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Onto interpolation for the Dirichlet space and for \(H_1 (\mathbb{D})\). (English) Zbl 1467.30022

Consider a sequence \(\mathcal{Z}=\{z_i\}\) of points in the unit disk \(\mathbb{D}\) and a reproducing kernel space \(\mathcal{H}\) of functions holomorphic in \(\mathbb{D}\). If the reproducing kernel is \(k_z(\cdot)\), then \(\mathcal{Z}\) is onto interpolating (OI) if \(T_\mathcal{H}(f):\mathcal{H}\to\ell^2: f\mapsto\{f(z_i)/\|k_{z_i}\|_{\mathcal{H}}\}\) is surjective and it is universal interpolating (UI) if \(T_{\mathcal{H}}\) is bounded. The problem is to characterize all the \(\mathcal{Z}\) that are OI or UI. First the author recalls known results and techniques from the literature for the simple case where \(\mathcal{H}\) is the Hardy space \(H^2\). A combination of these techniques is used and generalized to analyse the OI or UI property of \(\mathcal{Z}\), first when \(\mathcal{H}\) is the Dirichlet space \(\mathcal{D}\) and then, when analyticity is removed, for the Sobolev space \(H_1(\mathbb{D})\) consisting of \(L^2(\mathbb{D})\) functions whose first order partial derivatives are also in \(L^2(\mathbb{D})\). Several ingredients are used to characterize \(\mathcal{Z}\) by controling its distribution in the Poincaré disk: the hyperbolic distance to define besides strong separation also a weak separation property while weighted counting (i.e., Carleson) measures and logarithmic potentials are used to formulate a one-box sub-capacity condition (see also [B. Böe, J. Funct. Anal. 192, No. 2, 319–341 (2002; Zbl 1018.46018)]), and most importantly, a capacity condition that specifies a bound for the capacity of a condenser for two disjoint subsets of te closed unit disk. The capacity condition is essential to characterize in the main theorem all the OI sequences for \(\mathcal{D}\). The approach for solving the interpolation problem is constructive in the sense that \(f\) is approximated iteratively by adding bump-like functions. If the measure associated with the distribution of \(\mathcal{Z}\) in \(\mathbb{D}\) is finite, then finding an analytic \(f\) is reduced to solving the problem in \(H_1(\mathbb{D})\) by ensuring that the bump functions are small enough outside some region.

MSC:

30E05 Moment problems and interpolation problems in the complex plane
30H25 Besov spaces and \(Q_p\)-spaces
30J99 Function theory on the disc

Citations:

Zbl 1018.46018
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References:

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