Necessary and sufficient condition for existence and uniqueness of the solution of Cauchy problem for holomorphic Fuchsian operators (Q2740878)

Let \(P(x;D)\) be a Fuchsian operator of order \(m\) and weight \(m-k\) with respect to \(x_0\) at \(a=(0,\dots,0)\): \(P(x;D)= P_m(x;D_0)- Q(x;D)\), where NEWLINE\[NEWLINEP_m(x;D_0) =\sum^k_{p=0} a_{m-p}(x') x_0^{k-m} D_0^{m-p},NEWLINE\]NEWLINE NEWLINE\[NEWLINEQ(x;D)=-\sum_{\alpha_0 <m,|\alpha|<m}x_0^{\mu (\alpha_0)} D_0^{\alpha_0} \bigl( a_{\alpha_0, \alpha'} (x_0,x') D_{x'}^{\alpha'} \bigr)NEWLINE\]NEWLINE with \(a_m= 1\), \(mu(\alpha_0)=[\alpha_0 +1-(m-k)]_+\). When coefficients of \(P_m(x;D_0)\), \(Q(x;D)\) are holomorphic functions near the origin in \(\mathbb{C}^{n+1}\), then the Baouerdi-Goulaouic theorem says that for all integers \(\lambda \geq m-k\neq 0\), for any holomorphic Cauchy data \(u_j\), \(0\leq j\leq m-k-1\), near the origin in \(\mathbb{C}^n\) and for each holomorphic function \(f\) near the origin in \(\mathbb{C}^{n+1}\), there exists a unique holomorphic solution \(u\) near the origin in \(\mathbb{C}^{n+1}\) solving Cauchy problem: \(P(x;D)u(x)=f(x)\), \(D_0^j u(0, x')= u_j(x')\), \(0\leq j\leq m-k-1\). The author of this paper extends this theorem to the general situation: Let \(h\) be a Fuchsian holomorphic differential operator of weight \(\tau_{n,s}(a)\) in \(a\) with respect to a holomorphic hypersurface \(S\) pasing through \(a\), of a holomorphic differential manifold \(E\) of dimension \(n+1\) and \(\varphi\) a local equation of \(S\) in some neighborhood of \(a\). Then for all \(\lambda\geq \tau_{h,S}(a)\), \(C(\lambda,x')\neq 0\) and for all holomorphic functions \(f\) and \(v\) in a neighborhood of \(a\), there exists a unique holomorphic function \(u\) solving the Cauchy problem \(h(u)=f\), \(u-v=O (\varphi^{\tau_{h,S} (a)})\).