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Interpolation between Sobolev and between Lipschitz spaces of analytic functions on starshaped domains. (English) Zbl 0695.46034

Let \(\Omega\) be a bounded domain in \({\mathbb{C}}^ n\) starshaped with respect to the origin, with Lipschitz boundary, \(0<p<\infty\), \(s\in {\mathbb{R}}.\)
O(\(\Omega)\) denotes the space of analytic functions of \(\Omega\), k(s) the smallest nonnegative integer \(\geq s\) \[ A^{s,p}=\{f\in O(\Omega):\quad \| f\|_{s,p}<\infty \},\quad where \]
\[ \| f\|^ p_{s,p}=\int_{\Omega}| \delta^{k(s)-s} \nabla^{k(s)}f|^ p+\sum^{k(s)-1}_{j=0}| \nabla^ jf(0)|, \] (\(\delta\) denotes the distance function to the boundary of \(\Omega\) and \(\nabla^ jf\) denotes the vector of all derivatives of order j).
The main result of the paper is that the \({\mathcal L}^ p\)-sobolev spaces of analytic functions form an interpolation scale for both real and complex methods, i.e. \[ (A^{s_ 0,p},A^{s_ 1,p})_{\theta,p}=(A^{s_ 0,p},A^{s_ 1,p})_{[\theta]}=(A^{s_ 0,p},A^{s_ 1,p})^{[\theta]}=A^{s,p}, \] where \(0<\theta <1\), \(s_ 0,s_ 1\in {\mathbb{R}}\), \(s=(1-\theta)s_ 0+\theta s_ 1\). Similar results for \(p=\infty\).
Reviewer: V.Anisiu

MSC:

46M35 Abstract interpolation of topological vector spaces
46E15 Banach spaces of continuous, differentiable or analytic functions
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
32A99 Holomorphic functions of several complex variables
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
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