LP estimates for nonvariational hypoelliptic operators with VMO coefficients

Transactions of the American Mathematical Society

Volumn 352, Issue 2, 2000, Pages 781-822

  Bramanti, Marco ^a   Brandolini, Luca ^b  

^a UNIVERSITY OF CAGLIARI   (Italy)

^b UNIVERSITY OF CALABRIA   (Italy)

Author keywords

Discontinuous coefficients; Hypoelliptic operators

Indexed keywords

EID: 0012416550     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-99-02318-1     Document Type: Article

References (41)

1. S. Agmon-A. DougHs-L. Nircnberg: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I: Comm. Pure Appl. Math., 12 (1959), 623-727;
2. II: Comm. Pure Appl. Math., 17 (1964), 35-92. MR 23:A2610; MR 28:5252
3. J.-M. Bony: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénères. Ann. Inst. Fourier, Grenoble, 19, l (1969), 277-304. MR 41:7486
4. M. Bramanti: Commutators of integral operators with positive kernels. Le Matematiche, 49 (1994), fasc. I, 149-168. MR 97c:47059
5. M. Bramanti-L. Brandolini: £p-estimates for uniformly hypoelliptic operators with discontinuous coefficients on homogeneous groups. Preprint.
6. M. Bramanti-M. C. Cerutti: Wp'2-solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients. Comm. in Part. Diff, Eq., 18 (9&10) (1993), 1735-1763. MR 94j:35180
7. M. Bramanti-M. C. Cerutti: Commutators of singular integrals in homogeneous spaces. Boll. Unione Mat. Italiana, (7), 10-B (1996), 843-883. MR 99c:4202G
8. M. Bramanti-M. C. Cerutti: Commutators of fractional integrals in homogeneous spaces. Preprint
9. p-estimates for some ultraparabolic operators with discontinuous coefficients. Journal of Math. Anal, and Appl., 200, 332-354 (1996). MR 97a:35132
10. N. Burger: Espace des fonctions àvariation moyenne bornée sur un espace de nature homogène. C. R. Acd. Se. Paris, t. 286 (1978), 139-142. MR 57:7041
11. A. P. Calderôn-A. Zygmund: Singular integral operators and differential equations, Amer. Journ. of Math., 79 (1957), 901-921. MR 20:7196
12. 2,p-estimates for nondivergence elliptic equations with discontinuous coefficients. Ricerche di Mat. XL (1991), 149-168. MR 93k:35051
13. F. Chiarenza-M. Frasca-P. Longo: lK2'p-solvability of the Dirichlet problem for non divergence elliptic equations with VMO coefficients. Trans. Amer. Math. Soc., 336 (1993), n. l, 841-853. MR 93f:35232
14. M. Christ: Lectures on singular integral operators, CBMS Regional Conf. Ser. in Math., 77 (1990). MR 92f:42021
15. R. Coifman-R. Rochberg-G. Weiss: Factorization theorems for Hardy spaces in several variables. Annals of Math., 103 (1976) 611-635. MR 54:843
16. R. Coifman-G. Weiss: Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes. Lecture Notes in Mathematics, n. 242. Springer-Verlag, Berlin-Heidelberg-New York, 1971. MR 58:17690
17. E. B. Fabes-N. Rivière: Singular integrals with mixed homogeneity. Studia Mathematica, 27, (1966), 19-38. MR 35:683
18. C. Fefferman-D. H. Phong: Subelliptic eigenvalue problems, in Proceedings of the Conference on harmonic Analysis in honor of Antoni Zygmund, 590-606, Wadsworth Math. Series, 1981. MR 86c:35112
19. G. B. Folland: Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv for Math. 13, (1975), 161-207. MR 58:13215
20. b complex and analysis on the Heisenberg group, Comm. on Pure and Appl. Math. XXVII (1974), 429-522. MR 51:3719
