By Michael A. Arbib

This booklet makes an attempt to accumulate enough standpoint on class idea with no tough extra of the reader than a easy wisdom of units and matrix thought.

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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.