Derridj, Un problème aux limites pour une classe d'opérateurs du second ordre hypoelliptiques, Ann. Inst. Fourier Grenoble 21 (1971), 99-148. | Numdam | MR | Zbl

[20] M. Derridj, Sur un théorème de traces, Ann. Inst. Fourier Grenoble 22 (1972), 73-83. | Numdam | MR | Zbl

[21] M. J. Esteban - S. Müller, Sobolev maps with integer degree and applications to Skyrme's problem, Proc. Roy. Soc. London Ser. A 436 (1992), 197-201. | MR | Zbl

[22] C. Fefferman - D. H. Phong, Subelliptic eigenvalue problems, In: “Conference on harmonic analysis in honor of Antoni Zygmund”, Vol. I, II, Wadsworth, 1983, 590-606. | MR | Zbl

[23] B. Franchi, Trace theorems for anisotropic weighted Sobolev spaces in a corner, Math. Nachr. 127 (1986), 25-50. | MR | Zbl

[24] B. Franchi - R. Serapioni - F. Serra Cassano, Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields, Boll. Un. Mat. Ital. 11-B (1997), 83-117. | MR | Zbl

[25] L. Gallardo, “Capacités, mouvement Brownien et problème de l'épine de Lebesgue sur les groupes de Lie nilpotents”, Lecture Notes in Math. 928, Springer, Berlin-New York, 1982, 96-120. | MR | Zbl

[26] N. Garofalo - D. M. Nhieu, Isoperimetric and Sobolev inequalities for Carnot- Carathéodory spaces and the existence of minimal surfaces, Commun. Pure Appl. Math. 49 (1996), 1081-1144. | MR | Zbl

[27] M. Giaquinta - G. Modica, - J. Soucek, Remarks on the degree theory, J. Funct. Anal. 125 (1994), 172-200. | MR | Zbl

[28] L. Greco - T. Iwaniec - C. Sbordone - B. Stroffolini, Degree formulas for maps with nonintegrable Jacobian, Topol. Methods Nonlinear Anal. 6 (1995), 81-95. | MR | Zbl

[29] P. Hajlasz - P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688. | MR | Zbl

[30] D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), 503-523. | MR | Zbl

[31] D. Jerison - A. Sánchez-Calle, Subelliptic, second order differential operators, In: “Complex analysis III”, Lecture Notes in Math. 1277 Springer, Berlin, 1987, 46-77. | MR | Zbl

[32] E. Lanconelli - A. E. Kogoj, X-elliptic operators and X-control distances, Ricerche Mat. 49 (2000), 223-243. | MR | Zbl

[33] E. Lanconelli - D. Morbidelli, On the Poincaré inequality for vector fields, Ark. Mat. 38 (2000), 327-342. | MR | Zbl

[34] R. Monti - D. Morbidelli, Trace theorems for vector fields, to appear in Math. Z. | MR | Zbl

[35] R. Monti - F. Serra Cassano, Surface measures in Carnot-Carathéodory spaces, preprint. | Zbl

[36] D. Morbidelli, Fractional Sobolev norms and structure of Carnot-Carathéodory balls for Hörmander vector fields, Studia Math. 139 (2000), 213-244. | MR | Zbl

[37] U. Mosco, Alcuni aspetti variazionali dei mezzi discontinui, Boll. Un. Mat. Ital. (7) 7-A (1993), 149-198. | Zbl

[38] A. Nagel - E. M. Stein - S. Wainger, Balls and metrics defined by vector fields I: basic properties, Acta Math. 155 (1985), 103-147. | MR | Zbl

[39] P. Negrini - V. Scornazzani, Wiener criterion for a class of degenerate elliptic operators, J. Differential Equations 66 (1987), 151-164. | MR | Zbl

[40] L. Nirenberg, “Topics in nonlinear functional analysis”, Courant Institute Lecture Notes, New York, 1974. | MR | Zbl

[41] A. Sánchez-Calle, Fundamental solutions and geometry of the sum of squares of vector fields, Invent. Math. 78 (1984), 143-160. | MR | Zbl

[42] F. Uguzzoni - E. Lanconelli, On the Poisson kernel for the Kohn Laplacian, Rend. Mat. Appl. 17 (1997), 659-677. | MR | Zbl