Since M.S. Baouendi and C. Goulaouic ([2], [3]) defined partial differential operators of Fuchs type and proved theorems of Cauchy-Kowalevskaya type and Holmgren type, many authors have investigated operators of Fuchs type in various categories, that is, real-analytic, C∞ and so on. (Cf. [1], [4], [6], [8], [9], [11], [12], [17], [18], [19], [20], [21] etc.)

To Fuchsian partial di¨erential equations in the sense of M. S. Baouendi and C. Goulaouic, which is a natural extension of ordinary di¨erential equations with regular singularity at a point, all the solutions in a complex domain are constructed along the same line as the method of Frobenius to ordinary di¨erential equations, without any assumptions on the characteristic exponents. The same idea can be applied to Fuchsian hyperbolic equations considered by H. Tahara. Introduction.Let C be the set of complex numbers, t be a variable in C, and x x 1 ; ...; x n be variables in C n . We consider a Fuchsian partial di¨erential operator with weight 0 de®ned by M. S. Baouendi and C. Goulaouic [1]. P t m D m t P 1 t; x; D x t mÀ1 D mÀ1 t ÁÁÁP m t; x; D x ; 1:1where m is a positive integer, and D t : q=qt, D x :D x 1 ; ...; D x n , D x j : q=qx j .Assume that the coe½cients a j; a jaj U j U m are holomorphic in a neighborhood of t; x0; 0. M. Kashiwara and T. Oshima ([3], De®nition 4.2) called such an operator ``an operator which has regular singularity in a weak sense along S 0 :ft 0g.'' For such operators, M. S. Baouendi and C. Goulaouic [1] showed fundamental theorems that are extensions of the Cauchy-Kowalevsky theorem and the Holmgren theorem (see Theorems 4.1, 4.2 given later).

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An orthonormal wavelet basis in L 2 ðR n Þ used for microlocal filters, which decompose signals into microlocal contents, is shown to be a ''stepwise'' unconditional basis in L p ðR n Þ (1 < p < 1). Other related spaces are also treated. As part of the proof, an elementary proof of the L p version of the sampling theorem with unconditional convergence is given. Finally, an application is given to the expression of some distributions as sums of boundary values of holomorphic functions.

+c, then the regularity-loss depends on |Re6(0, x)\. We want also to deal Fuchsian operators. So, we consider the operators whose coefficients may have fractional or negative powers of t. For these operators, we can consider the flat Cauchy problem to which the non-characteristic Cauchy problem for operators with C°°-coefficients can be easily reduced.Our program is as follows. In Section 1, we state definitions, the result and some examples. Our result consists of three theorems. In Section 2, we consider two transformations of operators which reduce the theorems to easier situation. In Section 3, we study an elementary fact on Newton polygons and apply it. In Sections 4, 5 and 6, we prove the theorems.