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By Maxime Bocher

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5 the definition of does not depend on the paratlas on F. Indeed, ticular choice of the could equivalently be defined as the linear space of functions n = uI r with is c u=0 on 8S2\P. r = 0). H). We leave the details to the reader. 3. Lebesgue Spaces We assume that the reader is familiar with the basic theory of Lebesgue measure and integral, such as can be found, for instance in W. Rudin [133]. a. , respectively, for "almost any" and "almost 17 Function Spaces everywhere" with respect to Lebesgue measure measx on RN.

2, u> = _ _ (e fu)(x) for x e V. 9 with co D supp u: we obtain Da(oa u)I,, -. Daula in 11(w) for I a I < k, hence e * it -. v(RN). Note that for n large enough ea a it e Cp (RN) because supp u is compact. 27 below). P(Q) [or even it e Hj' (Q) only] and w cc Q. 9 yields (ea * u)I0 -. ula, in Hk,n(ru). Chapter 1 48 If supp u is a compact subset of S2 we are of course in the same situa- tion as in the proof of Lemma 1:24, so that per s u u in Hk,p(Q), p * u being an element of Cp(Q) for n large enough.

Then there exists fe LP'(S2), with I f Ip':o = I F I(e°cmr', such that = fn fv dx for v e LP(S2). 10). Finally we have the following theorem. S. C°(S2) is dense in L9(Q) for 1

r; Chapter 1 20 the definition of p * u is trivially extended to the whole of S5 if dist(supp u, OQ) > 2r.

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