New Criteria for Composition Operators into Weighted Harmonic Bloch Spaces

In this paper, we obtain new criteria for the norm and the essential norm of composition operators acting from the space \(\mathcal{H}^{\infty}\) of bounded harmonic mappings into weighted harmonic Bloch spaces \(\mathcal{B}^{\alpha}_H\), \(0< \alpha< \infty\), on the unit disk. These criteria are expressed in terms of the asymptotic behavior of the sequence \(\{\|C_{\psi}q_k\|_{\mathcal{B}^{\alpha}_H}\}_{k\ge0}\) and of a family of M\"obius-invariant test functions, yielding symbol-based characterizations of boundedness and of compactness, and hence of the essential norm for \(C_{\psi}:\mathcal{H}^{\infty} \to \mathcal{B}^{\alpha}_H\).