Aljuaid, Munirah; Colonna, Flavia Characterizations of Bloch-type spaces of harmonic mappings. (English) Zbl 1437.46032 J. Funct. Spaces 2019, Article ID 5687343, 11 p. (2019). Summary: We study the Banach space \(\mathcal{B}_H^\alpha\) (\(\alpha > 0\)) of the harmonic mappings \(h\) on the open unit disk \(\mathbb{D}\) satisfying the condition \(\sup_{z \in \mathbb{D}}(1 - \left|z\right|^2)^\alpha(\left|h_z \left(z\right)\right| + \left|h_{\overline{z}} \left(z\right)\right|) < \operatorname{\infty},\) where \(h_z\) and \(h_{\overline{z}}\) denote the first complex partial derivatives of \(h\). We show that several properties that are valid for the space of analytic functions known as the \(\alpha\)-Bloch space extend to \(\mathcal{B}_H^\alpha\). In particular, we prove that for \(\alpha > 0\) the mappings in \(\mathcal{B}_H^\alpha\) can be characterized in terms of a Lipschitz condition relative to the metric defined by \(d_{H, \alpha}(z, w) = \sup \{\left|h \left(z\right) - h \left(w\right)\right| : h \in \mathcal{B}_H^\alpha, \|h\|_{\mathcal{B}_H^\alpha} \leq 1 \}\). When \(\alpha > 1\), the harmonic \(\alpha\)-Bloch space can be viewed as the harmonic growth space of order \(\alpha - 1\), while for \(0 < \alpha < 1\), \(\mathcal{B}_H^\alpha\) is the space of harmonic mappings that are Lipschitz of order \(1 - \alpha\). Cited in 3 Documents MSC: 46E15 Banach spaces of continuous, differentiable or analytic functions Keywords:spaces of harmonic functions; Bloch space PDFBibTeX XMLCite \textit{M. Aljuaid} and \textit{F. Colonna}, J. Funct. Spaces 2019, Article ID 5687343, 11 p. (2019; Zbl 1437.46032) Full Text: DOI References: [1] Colonna, F., Bloch and normal functions and their relation, Rendiconti del Circolo Matematico di Palermo, Serie II, 38, 2, 161-180 (1989) · Zbl 0685.30027 [2] Zhu, K., Operator Theory in Function Spaces. Operator Theory in Function Spaces, Mathematical Surveys and Monographs (2007), Providence, RI, USA: American Mathematical Society, Providence, RI, USA · Zbl 1123.47001 [3] Colonna, F., The Bloch constant of bounded harmonic mappings, Indiana University Mathematics Journal, 38, 4, 829-840 (1989) · Zbl 0677.30020 [4] Zhu, K. H., Bloch type spaces of analytic functions, Rocky Mountain Journal of Mathematics, 23, 3, 1143-1177 (1993) · Zbl 0787.30019 [5] Yoneda, R., A characterization of the harmonic Bloch space and the harmonic Besov spaces by an oscillation, Proceedings of the Edinburgh Mathematical Society, 45, 1, 229-239 (2002) · Zbl 1032.46039 [6] Chen, S. H.; Ponnusamy, S.; Wang, X., Landau’s theorem and Marden constant for harmonic \(ν\)-Bloch mappings, Bulletin of the Australian Mathematical Society, 84, 1, 19-32 (2011) · Zbl 1220.30033 [7] Chen, S.; Ponnusamy, S.; Wang, X., On planar harmonic Lipschitz and planar harmonic Hardy classes, Annales- Academiae Scientiarum Fennicae Mathematica, 36, 2, 567-576 (2011) · Zbl 1279.30035 [8] Chen, S.; Ponnusamy, S.; Rasila, A., Lengths, areas and Lipschitz-type spaces of planar harmonic mappings, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, 115, 62-70 (2015) · Zbl 1307.30044 [9] Chen, S.; Wang, X., On harmonic Bloch spaces in the unit ball of \(C^n\), Bulletin of the Australian Mathematical Society, 84, 1, 67-78 (2011) · Zbl 1222.31002 [10] Stroethoff, K., Harmonic Bergman spaces, Holomorphic spaces. Holomorphic spaces, Math. Sci. Res. Inst. Publ., 33, 51-63 (1998), Cambridge Univ. Press, Cambridge · Zbl 0997.46036 [11] Chen, S.; Ponnusamy, S.; Wang, X., Harmonic mappings in Bergman spaces, Monatshefte für Mathematik, 170, 3-4, 325-342 (2013) · Zbl 1267.31002 [12] Fu, X.; Liu, X., On characterizations of Bloch spaces and Besov spaces of pluriharmonic mappings, Journal of Inequalities and Applications, 2015, article 360 (2015) · Zbl 1336.31019 [13] Duren, P., Harmonic Mappings in the Plane (2004), London, UK: Cambridge University Press, London, UK · Zbl 1055.31001 [14] Aljuaid, M.; Colonna, F., On the harmonic Zygmund spaces, In press · Zbl 1439.30082 [15] Hedenmalm, H.; Korenblum, B.; Zhu, K., Theory of Bergman Spaces (2000), New York, NY, USA: Springer-Verlag, New York, NY, USA · Zbl 0955.32003 [16] Conway, J. B., Functions of One Complex Variable I (2001), New York, NY, USA: Springer-Verlag, New York, NY, USA [17] Colonna, F.; Tjani, M., Operator norms and essential norms of weighted composition operators between Banach spaces of analytic functions, Journal of Mathematical Analysis and Applications, 434, 1, 93-124 (2016) · Zbl 1338.30049 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.