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Fundamental solutions for non-divergence form operators on stratified groups

Authors: Andrea Bonfiglioli, Ermanno Lanconelli and Francesco Uguzzoni
Journal: Trans. Amer. Math. Soc. 356 (2004), 2709-2737
MSC (2000): Primary 35A08, 35H20, 43A80; Secondary 35A17, 35J70
Published electronically: October 21, 2003
MathSciNet review: 2052194
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Abstract: We construct the fundamental solutions $\Gamma$ and $\gamma$for the non-divergence form operators $\,{\textstyle\sum_{i,\,j}\,} a_{i,\,j}(x,t)\,X_iX_j\,-\,\partial_t\,$ and ${\,\textstyle\sum_{i,\,j}}\,a_{i,\,j}(x)\,X_iX_j$, 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 also provide Gaussian estimates of $\Gamma$ and its derivatives and some results for the relevant Cauchy problem. Suitable long-time estimates of $\Gamma$ allow us to construct $\gamma$ using both $t$-saturation and approximation arguments.

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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

Ermanno Lanconelli
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

Keywords: Non-divergence sub-elliptic operators, stratified groups, fundamental solutions
Received by editor(s): November 21, 2002
Published electronically: October 21, 2003
Additional Notes: Investigation supported by University of Bologna, Funds for selected research topics
Article copyright: © Copyright 2003 American Mathematical Society

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