Fourier Integral Operators and Partial Differential by Louis Boutet de Monvel (auth.), Prof. Jacques Chazarain

By Louis Boutet de Monvel (auth.), Prof. Jacques Chazarain (eds.)

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Additional resources for Fourier Integral Operators and Partial Differential Equations: Colloque International, Université de Nice, 1974

Example text

Hand s i d e r e d u c e s to L W(S, O) where T = ~Ho(S, for some B ~ C O O . 8 ) ) ) W(S, 6) -- 8 z)l~ z D s + B, By hypothesis z = ~o'(S), the coefficient of D S has no zero in I. 21) for some R vanishing of infinite order when 8 = 0 $7 is equivalent to a system of equations W(S, O) = O, . . , L W(J)(S, O) +Ej - O, . . t W (j-l). These can be solved successively, and W(S, O) can be chosen with no zero in I. Borel there exists a C ~ function W(S, 8) with these derivatives when 8 ~ O, which proves the first part of the lemma.

12 in a fixed interval we must now introduce a new variable S through s = 6 + S 6 ~+I. 22),, (x(~)(a(6~)P-b(5~)a)(~(8~) +6-~-~Ds~) + + C(S, 6) b(6 f) Q ( ~ ( 6 f) + 6 - ~ f - I D s N ) ) V6(S ) - O. 59 Here C(S, 8) -K(6~)(I-A(sF))/A(8~)--~-~JoS , 6--~0. It is clear that C(S, 6) is analytic for small 8. 22)" become smooth after multiplication by some power of 6. 23) r(z) - ~ jO S q(z) when 6--YO. 17)' if ~ > 1, as we assume from now on. 12 (and the remark following its proof). 18)" is fulfilled for some z O.

22),, (x(~)(a(6~)P-b(5~)a)(~(8~) +6-~-~Ds~) + + C(S, 6) b(6 f) Q ( ~ ( 6 f) + 6 - ~ f - I D s N ) ) V6(S ) - O. 59 Here C(S, 8) -K(6~)(I-A(sF))/A(8~)--~-~JoS , 6--~0. It is clear that C(S, 6) is analytic for small 8. 22)" become smooth after multiplication by some power of 6. 23) r(z) - ~ jO S q(z) when 6--YO. 17)' if ~ > 1, as we assume from now on. 12 (and the remark following its proof). 18)" is fulfilled for some z O. 23) has a unique C OO zero z(S) with z(O) ~ z0 defined for S in a neighborhood of O.

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