  We study an operator analogue of the classical problem of finding the rate of
decay of an oscillatory integral on the real line. This particular problem
arose in the analysis of oscillatory Riemann-Hilbert problems associated with
partial differential equations in the Ablowitz-Kaup-Newell-Segur hierarchy, but
is interesting in its own right as a question in harmonic analysis and
oscillatory integrals. As was the case in earlier work of the first author, the
approach is general and purely real-variable. The resulting estimates we
achieve are strongly uniform as a function of the phase and can simultaneously
accommodate phases with low regularity (as low as $C^{1,\alpha}$), local
singularities, and essentially arbitrary sets of stationary points that
degenerate to finite or infinite order.
