Geometric Analysis and PDEs
Cetraro, June 11-76, 2007

Heat Kernels in Sub-Riemannian Settings

Prof. Ermanno Lanconelli, Università di Bologna, Italia


Abstract


(I) Heat Kernels, Gaussian bounds and Harnack inequalities in Riemannian settings: a brief survey.

(II) A motivation for studying Heat-type operators in sub-Riemannian settings.
a.) Real hypersurfaces in complex Euclidean spaces. Normalized Levi form.
     Levi-curvatures.
b.) Structure of the Levi-curvature operators (LC operators).
c.) Motion by Levi-curvatures and related evolution operators ( MLC operators).
d.) Structure of the LMC operators: Heat-type operators modeled on nonlinear
     step-two vector fields.
(III) Heat-type operators modeled on Hormander's vector fields.
a.) Heat kernel: existence and Gaussian bounds. The Levi parametrix method
     in sub-Riemannian settings.
b.) Harnack inequality: the Fabes and Strook implementation of the
     "old idea by Nash".
c.) Some application to LC and MLC equations.
d.) Open problems.

Main References

[1] L. Saloff-Coste, Aspects of Sobolev-type inequalities, London Mathematical Society, Lectures Note Series, 289 Cambridge University Press (2002)

[2] A.Montanari-E.Lanconelli, Pseudoconvex Fully Nonlinear Partial Differential Operators. Strong Comparison Theorems, J. Differential Equations 202 (2004) 306-331

[3] M. Bramanti-L.Brandolini-E.Lanconelli-F.Uguzzoni, Heat kernel for non-divergence operators of Hormander type, C.R. Acad. Sci. Paris, Mathematique, 343(2006) 463-466.

[4] M. Bramanti-L.Brandolini-E.Lanconelli-F.Uguzzoni, Non-divergence equations structured on H\"{o}rmander vector fields: heat kernels and Harnack inequalities, preprint.

No specific background knowledge is required. However, to better follow the lectures some familiarity with the following topics would be helpful:

a.) Elementary notions from the theory of functions of several complex variables, as presented e.g. in the first Chapter of: R.M.Range, Holomorphic Functions and Integral Representation in Several Complex Variables, Graduate Text in Mathematics, Springer (1998)
b.) Vector fields in Rn and Carnot-Carathéodory metric spaces, as presented.e.g in P.Hajlasz - P.Koskela, Sobolev met Poincaré, Memoirs of the Amer. Math. Soc. 688 (2000)