Levi’s parametrix for some sub-elliptic non-divergence form operators
Authors:
Andrea Bonfiglioli, Ermanno Lanconelli and Francesco Uguzzoni
Journal:
Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 10-18
MSC (2000):
Primary 35A08, 35H20, 43A80; Secondary 35A17, 35J70
DOI:
https://doi.org/10.1090/S1079-6762-03-00107-0
Published electronically:
January 31, 2003
MathSciNet review:
1988867
Full-text PDF Free Access
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Abstract: We construct the fundamental solutions for the sub-elliptic operators in non-divergence form ${\textstyle \sum _{i,j}} a_{i,j}(x,t) X_iX_j-\partial _t$ and ${\textstyle \sum _{i,j}}a_{i,j}(x) X_iX_j$, where the $X_i$’s form a stratified system of Hörmander vector fields and $a_{i,j}$ are Hölder continuous functions belonging to a suitable class of ellipticity.
BLU Bonfiglioli, A., Lanconelli, E., Uguzzoni, F., Uniform Gaussian estimates of the fundamental solutions for heat operators on Carnot groups, Adv. Differential Equations 7 (2002), 1153–1192.
BLU2 Bonfiglioli, A., Lanconelli, E., Uguzzoni, F., Fundamental solutions for non-divergence form operators on stratified groups, preprint.
BU1 Bonfiglioli, A., Uguzzoni, F., A note on lifting of Carnot groups, preprint.
BU2 Bonfiglioli, A., Uguzzoni, F., Families of diffeomorphic sub-Laplacians and free Carnot groups, to appear in Forum Math.
- Marco Bramanti and Luca Brandolini, $L^p$ estimates for nonvariational hypoelliptic operators with VMO coefficients, Trans. Amer. Math. Soc. 352 (2000), no. 2, 781–822. MR 1608289, DOI https://doi.org/10.1090/S0002-9947-99-02318-1
- Luca Capogna, Regularity for quasilinear equations and $1$-quasiconformal maps in Carnot groups, Math. Ann. 313 (1999), no. 2, 263–295. MR 1679786, DOI https://doi.org/10.1007/s002080050261
- Luca Capogna, Donatella Danielli, and Nicola Garofalo, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math. 118 (1996), no. 6, 1153–1196. MR 1420920
CapognaHan Capogna, L., Han, Q., Pointwise Schauder estimates for second order linear equations in Carnot groups, preprint.
- Giovanna Citti, Nicola Garofalo, and Ermanno Lanconelli, Harnack’s inequality for sum of squares of vector fields plus a potential, Amer. J. Math. 115 (1993), no. 3, 699–734. MR 1221840, DOI https://doi.org/10.2307/2375077
CLM Citti, G., Lanconelli, E., Montanari, A., Smoothness of Lipschitz continuous graphs with nonvanishing Levi curvature, Acta Math. 188 (2002), 87–128.
- G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161–207. MR 494315, DOI https://doi.org/10.1007/BF02386204
- B. Franchi, G. Lu, and R. L. Wheeden, Weighted Poincaré inequalities for Hörmander vector fields and local regularity for a class of degenerate elliptic equations, Potential Anal. 4 (1995), no. 4, 361–375. Potential theory and degenerate partial differential operators (Parma). MR 1354890, DOI https://doi.org/10.1007/BF01053453
- Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. MR 222474, DOI https://doi.org/10.1007/BF02392081
- Gerhard Huisken and Wilhelm Klingenberg, Flow of real hypersurfaces by the trace of the Levi form, Math. Res. Lett. 6 (1999), no. 5-6, 645–661. MR 1739222, DOI https://doi.org/10.4310/MRL.1999.v6.n6.a5
- David Jerison and John M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), no. 2, 167–197. MR 880182
- David S. Jerison and Antonio Sánchez-Calle, Estimates for the heat kernel for a sum of squares of vector fields, Indiana Univ. Math. J. 35 (1986), no. 4, 835–854. MR 865430, DOI https://doi.org/10.1512/iumj.1986.35.35043
- S. Kusuoka and D. Stroock, Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator, Ann. of Math. (2) 127 (1988), no. 1, 165–189. MR 924675, DOI https://doi.org/10.2307/1971418
LancPascPol Lanconelli, E., Pascucci, A., Polidoro, S., Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance, Nonlinear Problems in Mathematical Physics and Related Topics, II, in Honor of Professor O. A. Ladyzhenskaya, International Mathematical Series, 2, to appear.