21. B. Franchi-G. Lu-R. Wheeden: Weighted Poincaré Inequalities for Hormander vector fields and local regularity for a class of degenerate elliptic equations. Potential Analysis 4 (1995), 361-375. MR 97e:35018
22. B. Franchi-R. Serapioni-F. Serra Cassano: Approximation and embedding theorems for weighted Sobolev spaces with Lipschitz continuous vector fields. Boll. Un. Mat. Ital. B (7) 11 (1997) no. I, 83-117. MR 98c:46062
23. D. Gilbarg-N. S. Trudingcr: Elliptic Partial Differential Equations of Second Order. Second Edition. Springer-Verlag, Berlin-Heidelberg-Neu- York, Tokyo, 1983. MR 86c:35035
24. A. E. Gatto-S. Vagi: Fractional Integrals on Spaces of Homogeneous Type. Analysis and Partial Differential Equations, ed. by Cora Sadosky. Lecture Notes in Pure and Applied Math., vol 122, (1990), 171-216. MR 91e:42032
25. L. Hormander: Hypoelliptic second order differential equations. Acta Mathematica, 119 (1967), 147-171. MR 36:5526
26. F. John-L. Nirenberg: On functions of bounded mean oscillation. Comm. Pure Appl. Math., 14 (1961), 175-188. MR 24:A1348
27. J. J. Kohn: Pseudo-differential operators and hypoellipticity. Proc. Symp. Pure Math., 23, Amer. Math. Soc., 1973, 61-69. MR 49:3356
28. E. Lanconelli: Soluzioni deboli non variazionali per una classe di equazioni di tipo Kolmogorov-Fokker-Planck. Sem. di An. Matem. del Dip. di Matcmatica dell'Univ. di Bologna, 1992-'93.
29. E. Lanconelli-S. Polidoro: On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Polit. Torino 51.4 (1993), 137-171. MR 95h:35044
30. G. Lu: Weighted Poincarc and Sobolev inequalities for vector fields satisfying Hörmander's condition and applications. Revista Matematica Iberoamericana, 8, n. 3 (1992), 367-439. MR 94c:35061
31. G. Lu: Existence and size estimates for the Green's function of differential operators constructed from degenerate vector fields. Comm. in Part. Diff. Eq., 17 (7fc8), (1992), 1213-1251. MR 931:35030
32. A. Nagel: Vector fields and nonisotropic metrics. Beijing lectures in harmonic analysis, ed. by E. M. Stein, Princeton Univ. Press, 1986, 241-306. MR 88f:42045
33. A. Nagel-E. M. Stein-S. Wainger: Boundary behavior of functions holomorphic in domains of finite type. Proc. Natl. Acad. Sci. U.S.A., vol. 78, No. 11, pp. 6596-6599, november 1981, mathematics. MR 82k:32027
34. A. Nagel-E. M. Stein-S. Wainger: Balls and metrics defined by vector fields I: Basic properties. Acta Mathematica, 155 (1985), 130-147. MR 86k:46049
35. S. Polidoro: II metodo della parametrice per operatori di tipo Fokker-Planck. Sem. di An. Matem., Dip. di matematica dell'Univ. di Bologna, 1992-93.
36. L. P. Rothschild-E. M. Stein: Hypoelliptic differential operators and nilpotent groups. Acta Math., 137 (1976), 247-320. MR 55:9171
37. D. Sarason: Functions of vanishing mean oscillations. Trans. Amer. Math. Soc., 207 (1975), 391-405. MR 51:13690
38. A. Sanchez-Calle: Fundamental solutions and geometry of sum of squares of vector fields. Inv. Math., 78 (1984), 143-160. MR 86e:58078
39. E. M. Stein: Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton Univ. Press. Princeton, New Jersey, 1993. MR 95c:42002
40. G. Talenti: Equazioni linear! ellittiche in due variabili. Le Matematiche, vol. 21 (1966), 339376. MR 34:4681
41. C.-J. Xu: Regularity for quasilinear second order subelliptic equations. Comm. in Pure Appl. Math., vol. 45 (1992), 77-96. MR 93b:35042