## Journal of the Mathematical Society of Japan

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

Sunao ŌUCHI

#### Abstract

Consider a linear partial differential equation in $C^{d+1}P(z,\partial)u(z)=f(z)$, where $u(z)$ and $f(z)$ admit singularities on the surface $\{z_{0}=0\}$. We assume that $|f(z)|\leq A|z_{0}|^{c}$ 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 $\forall\epsilon>0\exists C_{\epsilon}$ such that $|u(z)|\leq C_{\epsilon}\exp(\epsilon|z_{0}|^{-\gamma^{*}})$ in the sectorial region, then $|u(z)|\leq C|z_{0}|^{c^{\prime}}$ for some constants $c^{\prime}$ and $C$.

#### Article information

Source
J. Math. Soc. Japan, Volume 52, Number 4 (2000), 767-792.

Dates
First available in Project Euclid: 10 June 2008

https://projecteuclid.org/euclid.jmsj/1213107109

Digital Object Identifier
doi:10.2969/jmsj/05240767

Mathematical Reviews number (MathSciNet)
MR1775389

Zentralblatt MATH identifier
0966.35006

#### Citation

ŌUCHI, Sunao. Growth property and slowly increasing behaviour of singular solutions of linear partial differential equations in the complex domain. J. Math. Soc. Japan 52 (2000), no. 4, 767--792. doi:10.2969/jmsj/05240767. https://projecteuclid.org/euclid.jmsj/1213107109