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.


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