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Harnack inequality for non-divergence form operators on stratified groups

Authors: Andrea Bonfiglioli and Francesco Uguzzoni
Journal: Trans. Amer. Math. Soc. 359 (2007), 2463-2481
MSC (2000): Primary 35B05, 35A08, 43A80; Secondary 35H20, 35J70
Published electronically: January 19, 2007
MathSciNet review: 2286040
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Abstract: We prove lower bounds for the fundamental solutions of the non-divergence form operators

$\displaystyle {\textstyle\sum_{i,j}} a_{i,j}(x,t)\,X_iX_j-\partial_t$   and$\displaystyle \quad {\textstyle\sum_{i,j}}a_{i,j}(x)\,X_iX_j,$

where the $ X_i$'s are Hörmander vector fields generating a stratified group $ \mathbb{G}$ and $ (a_{i,j})_{i,j}$ is a positive-definite matrix with Hölder continuous entries. We then prove an invariant Harnack inequality for such operators. As a byproduct we also study some relevant properties of the Green functions on bounded domains.

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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

Francesco Uguzzoni
Affiliation: Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5 - 40126 Bologna, Italy

Received by editor(s): September 26, 2003
Published electronically: January 19, 2007
Additional Notes: This work was supported by the University of Bologna, Funds for selected research topics.
Article copyright: © Copyright 2007 American Mathematical Society

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