Fundam Struct Analysis by Leet, Uang, Gilbert

By Leet, Uang, Gilbert

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6 (ii) is from Oshima and Sekiguchi raj. For different approaches to partial differential equations with regular singularities we refer to Harish-Chandra [g J,. Cas selman and Mili~i~ raJ, Wallach [cJ and the appendix section of Knapp [bJ. In Oshima [eJ a simpler but not yet as powerful theory is presented. 8 is taken) the theory of Kashiwara and Oshima raJ is generalized. For instance, the restriction on the characteristic exponents is removed. 4 is also considered in Section 0 of Kashiwara et al.

13), the remaining claims being immediate. 0 See Bj8rk [a] Ch. 4. Let P be a micro-differential operator. With the product defined it makes sense to apply to plex coefficients: _ ""m f(P) - ~j=O ajP j If P a polynomial in one variable with com_ m j E ~j=O ajs (aO, ••• ,am E) we define This kind of operation on P can in fact be f(s) - extended to all functions f holomorphic in a neighborhood of 0 in j a:, provided P is sufficiently nice. 3 S = ~j:O ajp j The expression differential operator of order < 0 k < 0 that for each the series and assume that near (zO' PO(zO' '0) O.

Be analytic differential operators on respectively. u = J ° (j=l, ••• ,L) Though the theory could be done in higher generality, it is assumed that the Pj 's mutually commute. The system ~ has regular singularities along the Defini tion. e with the edge (I) Pj is of the form point and J ~nd ••• , The degree of and for each x o The polynomial aj(x,s) where, for each is a polynomial in 1. (x,s) Pj(x,O,O,s) J only s = solves 0 where 0 v E [Ln by definition t riO = 0 x rI '. • , n o t 0 t. 1 ~)i-1 nX.

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