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Fundamental solutions for non-divergence form operators on stratified groups. (English) Zbl 1072.35011
The aim of the paper is to provide a fundamental solution \(\Gamma\) for operators of the form \[ H:= \sum^m_{i,j=1} a_{ij}(x,t) X_i X_j-\partial_t\qquad ((x,t)\in \mathbb{R}^{N+1}), \] where the \(X_i\) generate a Lie algebra with an additional homogeneous structure and the Hölder continuous functions \(a_{ij}\) form a matrix which belongs to the set \(M_\Lambda\) of symmetric \((m,m)\)-matrices \(A\) with \[ \Lambda^{-1}|\xi|^2\leq \langle A\xi,\xi\rangle\leq \Lambda|\xi|^2\qquad (\xi\in\mathbb{R}^m), \] \(\Lambda> 1\) being fixed once and for all. Constant coefficient operators admit of fundamental solutions that satisfy Gaussian bounds which are independent of \(A\in M_\Lambda\). This was proved by the authors in [Adv. Differ. Equ. 7, 1153–1192 (2002; Zbl 1036.35061)]. This result yields fundamental solutions \(\Gamma_\varepsilon\) for operators with suitably regularised functions \(a^\varepsilon_{ij}\) which approximate the \(a_{ij}\) uniformly on compact sets. The desired \(\Gamma\) is then obtained by means of a limiting procedure. After giving appropriate bounds for \(\Gamma\), a fundamental solution is obtained for the elliptic operator corresponding to \(H\) by integrating \(\Gamma\) over the time variable.

MSC:
35A08 Fundamental solutions to PDEs
35H20 Subelliptic equations
43A80 Analysis on other specific Lie groups
35A17 Parametrices in context of PDEs
35J70 Degenerate elliptic equations
Keywords:
Gaussian bounds
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[1] Georgios K. Alexopoulos, Sub-Laplacians with drift on Lie groups of polynomial volume growth, Mem. Amer. Math. Soc. 155 (2002), no. 739, x+101. · Zbl 0994.22006
[2] A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Uniform Gaussian estimates of the fundamental solutions for heat operators on Carnot groups, Adv. Differential Equations 7 (2002), 1153-1192. · Zbl 1036.35061
[3] A. Bonfiglioli and F. Uguzzoni, Families of diffeomorphic sub-Laplacians and free Carnot groups, to appear in Forum Math. · Zbl 1065.35102
[4] A. Bonfiglioli and F. Uguzzoni, A note on lifting of Carnot groups, preprint. · Zbl 1100.35029
[5] A. Bonfiglioli and F. Uguzzoni, Harnack inequality for non-divergence form operators on stratified groups, preprint. · Zbl 1113.35032
[6] Jean-Michel Bony, Principe du maximum, inégalite de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier (Grenoble) 19 (1969), no. fasc. 1, 277 – 304 xii (French, with English summary). · Zbl 0176.09703
[7] Marco Bramanti and Luca Brandolini, \?^{\?} estimates for nonvariational hypoelliptic operators with VMO coefficients, Trans. Amer. Math. Soc. 352 (2000), no. 2, 781 – 822. · Zbl 0935.35037
[8] M. Bramanti and L. Brandolini, \(L^p\) estimates for uniformly hypoelliptic operators with discontinuous coefficients on homogeneous groups, to appear in Rend. Sem. Mat. Univ. Politec. Torino. · Zbl 1072.35058
[9] Luca Capogna, Regularity of quasi-linear equations in the Heisenberg group, Comm. Pure Appl. Math. 50 (1997), no. 9, 867 – 889. , https://doi.org/10.1002/(SICI)1097-0312(199709)50:93.0.CO;2-3 · Zbl 0886.22006
[10] Luca Capogna, Regularity for quasilinear equations and 1-quasiconformal maps in Carnot groups, Math. Ann. 313 (1999), no. 2, 263 – 295. · Zbl 0927.35024
[11] Luca Capogna, Donatella Danielli, and Nicola Garofalo, An embedding theorem and the Harnack inequality for nonlinear subelliptic equations, Comm. Partial Differential Equations 18 (1993), no. 9-10, 1765 – 1794. · Zbl 0802.35024
[12] 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. · Zbl 0878.35020
[13] L. Capogna and Q. Han, Pointwise Schauder estimates for second order linear equations in Carnot groups, preprint. · Zbl 1330.35056