- Guozhen Lu, Existence and size estimates for the Green’s functions of differential operators constructed from degenerate vector fields, Comm. Partial Differential Equations 17 (1992), no. 7-8, 1213–1251. MR 1179284, DOI https://doi.org/10.1080/03605309208820883
- A. Montanari, Real hypersurfaces evolving by Levi curvature: smooth regularity of solutions to the parabolic Levi equation, Comm. Partial Differential Equations 26 (2001), no. 9-10, 1633–1664. MR 1865940, DOI https://doi.org/10.1081/PDE-100107454
- Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. MR 1867362
- Jean Petitot and Yannick Tondut, Vers une neurogéométrie. Fibrations corticales, structures de contact et contours subjectifs modaux, Math. Inform. Sci. Humaines 145 (1999), 5–101 (French, with English and French summaries). MR 1697185
- Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247–320. MR 436223, DOI https://doi.org/10.1007/BF02392419
- Zbigniew Slodkowski and Giuseppe Tomassini, Weak solutions for the Levi equation and envelope of holomorphy, J. Funct. Anal. 101 (1991), no. 2, 392–407. MR 1136942, DOI https://doi.org/10.1016/0022-1236%2891%2990164-Z
- N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, Cambridge, 1992. MR 1218884
- Chao Jiang Xu, Regularity for quasilinear second-order subelliptic equations, Comm. Pure Appl. Math. 45 (1992), no. 1, 77–96. MR 1135924, DOI https://doi.org/10.1002/cpa.3160450104
BLU Bonfiglioli, A., Lanconelli, E., Uguzzoni, F., Uniform Gaussian estimates of the fundamental solutions for heat operators on Carnot groups, Adv. Differential Equations 7 (2002), 1153–1192.
BLU2 Bonfiglioli, A., Lanconelli, E., Uguzzoni, F., Fundamental solutions for non-divergence form operators on stratified groups, preprint.
BU1 Bonfiglioli, A., Uguzzoni, F., A note on lifting of Carnot groups, preprint.
BU2 Bonfiglioli, A., Uguzzoni, F., Families of diffeomorphic sub-Laplacians and free Carnot groups, to appear in Forum Math.
BB1 Bramanti, M., Brandolini, L., $L^p$ estimates for nonvariational hypoelliptic operators with VMO coefficients, Trans. Amer. Math. Soc. 352 (2000), no. 2, 781–822.
Cap1 Capogna, L., Regularity for quasilinear equations and $1$-quasiconformal maps in Carnot groups, Math. Ann. 313 (1999), no. 2, 263–295.
CDG Capogna, L., Danielli, D., Garofalo, N., Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations, Amer. J. Math. 118 (1996), no. 6, 1153–1196.
CapognaHan Capogna, L., Han, Q., Pointwise Schauder estimates for second order linear equations in Carnot groups, preprint.
CGL G. Citti, N. Garofalo, E. Lanconelli, Harnack’s inequality for sum of squares of vector fields plus a potential, Amer. J. Math. 115 (1993), 699–734.
CLM Citti, G., Lanconelli, E., Montanari, A., Smoothness of Lipschitz continuous graphs with nonvanishing Levi curvature, Acta Math. 188 (2002), 87–128.
F Folland, G. B., Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), 161–207.
FLW Franchi, B., Lu, G., Wheeden, R. L., Weighted Poincaré inequalities for Hörmander vector fields and local regularity for a class of degenerate elliptic equations, Potential Anal. 4 (1995), 361–375.
Ho Hörmander, L., Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171.
HuK Huisken, G., Klingenberg, W., Flow of real hypersurfaces by the trace of the Levi form, Math. Res. Lett. 6 (1999), 645–661.
JL Jerison, D., Lee, J. M., The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), 167–197.
JS Jerison, D., Sánchez-Calle, A., Estimates for the heat kernel for a sum of squares of vector fields, Indiana Univ. Math. J. 35 (1986), 835–854.
KS2 Kusuoka, S., Stroock, D., Long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator, Ann. of Math. 127 (1988), 165–189.
LancPascPol Lanconelli, E., Pascucci, A., Polidoro, S., Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance, Nonlinear Problems in Mathematical Physics and Related Topics, II, in Honor of Professor O. A. Ladyzhenskaya, International Mathematical Series, 2, to appear.
L2 Lu, G., 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.
M Montanari, A., Real hypersurfaces evolving by Levi curvature: smooth regularity of solutions to the parabolic Levi equation, Comm. Partial Differential Equations 26 (2001), 1633–1664.
Montgomery Montgomery, R., A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs 91, American Mathematical Society, Providence, RI, 2002.
Petitot Petitot, J., Tondut, Y., Vers une neurogéométrie. Fibrations corticales, structures de contact et contours subjectifs modaux, Math. Inform. Sci. Humaines 145 (1999), 5–101.
RS Rothschild, L. P., Stein, E. M., Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247–320.
ST Slodkowski, Z., Tomassini, G., Weak solutions for the Levi equation and envelope of holomorphy, J. Funct. Anal. 101 (1991), 392–407.
VSC Varopoulos, N. T., Saloff-Coste, L., Coulhon, T., Analysis and geometry on groups, Cambridge Tracts in Mathematics 100, Cambridge University Press, Cambridge, 1992.
X Xu, C. J., Regularity for quasilinear second-order subelliptic equations, Comm. Pure Appl. Math. 45 (1992), 77–96.
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Additional Information
Andrea Bonfiglioli
Affiliation:
Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
Email:
bonfigli@dm.unibo.it
Ermanno Lanconelli
Affiliation:
Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
Email:
lanconel@dm.unibo.it
Francesco Uguzzoni
Affiliation:
Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
Email:
uguzzoni@dm.unibo.it
Keywords:
Non-divergence sub-elliptic operators,
stratified groups,
fundamental solutions,
parametrix method
Received by editor(s):
November 11, 2002
Published electronically:
January 31, 2003
Additional Notes:
Investigation supported by the University of Bologna Funds for selected research topics.
Communicated by:
Michael Taylor
Article copyright:
© Copyright 2003
American Mathematical Society
