Georgia Institute of Technology College of Sciences

I'll introduce the Hilbert transform in a natural way justifying it as a canonical operation. In fact, it is such a basic operation, that it arises naturally in a range of settings, with the important complication that the measure spaces need not be Lebesge, but rather a pair of potentially exotic measures. Does the Hilbert transform map L^2 of one measure into L^2 of the other? The full characterization has only just been found. I'll illustrate the difficulties with a charming example using uniform measure on the standard 1/3 Cantor set.