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

Dedicated to Professor Daisuke Fujiwara on his sixtieth birthday

抄録

Consider a linear partial differential equation in C^d+1 P(z, ∂)u(z)=f(z), where u(z) and f(z) admit singularities on the surface {z₀=0}. We assume that |f(z)|≤ A|z₀|^c in some sectorial region with respect to z₀. We can give an exponent γ^*>0 for each operator P(z, ∂) and show for those satisfying some conditions that if ∀ε>0∃ C_ε such that |u(z)|≤ C_εexp(ε|z₀|^{-γ^*}) in the sectorial region, then |u(z)|≤ C|z₀|^{c^{'}} for some constants c^{'} and C.