Essential Norm of Generalized Integral-Type Operator from the Fractional Cauchy Transform Space into Weighted Bloch and Dirichlet Spaces

In this paper, we investigate the generalized integral-type operator acting between the fractional Cauchy transform space and two classical analytic function spaces: the weighted Bloch space and the weighted Dirichlet space. For the operator acting into the weighted Bloch space, we obtain two equivalent exact formulas for its operator norm. Furthermore, an estimate for its essential norm is provided, which leads to a necessary and sufficient condition for compactness. For the operator acting into the weighted Dirichlet space, we derive the exact operator norm and fully characterize its compactness.