## Extension of holomorphic $$L^ 2$$-functions with weighted growth conditions.(English)Zbl 0759.32002

Let $$\Omega\Subset\mathbb{C}^ n$$ be a smooth bounded pseudoconvex domain, with the origin as a boundary point, and $$H^ k\subset H^{k+1}$$ linear subspaces of $$\mathbb{C}^ n$$ of dimensions $$k$$ and $$k+1$$ respectively. We assume that $$H^ k$$ intersects $$\partial\Omega$$ transversally near 0. Furthermore, we suppose that $$\Omega$$ is uniformly extendable in a pseudoconvex way along $$H^ k$$ near 0 of $$N$$th order. Finally, let $$D\subset\Omega$$ be a smooth pseudoconvex domain, such that $$\partial D\cap B(0,2R)=\partial\Omega\cap B(0,2R)$$ for some positive $$R$$, and let $$\rho_ D$$ denote a defining function for $$D$$. We prove that, given a number $$\delta\in(-1+{2\over N},{2\over N})$$ and a holomorphic function $$f$$ on $$H^ k\cap D$$ which is square-integrable with respect to $$|\rho_ D|^ \delta d\lambda_ k$$ (where $$d\lambda_ k$$ is the Lebesgue measure in complex dimension $$k)$$, there exists a holomorphic extension $$\hat f$$ of $$f$$ on $$H^{k+1}\cap D$$ which is square-integrable even with respect to $$|\rho_ D|^{\delta- {2\over N}}|\log|\rho_ D||^{-3}d\lambda_{k+1}$$ such that the weighted estimate $\|\hat f\|_{L^ 2(H^{k+1}\cap D,|\rho_ D|^{\delta-{2\over N}}|\log|\rho_ D||^{-3}d\lambda_{k+1})}\leq C\cdot\| f\|_{L^ 2(H^ k\cap D,|\rho_ D|^ \delta d\lambda_ k)}$ holds wirth a universal constant $$C>0$$. We construct $$\hat f$$ in such a way that $$f\to\hat f$$ becomes a continuous linear operator.

### MSC:

 32A10 Holomorphic functions of several complex variables 32A40 Boundary behavior of holomorphic functions of several complex variables 32T99 Pseudoconvex domains
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### References:

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