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Fundamental solutions and local solvability for nonsmooth Hörmander’s operators
About this Title
Marco Bramanti, Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy, Luca Brandolini, Dipartimento di Ingegneria Gestionale, dell’Informazione e della Produzione, Università di Bergamo, Viale Marconi 5, 24044 Dalmine, Italy, Maria Manfredini, Dipartimento di Matematica, Università di Bologna, Piazza Porta S. Donato 5, 40126 Bologna, Italy and Marco Pedroni, Dipartimento di Ingegneria Gestionale, dell’Informazione e della Produzione, Università di Bergamo, Viale Marconi 5, 24044 Dalmine, Italy
Publication: Memoirs of the American Mathematical Society
Publication Year:
2017; Volume 249, Number 1182
ISBNs: 978-1-4704-2559-3 (print); 978-1-4704-4131-9 (online)
DOI: https://doi.org/10.1090/memo/1182
Published electronically: August 8, 2017
Keywords: Nonsmooth Hörmander’s vector fields,
fundamental solution,
solvability,
Hölder estimates.
MSC: Primary 35A08, 35A17, 35H20
Table of Contents
Chapters
- 1. Introduction
- 2. Some known results about nonsmooth Hörmander’s vector fields
- 3. Geometric estimates
- 4. The parametrix method
- 5. Further regularity of the fundamental solution and local solvability of $L$
- 6. Appendix. Examples of nonsmooth Hörmander’s operators satisfying assumptions A or B
Abstract
We consider operators of the form $L=\sum _{i=1}^{n}X_{i}^{2}+X_{0}$ in a bounded domain of $\mathbb {R}^{p}$ where $X_{0},X_{1},\ldots ,X_{n}$ are nonsmooth Hörmander’s vector fields of step $r$ such that the highest order commutators are only Hölder continuous. Applying Levi’s parametrix method we construct a local fundamental solution $\gamma$ for $L$ and provide growth estimates for $\gamma$ and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients we prove that $\gamma$ also possesses second derivatives, and we deduce the local solvability of $L$, constructing, by means of $\gamma$, a solution to $Lu=f$ with Hölder continuous $f$. We also prove $C_{X,loc}^{2,\alpha }$ estimates on this solution.- Andrea Bonfiglioli, Ermanno Lanconelli, and Francesco Uguzzoni, Fundamental solutions for non-divergence form operators on stratified groups, Trans. Amer. Math. Soc. 356 (2004), no. 7, 2709–2737. MR 2052194, DOI 10.1090/S0002-9947-03-03332-4
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