A Note on Harnack Inequalities and Propagation Sets for a Class of Hypoelliptic Operators
2010 (English)In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 33, no 4, 341-354 p.Article in journal (Refereed) Published
In this paper we are concerned with Harnack inequalities for non-negative solutions u:Ω→ℝ to a class of second order hypoelliptic ultraparabolic partial differential equations in the form Lu:=∑j=1mX2ju+X0u−∂tu=0 where Ω is any open subset of ℝN + 1, and the vector fields X1, ..., Xm and X0−∂t are invariant with respect to a suitable homogeneous Lie group. Our main goal is the following result: for any fixed (x0, t0) ∈ Ω we give a geometric sufficient condition on the compact sets K⊆Ω for which the Harnack inequality supK u≤CK u(x0, t0) holds for all non-negative solutions u to the equation Lu=0. We also compare our result with an abstract Harnack inequality from potential theory.
Place, publisher, year, edition, pages
Springer Netherlands, 2010. Vol. 33, no 4, 341-354 p.
Harnack inequality, Hypoelliptic operators, Potential theory
IdentifiersURN: urn:nbn:se:umu:diva-109733DOI: 10.1007/s11118-010-9172-2ISI: 000282186900002OAI: oai:DiVA.org:umu-109733DiVA: diva2:859033