Schauder estimates for sub-elliptic equations |
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Authors: | Cristian E Gutiérrez Ermanno Lanconelli |
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Institution: | 1. Department of Mathematics, Temple University, Philadelphia, PA, 19122, USA 2. Dipartimento di Matematica, Universita di Bologna, Piazza Porta S. Donato, 5, Bologna, Italy
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Abstract: | We establish Hölder estimates of second derivatives for a class of sub-elliptic partial differential operators in ${\mathbb{R}^{N}}$ of the kind $$\mathcal L=\sum_{i,j=1}^{m}a_{ij}(x)X_{i}X_{j}+X_{0},$$ where the X j ’s are smooth vector fields in ${\mathbb{R}^{N}}$ , and a ij is a uniformly elliptic matrix. It is assumed that the X j ’s satisfy homogeneity conditions with respect to a group of dilations δ r which yield the existence of a composition law ${\circ}$ in ${\mathbb{R}^{N}}$ making the triplet ${\mathbb G=(\mathbb{R}^{N},\circ,\delta_{r})}$ an homogeneous Lie group on which the X j ’s are left translation invariant. The Hölder norms are defined in terms of this composition law. The main tools used are the Taylor formula for smooth functions on ${\mathbb{G}}$ , some properties of the corresponding Taylor polynomials, and an orthogonality theorem that extends to homogeneous Lie groups a classical theorem of Calderón and Zygmund in the Euclidean setting. |
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