On Volterra integral operators with highly oscillatory kernels

We study the high-oscillation properties of solutions to integral equationsassociated with two classes of Volterra integral operators: compact operatorswith highly oscillatory kernels that are either smooth or weakly singular, andnoncompact cordial Volterra integral operators with highly oscillatory kernels.In the latter case the focus is on the dependence of the (uncountable) spectrumon the oscillation parameter. It is shown that the results derivedin this paper merely open a window to a general theory of solutions of highlyoscillatory Volterra integral equations, and many questions remain to beanswered.