New Criteria for Composition Operators into Weighted Harmonic Bloch Spaces

In this paper, we obtain new criteria for the boundedness, the compactness and the essential norm of composition operators acting from the space H∞ of bounded harmonic mappings into weighted harmonic Bloch spaces BHα, 0<α<∞, on the unit disk. These criteria are expressed in terms of the asymptotic behavior of the sequence {∥Cψqk∥BHα}k≥0, where qk(z)=zk+z¯k, and of a family of test functions Fa,n normalized by Fa,n(a)=2. We further relate these quantities to an explicit function of the symbol ψ, which yields symbol-based characterizations of boundedness and compactness, and two-sided estimates with absolute constants for the essential norm of Cψ:H∞→BHα.