# Download A course in mathematical analysis. - Derivatives and by Goursat E. PDF

By Goursat E.

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Arguing exactly in the ~ame way as in Step B of the Ist proof of Lemma 7 we conclude that it is sufficient to consider functions of this subclass. 'E) (~-z)Jg(~)~-zJd~d~, D(z;~) =D\AS(z). 2) IIS*gllps ½~Apllgl~, where Ap is the same as in Lemma 6. 3) g(z) =0 for z ~D and changing the variables of integration according to the formulae ~=z+re i(t-~), -g

Hence, ~iven ~, there i_gs~ sequence (~n) of C1-functions which aDproximates ~ in the above sense. S t e p ~. 1) with ~ replaced by ~n and = i d has an LP-solution ~=gn(Z) with spt g n C A r, where Ip-21 <~, and since II~nll~ ~ q for all n, we may take s independent of n. 2) with ~ replaced by ~n" We claim that there is an LP-func tion g with spt g c A r such that Ilg - gnI~--~0. 1) with ~ replaced by ~n' ~ = g n (z)' and @ = i d , and by Lemma 8, we have IlgnIIp < II~nll~II sI~II gnIIp + II~nllp ~ qll SIIpllgnI~ + q(~r2) ~/p, llSllp<+ ~, since tions plies II~nll~ q and spt~ n c A r .

Thus f is a sense-preserving homeomorphism. Therefore, by 49 11. 1) holds. Suppose now that f is a sense-preserving homeomorphism which possesses distributional while D is, as before, L2-derivatives f~ and of finite connectivity. e. z for which there exist partial derivatives fz(Z), f~(z) we have fn~(Z)/fnz(Z)--~f~(z)/fz(Z), as desired. The case of infinitely connected D can be treated exactly in the same way as the preceding case provided we take into account the remark after the proof of Theorem 3.