Characterize bounded weighted composition operators on the Hardy space

Determine a simple necessary-and-sufficient condition for boundedness of general weighted composition operators on the Hardy space H^2 of the unit disk, analogous to the self-map criterion for ordinary composition operators.

Background

The paper introduces weighted composition operators W_{\psi,\phi}=M_\psi C_\phi on the classical Hardy space H2 of the unit disk as an auxiliary tool for analyzing the weighted Hankel operators H_t. Although sufficient conditions and partial characterizations for their boundedness are available, the authors emphasize that the general boundedness problem lacks a simple criterion comparable to the ordinary composition-operator case.

This unresolved issue is presented as background rather than as a problem specific to the finite-prime composition operators on the Hardy space of Dirichlet series. The paper resolves the boundedness and norm calculation only for the particular Möbius symbol and weight used to represent H_t.

References

The boundedness and norm of weighted composition operators are not well understood. Although various sufficient conditions and partial characterizations for boundedness are known, a simple or explicit necessary and sufficient condition analogous to the case of composition operators is not known in general. In particular, no explicit formula for the operator norm of $W_{\psi,\phi}$ on $H2$ is available in full generality, and obtaining sharp norm estimates remains a challenging problem.

— Critical Norm Profiles for Finite-Prime Composition Operators on the Hardy Space of Dirichlet Series  (2608.26041 - Fang et al., 26 Aug 2026) in Section 2, immediately before Corollary 2.4