Characterizations of Bloch-type spaces of harmonic mappings (Q2323537)

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 \(1 - \alpha\).