Boundedness of the composition R^{2T}J^{2T} in dimensions n≥2

Determine whether the operator R^{2T}J^{2T}, where R^{2T} is the response operator of the dynamical system \alpha^{2T} for the wave equation u_{tt}-\Delta u+qu=0 and J^{2T} is the time‑integration operator (J^{2T}f)(\cdot,t)=\int_0^t f(\cdot,s)\,ds acting on \mathscr F^{2T}=L_2(\Sigma^{2T}), is a bounded linear operator on \mathscr F^{2T} in spatial dimensions n\ge 2.

Background

The connecting operator C^T=W^{T*}W^T admits the formal identity C^T=\tfrac12 S^{T*}R^{2T}J^{2T}S^T, where S^T is the odd extension in time, R^{2T} is the response (Dirichlet‑to‑Neumann) operator on \Sigma^{2T}, and J^{2T} integrates in time. This identity is rigorously valid only for controls f such that S^T f lies in the domain of R^{2T}J^{2T}.

In one spatial dimension the composition R^{2T}J^{2T} is bounded, but for n\ge 2 the boundedness (or unboundedness) is not known. Clarifying this would sharpen the functional‑analytic foundation of the connecting‑operator representation and its use in reconstruction.