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By Mahowald M., Priddy S. (eds.)

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By definition X- (0) = 0 . )>dVol 3Xpr- = 2 Re J F r* 5 \ — = 2Re J ( 2~ ~ I 1> r* + j dVol By evaluating this at ? = 0 we obtain: S2X. 2-(0) = 2 J* B ? F . dVol . We 50 Charles L. Epstein The other term vanishes as (L (0) = " /Vol F . (0) E 0 . 1 r Dpi (0) is ^ multiple of satisfies and set \L . ) ? ) and set ? = 0 we obtain: /voiH3/r* ^ A Hence - : s L ,3 ,-* — D-<)>-. (p,0) is a harmonic function which transforms correctly.

GTn) = e" i n § is a unitary character. ) is in H (F . , + || grad f || 2 ] dVol d§ < » , p € BF"^. ' -functions on hypersurfaces. H1 is clearly a complete Hilbert space. 4 U 12 3 extends to a unitary isomorphism of W ' (K IT) onto H (F ^ X [0,2*17)) . Furthermore U grad F = grad UF . Proof: If F € C * O H 3 / T ) then: (UF)(p,S) = F(T n p) 6 £ in? ) = « in? ) . Taking limits we obtain this formula for all oa 12 3 F € W ' (H /T) . ) Let F € C* (H3/T) ; the limit of 2 L -gradient for almost every 2TT J 0 Thus the restriction of for UF implies that P € SF" r* .

And note that "p** p** The continuity of U f(p) implies that P* J* (M^ *) r U*f(p)Y-VdS + + P* J* (ST U*f(p)Y-N*lS = 0 . n+1 F r # )- P* Spectral Theory of 3-Manifolds 23 We obtain: Jo 3 dVol = J\ 3 n /r n /r U*f(p)div Y dVol , * 2 ^ thus X(p) is the weak gradient of U f (p) as a function in L (H /T) . We have already shown that X(p) is square integrable thus U f is in W 192 3 (H /T) . Taking limits, we obtain the lemma. Note that "k "k ' U grad f = grad U f holds for any Remark: Since f € H U X .

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