1.3. THE NAGEL-STEIN-WAINGER POLYNOMIAL AND THE SIZE OF THE CC BALLS 11 and therefore, if r 7r/2, we have B(0, r) = R. This global aspect is intimately con- nected to the powerful extension of the Theorem of Hopf-Rinow due to Cohn-Vossen [CV35], and we refer the reader to the forthcoming book [G02] for a detailed dis- cussion. In the present paper we are solely concerned with local questions. Thereby, in order to eliminate all the topological complications connected to the growth of the vector fields at infinity, we will henceforth make the following hypothesis: (1.16) The vector fields Xi,...,Xm have coefficients in Lip(Mn) . Such assumption will be in force throughout the paper in the purely Hormander case. It is instead unnecessary for Carnot groups since, in that framework, the compactness of balls holds irregardless of the radius, and of the growth of the Xj's at infinity, see [G02]. A basic consequence of (1.16) is the following result established in [GN98]. PROPOSITION 1.9. Under the hypothesis (1.16), for any x0 G Rn, and every r 0, the closed ball B(x0,r) is compact. Having squared the table of global aspects, we can now improve on Proposition 1.5, by replacing du(x,y) in the right-hand side, with the smaller quantity d(x,y). The price that we must pay is represented by the presence of a larger constant C. We stress that, as the above simple example shows, without (1.16) the proof of the next proposition would break down, since the boundedness of the set U would fail in general, see [G02]. PROPOSITION 1.10. Let U c l n be a bounded set. There exists a bounded set U, with U C U, such that for every x,y G U one has \x-y\ C d(x,y). Here, ( m max £l*j(*)l 2 zeu j=1 This gives for every x G U, and any r 0, (1.18) B(x,r/C) C Be(x,r). 1.3. The Nagel-Stein-Wainger polynomial and the size of the CC balls Let X {Xi,...,Xm} be a system of C°° vector fields in Rn, n 3, sat- isfying the finite rank condition (1.4), and denote by Yi,...,Y/ the collection of the Xj's and of those commutators which are needed to generate W1. A "degree" is assigned to each Yi, namely the corresponding order of the commutator. If / = (ii, ...,in), 1 ij /, is a n-tuple of integers, following [NSW84] one defines d(I) - E i = i degiYi ), and aj{x) = det (F2l, ...,y,J. DEFINITION 1.11. The Nagel-Stein- Wainger polynomial is defined by A(x,r) = Yl Mx)l rd{I) i r 0 ' j . / *
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