Singular Cauchy problems for second order partial differential operators with non-involutory characteristics (Q5966440)

The author studies the following singular Cauchy problem \[ (S.C.P.):\quad Pu(x,y)=0,\quad D^ i_ xu(0,y)=u^ 0_ i(y),\quad i=0,1 \] where \(u^ 0_ i(y)\) \((i=0,1)\) are multivalued holomorphic functions defined on \(\{y\in {\mathbb{C}}^ n:\quad | y_ i| <R;\quad y_ 1=0\}\) for some \(R>0\), satisfying the growth conditions: \[ | u^ 0_ j(y)| \leq C\quad \exp \{C\quad y_ 1^{-(q-1- q')/q+1}\} \] for some positive C, and \[ P=P(x,y,D_ x,D_ y)=\sum_{i+| \alpha | \leq 2}x^{k(i,\alpha)} a_{i\alpha}(x,y) D^ i_ x D_ y^{\alpha} \] where k(i,\(\alpha)\) is equal to \(q^{| \alpha |}\) for \(i+| \alpha | =2\), or to q' for \(i=0\), \(| \alpha | =1\) and 0 otherwise, where q,q' are integers, \(0\leq q'\leq q-2\). One supposes that \(a_{i\alpha}(x,y)\) for \(i+| \alpha | \leq 2\) are holomorphic at the origin, \(a_{2,0}=1\), and that the equation \[ \sum_{i+| \alpha | =2}x^{q^{| \alpha |}} a_{i\alpha}(x,y) \xi^ i \eta^{\alpha}=0 \] has two distinct roots \(\xi =x^ q\lambda_ i(x,y,\eta)\) with \(\lambda_ i(x,y,\eta)\) holomorphic at \(x=y=0\), \(\eta =(1,0;\infty,0)\) homogeneous of degree 1 with respect to \(\eta\). The main result of the author is the following: there exists a unique solution of SCP, holomorphic in a domain described explicitly, which satisfies the following estimate: \[ | u(x,y)| \leq C\sum_{i=1,2}\exp \{C\phi_ i(x,y)-(q-1-q')/q+1\} \] for some positive C. (Here \(\phi_ i(x,y)\) is defined by \(\partial_ x\phi_ i(x,y)-x^ q\lambda_ i(x,y,\text{grad}_ y \phi (x,y))=0\) \(\phi_ i(0,y)=y_ i\), \(i=1,2).\) The problem is reduced to a technical Main Lemma, which roughly asserts that there exist parameters \(U^{\pm}(x,y,D_ y)\) with suitable properties. The main part of the paper is dedicated to the proof of this Main Lemma, and uses an explicit correspondence between operators of infinite order and their symbols, which is more or less an explicit translation of results of \textit{T. Aoki} [Publ. Res. Inst. Math. Sci. 18, 421-449 (1982; Zbl 0512.35077)]. The construction of the \(U^{\pm}(x,y;\eta)\) is done using the classical WKB method. The very precise results are too intricated to be given. - Let us mention that in this paper the case of operators satisfying a Levi condition (i.e. with \(q'=q-1)\) is not treated.