= Weighted holomorphic norm on a shrinking time domain
{title2=$\|u\|_\alpha=\sup\limits_{0<s<1,\;0<|t|<\alpha(1-s),\;|x|\leq sR}|u(x,t)|\frac{\alpha(1-s)-|t|}{|t|}$}
Let $R,\alpha>0$ and $\mathcal C_\alpha=\{(x,t)\in\mathbb C^2:|x|<R,\;|t|<\alpha(1-|x|/R)\}$. The displayed <norm> makes the <holomorphic functions> with finite norm a <Banach space>. Finiteness forces $u(x,0)=0$: for fixed $x$, the weight grows like $1/|t|$ near zero. A Cauchy sequence in this norm converges uniformly on every compact subset of $\mathcal C_\alpha$, including at $t=0$. Its limit is holomorphic by <locally uniform convergence of holomorphic functions>, and the pointwise norm bounds show convergence in the original norm. The shrinking domain compensates for loss of an $x$-<derivative> in a <Cauchy-Kovalevskaya theorem> argument.
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