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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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##### References:
 [1] Avantaggiati A., Boll.Un.Mat.Ital 13 pp 1– (1976) [2] Baouendi, M.S. 1967.Sur une classe d’opérateurs elliptiques dégénérés,Bull.Soc.MathVol. 95, 45–87. France [3] Bony J.M., Ann,Inst. Fourier(Grenoble) 19 pp 277– (1969) · Zbl 0176.09703 [4] Bugrov Ja.S., TrudyMat.Inst.Steklov 77 pp 45– (1965) [5] Edmunds D.E., J.London Math.Soc. 5 (2) pp 21– (1972) · Zbl 0234.35030 [6] Fabes E.B., Comm.Partial Differential Equations 7 (2) pp 77– (1982) · Zbl 0498.35042 [7] Fefferman C., Séminaire Goulaouic-Meyer-Schwartz 23 (2) (1981) [8] Fefferman C., Conference on Harmonic Analysis (Chicago 1981) pp 590– (1981) [9] Franchi B., ”Linear Partial and Pseudo Differential Operators” pp 105– (1982) [10] Gilbart D., Elliptic Partial Differential Equations of Second Order, (1968) [11] Hermann R., Differential Geometry and the Calculus of Variations (1968) · Zbl 0219.49023 [12] HÖrmander L., Acta Math 119 pp 147– (1967) · Zbl 0156.10701 [13] Kolodii I.M., Ukrain Mat.Z 7 pp 320– (1965) [14] Kruzkov S.N., Dokl.Akad.Nauk SSSR 150 pp 470– (1963) [15] Kusainova L.K., Dokl.Akad.Nauk.SSSR 263 pp 1050– (1982) [16] Moser J., Comm.Pure Appl.Math 13 pp 457– (1960) · Zbl 0111.09301 [17] Moser J., Comm.Pure Appl.Math pp 577– (1961) · Zbl 0111.09302 [18] Murthy M.K.V., Ann.Mat.Pura Appl. 80 (4) pp 1– (1968) · Zbl 0185.19201 [19] Mynbaev K.T., Izv.Akad.Nauk Kazah.SSR 5er.Fiz.Mat 100 (4) pp 77– (1981) [20] Nagel A., Proc.Nat.Acad.Sci. 78 pp 6596– (1981) [21] Rothschild L.P., Acta Math. 137 pp 247– (1976) · Zbl 0346.35030 [22] Rybalov Yu.V., Sibirsk,Mat J. 20 pp 610– (1979) [23] Sussmann H.J., Trans.Amer.Math.Soc. 180 pp 171– (1973) [24] Triebel H., Differential Operators, (1978) [25] Trudinger N.S., Arch.Rational Mech.Anal 42 pp 50– (1971) · Zbl 0218.35035 [26] Trudinger N.S., Ann,Scuola Norm.Sup.Pisa 27 (3) pp 265– (1973)
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