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3) 5 / \ Lc„-uJ,( 1 - 1 )'(rc 0). (*) = e 6 * . . +„„,_, +i(x) = eixU,.. ,tpn{x) = t^x«*-\ ixU nm e x -\ Ai = A10, A2 = A n , . •• > A ni = A i > n , _ i , . . , An = A m n 1. APPLICATIONS TO PSEUDO-DIFFERENTIAL EQUATIONS 57 Clearly, Vn(Xo) Vi(zo) ■ det L^i '(x0) . ^'(xo) to. 3) has a unique solution ( A i , . . , A n ). 5. Boundary-value problems. 1) =

Denote by / the inverse Fourier transform of / . Then f(() is an analytic functional and supp/ C Br for some r £ (0, R). We shall show that in W£°°(IR") there exists the limit Hm Ln(D)f(x), aaDa. Ln(D) = £ For this purpose we define the analytic functional g(£) 6 A'[5 r ] by the formula 9(0 = £ ( £ ) / « ) . 8 u p p / c B r , r

S)"ds = £ fcj 2l*at)wUj^ S ds ? 3) where (^<^(x) = dk 0, as required. p( a 'jy^)> PD-OPERATORS WITH REAL ANALYTIC SYMBOLS 39 The computations for t < 0 are completely analogous. The fact that this integral takes the value tp{x) for t = 0 can be proved by classical arguments usually applied in the study of the Poisson integral.

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A course in mathematical analysis. - part.2 Differential equations by Goursat E.


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