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An embedding theorem for Sobolev spaces related to non-smooth vector fields and Harnack inequality. (English) Zbl 0589.46023
In the first part of this paper, differentiability properties are studied for functions belonging to the space generated by the vector fields \(X_ j=\lambda_ j\partial /\partial x_ j\), \(1\leq j\leq n\), acting on test functions on a connected open subset \(\Omega\) of \({\mathbb{R}}^ n\). The \(\lambda_ j\) are continuous and non-negative. In particular, under suitable geometrical hypotheses involving the integral curves of the \(\pm X_ j\), estimates are obtained which show that \(\overset\circ W^ p_{\lambda}(\Omega)\subset \overset\circ W^{p,\epsilon}(\Omega)\) for some \(\epsilon >0\). Here \(\lambda =(\lambda_ 1,...,\lambda_ n)\) and \(\overset\circ W^ p_{\lambda}(\Omega)\) is the completion of \(C_ 0^{\infty}(\Omega)\) with respect to the norm \(\| u\|_ p+\sum^{n}_{1}\| X_ ju\|_ p\), \(\| u\|_ p\) being the \(L^ p(\Omega)\) norm, \(1<p<\infty\); and \(\overset\circ W^{p,\epsilon}(\Omega)\) is the Sobolev space of order \(\epsilon\).
In the second part of the paper, a Harnack inequality is proved for solutions of a class of second order linear degenerate elliptic equations with measurable coefficients defined on \(\Omega\). In addition to the previous conditions, it is assumed that the quadratic form \(\sum^{n}_{1}\lambda_ j(x)^ 2\xi^ 2_ j\) is dominated by the quadratic form of the differential equation.
Reviewer: A.Pryde

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J70 Degenerate elliptic equations
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