Vol. 85, No. 1 DUKE MATHEMATICAL JOURNAL (C) October 1996 SOBOLEV INEQUALITIES AND MYERS’S DIAMETER THEOREM FOR AN ABSTRACT MARKOV GENERATOR D. BAKRY AND M. LEDOUX 1. Introduction. The classical theorem of (Bonnet-) Myers on the diameter [M] (see [C], [GHL]) states that if (M, g) is a complete, connected Riemannian manifold of dimension n (>2) such that Ric > (n-1)g, then its diameter D D(M) is less than or equal to r (and, in particular, M is compact). Equiv- alently, after a change of scale, if Ric > R# with R > 0, and if S n is the sphere of dimension n and constant curvature R (n- 1)/r 2 where r > 0 is the radius of sn, then the diameter of M is less than or equal to the diameter of S, that is, (1.1) D < rcr n R The aim of this work is to prove an analogue of Myers’s theorem for an abstract Markov generator and to provide at the same time a new analytic proof of this result based on Sobolev inequalities. In particular, we will show how to get exact bounds on the diameter in terms of the Sobolev constant. As an intro- duction, let us describe the framework, referring to [B2] for further details. On some probability space (E, d*, #), let L be a Markov generator associated to some semigroup (Pt)t >o continuous in L2(/). We will assume that L is invariant and symmetric with respect to/, as well as ergodic. We assume furthermore that we are given a nice algebra of bounded functions on E, containing the constants and stable by L and Pt (though this last hypothesis is not really needed but is used here for convenience) and by the action of C functions. We may then introduce, following P.-A. Meyer, the so-called carr du champ operator F as the symmetric bilinear operator on x defined by 2r(f, o) L(/0) -fL9 9Lf, as well as the iterated carr6 du champ operator F2 2FE(f, g) LF(f, g) F(f, Lg) F(g, Lf), f,g e . Finally, we assume that L is a diffusion: for every C function W on IRk, and every finite family F (f 1,..., fk) in , LV(F) VV(F). LF + VVV(F). F(F, F). Received 4 November 1994. 253