Let $L=\sum_{i=1}^{d} X_{i}(z) \partial_{z_{i}}$ be a holomorphic vector field degenerating at $z=0$ such that Jacobi matrix $((\partial X_{i}/\partial z_{j})(0))$ has zero eigenvalues. Consider $Lu=F(z,u)$ and let $\tilde{u}(z)$ be a formal power series solution. We study the Borel summability of $\tilde{u}(z)$ , which implies the existence of a genuine solution $u(z)$ such that $u(z)\sim\tilde{u}(z)$ as $z \rightarrow 0$ in some sectorial region. Further we treat singular equations appearing in finding normal forms of singular vector fields and study to simplify $L$ by transformations with Borel summable functions.
@article{1158242065,
author = {\=OUCHI, Sunao},
title = {Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields},
journal = {J. Math. Soc. Japan},
volume = {57},
number = {4},
year = {2005},
pages = { 415-460},
language = {en},
url = {http://dml.mathdoc.fr/item/1158242065}
}
ŌUCHI, Sunao. Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields. J. Math. Soc. Japan, Tome 57 (2005) no. 4, pp. 415-460. http://gdmltest.u-ga.fr/item/1158242065/