A note on the Cauchy-Bochner formula for a class of Beurling ultradistributions

Pilipovic, Stevan

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Let $L (z, \partial_{z}) = (\partial_{z_{0}})^{k} - A (z, \partial_{z})$ be a linear partial differential operator with holomorphic coefficients, where $A (z, \partial_{z}) = \sum_{j = 0}^{k - 1} A_{j} (z, \partial_{z^{'}}) (\partial_{z_{0}})^{j}, ord . A (z, \partial_{z}) = m > k$ and $z = (z_{0}, z^{'}) \in C^{n + 1} .$ We consider Cauchy problem with holomorphic data $L (z, \partial_{z}) u (z) = f (z), (\partial_{z_{0}})^{i} u (0, z^{'}) = {\hat{u}}_{i} (z^{'}) (0 \leq i \leq k - 1) .$ We can easily get a formal solution $\hat{u} (z) = \sum_{n = 0}^{\infty} {\hat{u}}_{n} (z^{'}) (z_{0})^{n} / n!$ , bu in general it diverges. We show under some conditions that for any sector $S$ with the opening less that a constant determined by $L (z, \partial_{z})$ , there is a function $u_{S} (z)$ holomorphic except on ${z_{0} = 0}$ such that $L (z, \partial_{z}) u_{S} (z) = f (z)$ and $u_{S} (z) \sim \hat{u} (z)$ as $z_{0} \to 0$ in $S$ .

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