Growth property and slowly increasing behaviour of singular solutions of linear partial differential equations in the complex domain

Abstract

Consider a linear partial differential equation in $C^{d+1}P(z,\partial )u\left(z\right)=f\left(z\right)$, where $u\left(z\right)$ and $f\left(z\right)$ admit singularities on the surface $\{z_0=0\}$. We assume that $\left|f\right(z\left)\right|\le A|z_0{|}^c$ in some sectorial region with respect to $z_0$. We can give an exponent ${\gamma }^{*}>0$ for each operator $P(z,\partial )$ and show for those satisfying some conditions that if $\forall ϵ>0\exists C_{ϵ}$ such that $\left|u\right(z\left)\right|\le C_{ϵ}\exp \left(ϵ\right|z_0{|}^{-{\gamma }^{*}})$ in the sectorial region, then $\left|u\right(z\left)\right|\le C|z_0{|}^{c^{\prime }}$ for some constants $c^{\prime }$ and $C$.