= Contraction estimate on a shrinking holomorphic domain
{title2=$\left\|\int_0^t\partial_xu(x,z)\,dz\right\|_\alpha\leq\frac{4\alpha}{R}\|u\|_\alpha$}
Write $N=\|u\|_\alpha$, $r=|t|$ and $A=\alpha(1-s)>r$. Along the straight integration segment, choose $\sigma(\tau)=s+(A-\tau)/(2\alpha)$. A <Cauchy estimate> on a disc of radius $R(\sigma-s)$ bounds the integrand by $4\alpha N\tau/[R(A-\tau)^2]$. Therefore
$$
\frac{A-r}{r}\left|\int_0^t\partial_xu(x,z)\,dz\right|
\leq\frac{4\alpha N}{R}\left[1+\frac{A-r}{r}\log(1-r/A)\right]
\leq\frac{4\alpha N}{R}.
$$
The integral is taken at fixed $x$, and the disc and segment lie inside the shrinking domain. Consequently $u\mapsto\int_0^t(iu_x+f)\,dz$ is a <contraction mapping> when $0<\alpha<R/4$ and the holomorphic source $f$ is bounded on the relevant closed polydisc. The source term has weighted norm at most $\alpha\sup|f|$, so the <Banach fixed-point theorem> gives a holomorphic solution with zero initial data.
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