A priori estimates for nonvariational operators modeled on Hörmander's vector fields with drift

Authors

  • Marco Bramanti

DOI:

https://doi.org/10.6092/issn.2240-2829/4155

Keywords:

Hormander's vector fields, Schauder estimates, Lp estimates, drift

Abstract

For a nonvariational operator structured on Hörmander's vector fields with
drift, where the matrix of coffiecients is real, symmetric and uniformly positive, we prove local a priori estimates on the second order derivatives with respect to the vector fields, in Hölder spaces if the coecients are Holder continuous, in Lp spaces if the coefficients are bounded, measurable and locally VMO.

References

M. Bramanti: Singular integrals in nonhomogeneous spaces: L2and Lpcontinuity from Hölder estimates. Revista Matematica Iberoamericana 26 (2010), no. 1, 347{366.

M. Bramanti, L. Brandolini: Lp-estimates for uniformly hypoelliptic operators with discontinuous coefficients on homogeneous groups. Rend. Sem. Mat. dell'Univ. e del Politec. di Torino. Vol. 58, 4

(2000), 389-433.

M. Bramanti, L. Brandolini: Lp-estimates for nonvariational hypoelliptic operators with VMO coefficients. Trans. Amer. Math. Soc. 352 (2000), no. 2, 781-822.

M. Bramanti-L. Brandolini: Estimates of BMO type for singular integrals on spaces of homogeneous type and applications to hypoelliptic PDES. Revista Matematica Iberoamericana, 21 (2005), no. 2,

{556.

M. Bramanti, L. Brandolini: Schauder estimates for parabolic nondivergence operator of Hormander type. J. Differential Equations 234 (2007) 177-245.

M. Bramanti, L. Brandolini, M. Pedroni: On the lifting and approximation theorem for nonsmooth vector fields. Indiana University Mathematics Journal. Issue 6 Volume 59 (2010), 1889{1934.

M. Bramanti, M. C. Cerutti, M. Manfredini: Lp-estimates for some ultraparabolic operators with discontinuous coefficients. Journal of Math. Anal. and Appl., 200, 332-354 (1996).

M. Bramanti, G. Cupini, E. Lanconelli, E. Priola: Global Lp-estimates for degenerate Ornstein-Uhlenbeck operators with variable coffiecients. Mathematische Nachrichten, 1-15 (2013)/ DOI 10.1002/mana.201200189.

M. Bramanti, M. Zhu: Lpand Schauder estimates for nonvariational operators structured on Hörmander vector fields with drift. To appear on Analysis and Partial Dierential Equations. ArXiv:

5116v1 26 Mar 2011.

M. Bramanti, M. Zhu: Local real analysis in locally doubling spaces. Manuscripta Mathematica 138, 477{528 (2012).

S. Campanato: Proprietà di Holderianità di alcune classi di funzioni. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963) 175{188.

M. Christ: A T(b) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60/61 (1990), no. 2, 601{628.

M. Di Francesco, S. Polidoro: Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form. Adv. Dierential Equations 11 (2006), no. 11,

{1320.

G. B. Folland: Subelliptic estimates and function spaces on nilpotent Lie groups. Arkiv for Math. 13, (1975), 161-207.

B. Franchi, E. Lanconelli: Holder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 4,

{541.

C. E. Gutierrez, E. Lanconelli: Schauder estimates for sub-elliptic equations. J. Evol. Equ. 9 (2009), no. 4, 707{726.

L. Hormander: Hypoelliptic second order differential equations. Acta Math., 119 (1967), 147-171.

E. Lanconelli, S. Polidoro: On a class of hypoelliptic evolution operators. Partial differential equations, II (Turin, 1993). Rend. Sem. Mat. Univ. Politec. Torino 52 (1994), no. 1, 29{63.

A. Nagel, E. M. Stein, S. Wainger: Balls and metrics dened by vector fields I: Basic properties. Acta Math., 155 (1985), 130-147.

F. Nazarov, S. Treil, A. Volberg: The Tb-theorem on non-homogeneous spaces. Acta Math. 190 (2003), no. 2, 151{239.

L. P. Rothschild-E. M. Stein: Hypoelliptic differential operators and nilpotent groups. Acta Math., 137 (1976), 247-320.

X. Tolsa: A proof of the weak (1,1)-inequality for singular integrals with non doubling measures based on a Calderon-Zygmund decomposition. Publ. Mat. 45 (2001), no. 1, 163{174.

Downloads

Published

2013-12-28

How to Cite

Bramanti, M. (2013). A priori estimates for nonvariational operators modeled on Hörmander’s vector fields with drift. Bruno Pini Mathematical Analysis Seminar, 4(1), 15–37. https://doi.org/10.6092/issn.2240-2829/4155

Issue

Section

Articles