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Fundamental solutions and local solvability for nonsmooth Hörmander’s operators


About this Title

Marco Bramanti, Luca Brandolini, Maria Manfredini and Marco Pedroni

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.

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Some known results about nonsmooth Hörmander’s vector fields ec known results
  • Chapter 3. Geometric estimatesec geometric
  • Chapter 4. The parametrix methodec parametrix
  • Chapter 5. Further regularity of the fundamental solution and local solvability of $L$
  • Chapter 6. Appendix. Examples of nonsmooth Hörmander’s operators satisfying assumptions A or B

Abstract


We consider operators of the form in a bounded domain of where are nonsmooth Hörmander's vector fields of step such that the highest order commutators are only Hölder continuous. Applying Levi's parametrix method we construct a local fundamental solution for and provide growth estimates for and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients we prove that also possesses second derivatives, and we deduce the local solvability of , constructing, by means of , a solution to with Hölder continuous . We also prove estimates on this solution.

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