By Edwin H. Spanier

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9 Nilpotent groups 47 with Ak = dLjJ(ad acting on the spaces Lp(CJ). 12. 6. The dilations Yu on 9 have the property yu(ad = uak for all u > 0 and k E {l, ... , d'}. Thus if r u denotes the corresponding dilations on G, then (H({J) for all u > 0 and ({J E 0 r;1 = u 2H«({J 0 r;l) C;:o(G). Moreover, - - Sr({J = (Su2t«({J 0 ru-I » 0 ru for all u > O. 28). In particular, if u = t -I /2, then (-» I K- r (-) g = t -V/2 K- 1(rrl / 2 g for all t > 0 and g E G. Hence the kernel bounds for large t bounds can ~e deduced from the bounds for t = 1.

In the real symmetric case the first step is straightforward. Since II K, 1100 = II S, III ~oo it is equivalent to establishing an appropriate bound on the crossnorm on the semigroup. , Nash or Sobolev inequalities. The calculation is simplified because the semigroup is contractive on L oo , or on Ll . 2. The derivation of uniform bounds for the kernel of complex operators is considerably more complicated. The perturbation procedure used in the second step, often called the Davies perturbation, is quite general.

The major difference is that the derivatives have an exponential decrease in the compact case. Similar results can be established for the direct product, or local direct product, of a compact group and a nilpotent group. More general results will, however, be derived in the analysis of groups of polynomial growth in Chapters V and VI. 1 if the group G is nilpotent but not homogeneous. Then K t can be asymptotically approximated by a semi group kernel on a related homogeneous group. 27) used to introduce the local approximant Go of the group G.

### Algebraic Topology by Edwin H. Spanier

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