Hilbert matrix on spaces of Bergman-type. (English) Zbl 1456.47011
Summary: It is well known (see [M. Jevtić and B. Karapetrović, “Libera operator on mixed norm spaces \(H_\nu^{p, q, \alpha}\) when \(0 <p < 1\)”, Filomat 31, No. 14, 4641–4650 (2017; doi:10.2298/FIL1714641J); M. Pavlović, “Definition and properties of the Libera operator on mixed norm spaces”, The Scientific World Journal 2014, Article ID 590656, 15 p. (2014; doi:10.1155/2014/590656)]) that the Libera operator \(\mathcal{L}\) is bounded on the Besov space \(H_\nu^{p, q, \alpha}\) if and only if \(0 < \kappa_{p, \alpha, \nu} : = \nu - \alpha - \frac{1}{p} + 1\). We prove unexpected results: the Hilbert matrix operator \(H\), as well as the modified Hilbert operator \(\tilde{H}\), is bounded on \(H_\nu^{p, q, \alpha}\) if and only if \(0 < \kappa_{p, \alpha, \nu} < 1\). In particular, \(H\), as well as \(\tilde{H}\), is bounded on the Bergman space \(A^{p, \alpha}\) if and only if \(1 < \alpha + 2 < p\) and is bounded on the Dirichlet space \(\mathcal{D}_\alpha^p = A_1^{p, \alpha}\) if and only if \(\max \{- 1, p - 2 \} < \alpha < 2 p - 2\). Our results are substantial improvement of [B. Łanucha et al., Ann. Acad. Sci. Fenn., Math. 37, No. 1, 161–174 (2012; Zbl 1258.47047), Theorem 3.1] and of [P. Galanopoulos et al., Ann. Acad. Sci. Fenn., Math. 39, No. 1, 231–258 (2014; Zbl 1297.47030), Theorem 5].
MSC:
| 47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
| 30H20 | Bergman spaces and Fock spaces |
References:
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