We study the Banach space () of the harmonic mappings on the open unit disk satisfying the condition where and denote the first complex partial derivatives of . We show that several properties that are valid for the space of analytic functions known as the -Bloch space extend to . In particular, we prove that for the mappings in can be characterized in terms of a Lipschitz condition relative to the metric defined by . When , the harmonic -Bloch space can be viewed as the harmonic growth space of order , while for , is the space of harmonic mappings that are Lipschitz of order .
In this work, we characterize the bounded and compact weighted composition operators from a large class of Banach space of analytic functions on the open unit disk into weighted Banach spaces, determine the operator norm exactly and obtain estimates on the essential norm. We obtain an exact formula of the essential norm when such operators map the Hardy spaces, the weighted Bergman spaces, the Bloch space and the little Bloch space into the Banach space H μ ∞ (where the weight μ is a continuous function). In this general setting we also study such operators when the target space is an α-Bloch space for any α > 0 and more generally, any Bloch-type space. As special cases, we derive norm and essential norm approximations for the operators acting on the Hardy spaces and the weighted Bergman spaces. Immediate corollaries are results on certain integral operators.
Berzelius failed to make use of Faraday's electrochemical laws in his laborious determination of equivalent weights.