Let $X=(X_1,X_2,\dots ,X_q)$ be a system of vector fields satisfying the Hör\-man\-der condition. We prove $L^{2,\lambda}_X$ local regularity for the gradient $Xu$ of a solution of the following strongly elliptic system $$ -X^{*}_{\alpha }(a^{\alpha \beta }_{ij}(x)X_{\beta } u^{j})= g_{i}-X^{*}_{\alpha } f^{\alpha }_{i}(x) \quad \forall i=1,2,\dots ,N, $$ where $a^{\alpha \beta }_{ij}(x)$ are bounded functions and belong to Vanishing Mean Oscillation space.
@article{119351,
author = {Giuseppe Di Fazio and Maria Stella Fanciullo},
title = {Gradient estimates for elliptic systems in Carnot-Carath\'eodory spaces},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {43},
year = {2002},
pages = {605-618},
zbl = {1090.35058},
mrnumber = {2045784},
language = {en},
url = {http://dml.mathdoc.fr/item/119351}
}
Di Fazio, Giuseppe; Fanciullo, Maria Stella. Gradient estimates for elliptic systems in Carnot-Carathéodory spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) pp. 605-618. http://gdmltest.u-ga.fr/item/119351/
On BMO regularity for linear elliptic systems, Ann. Mat. Pura Appl. (4) 161 (1992), 231-269. (1992) | MR 1174819 | Zbl 0802.35015
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math. XII (1959), 623-727. (1959) | MR 0125307 | Zbl 0093.10401
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Comm. Pure Appl. Math. XVII (1964), 35-92. (1964) | MR 0162050 | Zbl 0123.28706
$L^p$ estimates for nonvariational hypoelliptic operators with VMO coefficients, Trans. Amer. Math. Soc. 352 2 (2000), 781-822. (2000) | MR 1608289
$L^p$ estimates for uniformly hypoelliptic operators with discontinuous coefficients on homogeneous groups, to appear in Rend. Sem. Mat. Univ. Politec. Torino. | MR 1962808 | Zbl 0935.35037
Estimates of BMO type for singular integrals on spaces of homogeneous type and applications to hypoelliptic pdes, preprint. | MR 2174915 | Zbl 1082.35060
$W^{1,2}_p$ Solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Comm. Partial Differential Equations 18 (1993), 1735-1763. (1993) | MR 1239929 | Zbl 0816.35045
Espace des fonctions à variation moyenne bornée sur un espace de nature homogène, C.R. Acad. Sci. Paris Série A 286 (1978), 139-142. (1978) | MR 0467176 | Zbl 0368.46037
Equazioni ellittiche del II ordine e spazi ${\Cal L}^{(2,\lambda)}$, Ann. Mat. Pura Appl. (4) 69 (1965), 321-381. (1965) | MR 0192168
Sistemi ellittici in forma di divergenza. Regolarità all'interno, Quaderni SNS Pisa (1980). | MR 0668196
Zero order a priori estimates for solutions of elliptic differential equations, Proceedings of Symposia in Pure Mathematics IV (1961), 157-166. | MR 0146511 | Zbl 0178.46001
$L\sp p$ estimates for linear elliptic systems with discontinuous coefficients, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 5 (1994), 1 27-32. (1994) | MR 1273890 | Zbl 0803.35016
Interior $W^{2,p}$-estimates for nondivergence elliptic equations with discontinuous coefficients Ricerche Mat., XL (1991), 149-168. (1991) | MR 1191890 | Zbl 0772.35017
$W^{2,p}$-solvability of the Dirichlet problem for non divergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336 (1993), 1 841-853. (1993) | MR 1088476
A Fefferman-Phong type inequality and applications to quasilinear subelliptic equations, Potential Anal. 11 (1999), 387-413. (1999) | MR 1719837
Trace inequalities for Carnot-Carathéodory spaces and applications, Ann. SNS Pisa Cl. Sci. (4) 27 (1998), 195-252. (1998) | MR 1664688
