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 zer...