Fundamental solutions and geometry of the sum of squares of vector fields
type: Item
instance of: scholarly article
title: Fundamental solutions and geometry of the sum of squares of vector fields
zbMATH Open document ID: 0582.58004
DOI: 10.1007/BF01388721
author: Antonio Sánchez-Calle
published in: Inventiones Mathematicae
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143171
review text: Let \(X_ 1,...,X_ m\) be smooth vector fields on a smooth compact manifold M, endowed with a smooth positive measure \(\mu\). It is assumed that taking a sufficient number of commutators of \(X_ 1,...,X_ m\) they span the tangent of M at every point (Hörmander's condition). Then the operators like \(L=\sum^{m}_{j=1}X^ 2_ j+\sum^{m}_{i,j=1}f_{ij}X_ i,X_ j+\sum^{m}_{j=1}f_ jX_ j+f_ 0\) with smooth functions \(f_{ij}\), \(f_ j\), \(f_{ij}\) reals, are hypoelliptic and the solutions of \(Lu=f\) can be represented in the form \(u(x)=\int G(x,y)f(y)d\mu (y)\). In this paper estimates for the Green kernel G in terms of a distance \(d_ L\), canonically attached to L are found. Also an estimate for the fundamental solution K of \(\partial /\partial t-L\) is found.
Mathematics Subject Classification ID: 58A30
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 58J99
zbMATH DE Number: 3932035
zbMATH Keywords: hypoelliptic operator
zbMATH Keywords: Green kernel
zbMATH Keywords: vector fields
reviewed by: Stancho Dimiev
MaRDI profile type: Publication
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OpenAlex ID: W1567336650
description: scientific article; zbMATH DE number 3932035