247A, Notes 3: Singular integral operators

— 1. The Hilbert transform — In this set of notes we begin the theory of singular integral operators – operators which are almost integral operators, except that their kernel just barely fails to be integrable near the diagonal . (This is in contrast to, say, fractional integral operators such as , whose kernels go to infinity at the diagonal but remain locally integrable.) These operators arise naturally in complex analysis, Fourier analysis, and also in PDE (in particular, they include the zeroth order pseudodifferential operators as a special case, about which more will be said later). It is here for the first time that we must make essential use of cancellation: cruder tools such as Schur’s test which do not take advantage of the sign of the kernel will not be effective here. Before we turn to the general theory, let us study one of the simplest non-trivial singular integral operators, the Hilbert transform It is not immediately obvious that is well-defined even for nice functions…