[14] 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. · Zbl 0795.35018
[15] G. Citti, E. Lanconelli, and A. Montanari, Smoothness of Lipschitz continuous graphs with nonvanishing Levi curvature, Acta Math. 188 (2002), 87-128. · Zbl 1030.35084
[16] G. Citti, M. Manfredini, and A. Sarti, A note on the Mumford-Shah functional in Heisenberg space, preprint. · Zbl 1058.49010
[17] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161 – 207. · Zbl 0312.35026
[18] 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). · Zbl 0841.46018
[19] Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. · Zbl 0092.31002
[20] Piotr Hajłasz and Pekka Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101. · Zbl 0954.46022
[21] Lars Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147 – 171. · Zbl 0156.10701
[22] 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. · Zbl 0956.32034
[23] David Jerison and John M. Lee, The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), no. 2, 167 – 197. · Zbl 0661.32026
[24] 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. · Zbl 0639.58026
[25] Hubert Kalf, On E. E. Levi’s method of constructing a fundamental solution for second-order elliptic equations, Rend. Circ. Mat. Palermo (2) 41 (1992), no. 2, 251 – 294 (English, with French summary). · Zbl 0818.35022
[26] S. Kusuoka and D. Stroock, Applications of the Malliavin calculus. III, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 2, 391 – 442. · Zbl 0633.60078
[27] 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. · Zbl 0699.35025
[28] E. Lanconelli, Non-linear equations on Carnot groups and CR-curvature problems, Proceeding of the Conference “Renato Caccioppoli and Modern Analysis”, to appear in Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.
[29] E. Lanconelli, A. Pascucci, and S. Polidoro, 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. · Zbl 1032.35114
[30] Ermanno Lanconelli and Alessia Elisabetta Kogoj, \?-elliptic operators and \?-control distances, Ricerche Mat. 49 (2000), no. suppl., 223 – 243. Contributions in honor of the memory of Ennio De Giorgi (Italian). · Zbl 1029.35102
[31] Guozhen Lu, Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications, Rev. Mat. Iberoamericana 8 (1992), no. 3, 367 – 439. · Zbl 0804.35015
[32] 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. · Zbl 0798.35002
[33] Guozhen Lu, On Harnack’s inequality for a class of strongly degenerate Schrödinger operators formed by vector fields, Differential Integral Equations 7 (1994), no. 1, 73 – 100. · Zbl 0827.35032
[34] 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. · Zbl 1019.35056
[35] Carlo Miranda, Partial differential equations of elliptic type, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 2, Springer-Verlag, New York-Berlin, 1970. Second revised edition. Translated from the Italian by Zane C. Motteler.
[36] Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. · Zbl 1044.53022
[37] 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).
[38] Sergio Polidoro, On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type, Matematiche (Catania) 49 (1994), no. 1, 53 – 105 (1995). · Zbl 0845.35059
[39] Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247 – 320. · Zbl 0346.35030
[40] Zbigniew Slodkowski and Giuseppe Tomassini, Weak solutions for the Levi equation and envelope of holomorphy, J. Funct. Anal. 101 (1991), no. 2, 392 – 407. · Zbl 0744.35015
[41] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. · Zbl 0821.42001
[42] F. Uguzzoni, A note on a generalized form of the Laplacian and of sub-Laplacians, to appear in Arch. Math. (Basel). · Zbl 1083.31006
[43] 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. · Zbl 0813.22003
[44] Chao Jiang Xu, Regularity for quasilinear second-order subelliptic equations, Comm. Pure Appl. Math. 45 (1992), no. 1, 77 – 96. · Zbl 0827.35023
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