Sobolev Estimates for Singular Radon Transforms

Singular Radon transforms of order α are defined. For each α ∈ R, a model transform R̃α on RPn for n odd is considered. Generalizing earlier results by the author (Comm. Partial Differential Equations14, 1989, 1461-1470), sharp Sobolev estimates, both upper bounds and lower bounds, are given. In particular, R̃α is injective for all α not equal to a certain critical value. Using a result in microlocal analysis known as the "parabolic trick," Sobolev upper bounds are proved for quite general singular Radon transforms. These are consistent with the upper bounds found in the model case. In addition, a microlocal description of the critical value of α is found. For α a non-negative even integer, it is shown that the lower bounds continue to hold for transforms "close" to the model transform.