Abstract
This note describes the results of a joint research with L. Brandolini, M. Manfredini and M. Pedroni, contained in Bramanti et al. [Fundamental solutions and local solvability of nonsmooth Hörmander’s operators. Mem. Am. Math. Soc., in press. http://arxiv.org/abs/1305.3398], with some background. We consider operators of the form \(L =\sum _{ i=1}^{n}X_{i}^{2} + X_{0}\) in a bounded domain of \(\mathbb{R}^{p}\) (p ≥ n + 1) where X 0, X 1, …, X n are nonsmooth Hörmander’s vector fields of step r, such that the highest order commutators are only C 1, α. Applying Levi’s parametrix method we construct a local fundamental solution γ for L, provide growth estimates for γ and its first and second order derivatives with respect to the vector fields and deduce the local solvability of L in C X 2, β spaces (for any β < α).
Dedicated to Ermanno Lanconelli on the occasion of his 70th birthday
Mathematics Subject Classification: 35A08, 35A17, 35H20
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Bramanti, M. (2015). Local Solvability of Nonsmooth Hörmander’s Operators. In: Citti, G., Manfredini, M., Morbidelli, D., Polidoro, S., Uguzzoni, F. (eds) Geometric Methods in PDE’s. Springer INdAM Series, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-02666-4_8
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