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

The geometry and research that's mentioned during this booklet extends to classical effects for basic discrete or Lie teams, and the tools used are analytical, yet usually are not fascinated about what's defined nowadays as genuine research. many of the effects defined during this publication have a twin formula: they've got a "discrete model" concerning a finitely generated discrete team and a continual model relating to a Lie workforce. The authors selected to middle this booklet round Lie teams, yet may perhaps simply have driven it in numerous different instructions because it interacts with the idea of moment order partial differential operators, and likelihood concept, in addition to with workforce conception.

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2. 3 Remarks (a) The theorem holds more generally for symmetric semigroups that are equicontinuous on L' and L. 22 II Dimensional inequalities for semigroups of operators (b) The equivalence between (ii) and (iii) holds for any n > 0. (c) The implication (i) (ii) is a simple consequence of Holder's inequality. By contrast, we do not know how to prove (ii) = (i) without considering the semigroup. We are now going to study the stability of property Rn under perturbations of symmetric submarkovian semigroups.

3 Harnack inequalities for nilpotent Lie groups 45 which may be extended to all CG by Ot (X) = tiX if X E Vi. The maps Ot = exp oqt o exp-1: G -* G satisfy: `dt,s>0, Oto0s=1ts, 01=IdG, Vt > 0,V(g,9) E G2, `dt > 0, Ot(g9) _ 4t(g)0t(g'), dOt = Ot In other words, G admits a group of dilations which is adapted to its structure. Conversely, we can easily show that if a simply connected Lie group G admits a dilation group {Ot I t > 0} which is adapted to its structure, and if the space VI of the elements of Lc that are homogeneous of degree one with respect to Ot generates £G, then G is nilpotent, in fact stratified, with VI as first slice.

3 yields a constant C such that every positive solution u of Du = 0 in I x Q satisfies sup XEK I a I' ataxI,u(ti,x)I C f ]t,-E,tl+E[xn (1) at Moreover, since g* is the Green function associated with D, we have that is D7g('Y, -S£, V E W. , Z;) is a positive solution of Dv = 0 on Wl {e}. 2, it would vanish on (] - oo, t2 [ x V) fl W, an open subset containing . , and could not satisfy Dryg(y, Z;) = -6 . Finally inf{g(y, l;) I y E {t2} x K, Z; E [t1 - e, t1 + E] x S21 = C-1 > 0 and sup xEK \ at 01 ax (--) J u(t1i x)I C2 f]tl-E,tl+E[X$Z 9(Y, d d =C2(Cu)(y) < C2u(y), and this holds for all y E {t1} x K.