Characterizations of Bloch-Type Spaces of Harmonic Mappings. (5th August 2019) - Aljuaid, Munirah; Colonna, Flavia
Abstract : We study the Banach spaceB H α (α > 0 ) of the harmonic mappingsh on the open unit diskD satisfying the conditionsup z ∈ D ⁡ ( 1 - z 2 ) α ( h z z + h z ¯ z ) < ∞, whereh z andh z ¯ denote the first complex partial derivatives ofh . We show that several properties that are valid for the space of analytic functions known as theα -Bloch space extend toB H α . In particular, we prove that forα > 0 the mappings inB H α can be characterized in terms of a Lipschitz condition relative to the metric defined byd H, α ( z, w ) = sup ⁡ h z - h w : h ∈ B H α, h B H α ≤ 1 . Whenα > 1, the harmonicα -Bloch space can be viewed as the harmonic growth space of orderα - 1, while for0 < α < 1, B H α is the space of harmonic mappings that are Lipschitz of order1 - α .