$L\sp p$ estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Un. Mat. Ital. A (7) 10 (1996), 2 409-420. (1996) | MR 1405255
Oblique derivative problem for elliptic equations in non-divergence form with VMO coefficients, Comment. Math. Univ. Carolinae 37 (1996), 3 537-556. (1996) | MR 1426919
Oblique derivative problem for quasilinear elliptic equations with VMO coefficients, Bull. Austral. Math. Soc. 53 (1996), 3 501-513. (1996) | MR 1388600
Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal. 112 (1993), 2 241-256. (1993) | MR 1213138
Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients, J. Funct. Anal. 166 (1999), 2 179-196. (1999) | MR 1707751
Sobolev and isoperimetric inequalities for degenerate metrics, Math. Ann. 300 (1994), 4 557-571. (1994) | MR 1314734
Weighted Sobolev-Poincaré inequalities for Grushin type operators, Comm. Partial Differential Equations 19 (1994), 523-604. (1994) | MR 1265808
Regularité partielle pour une classe de systèmes elliptiques dégénérés, C.R. Acad. Sci. Paris Série I 316 (1993), 37-40. (1993) | MR 1198746
Recent Developments in the Theory of Subelliptic Equations and its Geometric Aspects, Birkhäuser, to appear.
Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. XLIX (1996), 1081-1144. (1996) | MR 1404326
Morrey-Campanato spaces on manifolds, Comment. Math. Univ. Carolinae 29 (1988), 2 309-318. (1988) | MR 0957401
Higher Integrability for quasi- minima of functionals depending on vector fields, Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 17 (1993), 209-227. (1993) | MR 1268631
Multiple integrals in calculus of variations and nonlinear elliptic systems, Ann. of Math. Stud. 105 (1983), Princeton University Press. (1983) | MR 0717034
Introduction to Regularity Theory for Nonlinear Elliptic Systems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1993. | MR 1239172 | Zbl 0786.35001
Carnot-Carathéodory spaces seen from within, Inst. Hautes Études Sci. Publ. Math. (1994).
General linear boundary value problems for elliptic operators with VMO coefficients, Math. Nachr. 237 (2002), 62-88. (2002) | MR 1894353 | Zbl 1009.35024
Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), 403-415. (1996)
Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688. | MR 1683160
Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171. (1967) | MR 0222474
Estimates on the Generalized Morrey spaces $L^{2,\lambda }_\varphi $ and BMO for linear elliptic systems, Indiana Univ. Math. J. 45 2 (1996), 397-439. (1996) | MR 1414336
Embedding theorems on Campanato-Morrey space for vector fields on Hörmander type, Approx. Theory Appl. (N.S.) 14 (1998), 1 69-80. (1998) | MR 1651473
Embedding theorems on Campanato-Morrey space for vector fields and applications, C.R. Acad. Sci. Paris Série. I 320 (1995), 4 429-434. (1995) | MR 1320116
Sulle equazioni ellittiche del secondo ordine a coefficienti discontinui, Ann. Mat. Pura Appl. 63 (1963), 353-386. (1963) | MR 0170090
Fundamental solutions and geometry of the sum of squares of vector fields, Invent. Math. 78 (1984), 143-160. (1984) | MR 0762360
Functions of vanishing mean oscillations, Trans. Amer. Math. Soc. 207 (1975), 391-405. (1975) | MR 0377518
On the weak continuity of elliptic operators and applications to potential theory, Amer. J. Math. 124 (2002), 2 369-410. (2002) | MR 1890997 | Zbl 1067.35023
Subelliptic variational problems, Bull. Soc. Math. France 118 (1990), 147-169. (1990) | MR 1087376
Higher interior regularity for quasilinear subellliptic systems, Calc. Var. 5 (1997), 323-343. (1997) | MR 1450714