By Nicholas T. Varopoulos, L. Saloff-Coste, T. Coulhon

The geometry and research that's mentioned during this booklet extends to classical effects for normal discrete or Lie teams, and the equipment used are analytical, yet aren't excited by what's defined nowadays as genuine research. many of the effects defined during this ebook have a twin formula: they've got a "discrete model" on the topic of a finitely generated discrete crew and a continual model regarding a Lie staff. The authors selected to heart this booklet round Lie teams, yet may well simply have driven it in numerous different instructions because it interacts with the speculation of moment order partial differential operators, and likelihood thought, in addition to with crew conception.

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It turned out that each of the two sides of the equivalence Rn Dn (cf. 1) had at least one ancestor: Yoshikawa ([158]) had essentially proved that Rn = Dn and Benilan ([14]), that Dn = Rn; see also [75]. 3 (c). For their part, Davies and Simon have studied in [44] the Ll - L°° regularization properties of the Schrodinger semigroup et(A+v) by means of logarithmic Sobolev inequalities of Gross [58]. Their theory has been generalized since, and is set forth in [42]. 5 axe due to T. 2, to T. Coulhon and Y.

1 is due to Chow ([22]). For more details on distances associated to vector fields, see [94]. CHAPTER IV THE HEAT KERNEL ON NILPOTENT LIE GROUPS At the end of this chapter, we shall know almost everything about the heat kernel and the Sobolev inequalities associated to a family of Hormander fields on a nilpotent Lie group. Later we shall obtain related results in much wider settings. Local questions will be studied on manifolds in Chapter V. Global questions will be studied on unimodular groups (Chapters VI, VII and VIII).

For simplicity, assume that a > 0. Consider the minimal heat kernel ho associated with D on S2, with Dirichlet boundary conditions. There are several ways to construct it: probability theory, potential theory or semigroup theory (see References and Comments). Let u be a positive solution of Du = 0 on I x Q. By minimality of hn, we have u(t2, x) ? fv ht -t(x, y)u(t, y) dy, Vt E]0, t2[ . It follows that u(t2, x) fK ht, -t(x, y)u(t, y) dy > f hi2_t(x, y)u(t, y) dt dy, 1 82 - 81 Kx[s1,s2] for 0 < sl < s2 < ti.