Per Sjölin: Lp-estimates for singular oscillatory integral operators.

Singular integral operators have been studied for a very long time and since approximately 1970 there has also been a lot of interest in oscillatory integral operators. We shall here consider singular oscillatory integral operators, that is operators defined by integrals containing both a singular kernel and an oscillating factor. Operators of this type have been much studied in the theory of convergence of Fourier series and also by for instance Phong and Stein. We shall continue this study.

We shall apply our results to obtain estimates for convolution operators and also to decay estimates for the eigenvalue problem corresponding to the Helmholtz equation. 

This is partly joint work with H. Aleksanyan and H. Shahgholian.