Hausdorff operators defined by measures on \(p\)-adic linear spaces and their norms
Let \(p\) be a prime number and let \(\mathbb{Q}_p\) be the field of \(p\)-adic numbers equipped with its canonical non-Archimedean \(p\)-adic valuation. The author introduces the Morrey and Herz spaces associated to \(L^{q} (\mathbb{Q}_{p}^{n})\), \(1 \leq q < \infty\), \(n \in \mathbb{N}\). Their classical versions were given by \textit{C. B. Morrey jun.} in [Trans. Am. Math. Soc. 43, 126--166 (1938; Zbl 0018.40501; JFM 64.0460.02)] and \textit{C. S. Herz} in [J. Math. Mech. 18, 283--323 (1968; Zbl 0177.15701)], respectively. Here, \(L^{q} (\mathbb{Q}_{p}^{n})\) is the space of all complex-valued \(\mu\)-measurable functions \(f\) (where \(\mu\) is the Haar measure on \(\mathbb{Q}_{p}^{n}\)) with the finite norm \(\| f \|_q := \left(\int_{\mathbb{Q}_{p}^{n}} | f(x)|^{q} \, d \mu(x)\right)^{1/q}\). Now, let \({\mathcal B}\) be the Borel algebra of \(\mathbb{Q}_{p}^{n}\) and let \(\nu\) be a measure on \(\mathbb{Q}_{p}^{n}\), that is, a set function defined on \({\mathcal B}\) which is countably additive and such that \(\nu(E)\) is finite if \(E\) is bounded in \(\mathbb{Q}_{p}^{n}\). The aim of the present paper is to study the non-Archimedean counterpart of the classical Hausdorff operators introduced by \textit{G. H. Hardy} in [Divergent series. Oxford: At the Clarendon Press (Geoffrey Cumberlege) (1949; Zbl 0032.05801)], more concretely, operators defined by \({\mathcal H}_{\nu} (f) (x) = \int_{\mathbb{Q}_{p}} f(tx) \, d \nu(t)\). The norms of \({\mathcal H}_{\nu}\) in Morrey spaces are calculated, while for the norms of \({\mathcal H}_{\nu}\) in Keller spaces some weak asymptotics are obtained. These results generalize some previous ones obtained by the author in [\(p\)-Adic Numbers Ultrametric Anal. Appl. 3, No. 2, 149--156 (2011; Zbl 1254.42011); ibid. 4, No. 3, 222--230 (2012; Zbl 1271.42038)]. We could say that the present paper has but a partial non-Archimedean character, in the sense that the author considers functions defined on the non-Archimedean space \(\mathbb{Q}_{p}^{n}\), but with complex values. For a future research work it would interesting to study the subject treated in this paper for functions on \(\mathbb{Q}_{p}^{n}\) and with values in a non-Archimedean valued field, e.g., \(\mathbb{Q}_{p}\). Then the resulting investigation will have a pure non-Archimedean character.
Hausdorff operators defined by a measure
scientific article; zbMATH DE number 6516131
scientific article