We consider operators sum of squareo vector fields in a bounded domain where the vector fields are nonsmooth Hormander's vector fields of step r such that the highest order commutators are only holder 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 holder continuous f. We also prove C^{2,alpha}_loc estimates on this solution.

Bramanti, M., Brandolini, L., Manfredini, M., Pedroni, M. (2017). Fundamental solutions and local solvability for nonsmooth Hormander's operators. MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY, 249(1182), 1-92 [10.1090/memo/1182].

Fundamental solutions and local solvability for nonsmooth Hormander's operators

Manfredini, Maria;
2017

Abstract

We consider operators sum of squareo vector fields in a bounded domain where the vector fields are nonsmooth Hormander's vector fields of step r such that the highest order commutators are only holder 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 holder continuous f. We also prove C^{2,alpha}_loc estimates on this solution.
2017
Bramanti, M., Brandolini, L., Manfredini, M., Pedroni, M. (2017). Fundamental solutions and local solvability for nonsmooth Hormander's operators. MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY, 249(1182), 1-92 [10.1090/memo/1182].
Bramanti, Marco; Brandolini, Luca; Manfredini, Maria; Pedroni, Marco
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/615689
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