Methods of Harmonic Analysis

I spent most of my research time this summer studying the basic methods of harmonic analysis. My goal was to write an explanation and elaboration of Fefferman’s 1973 paper, Pointwise Convergence of Fourier Series. I was attempting to fill in the details and clarify things to the degree that the paper would be accessible at the level of a beginning graduate student, and to add enough background that the paper could be read more or less without reference to the literature. The outcome of this summer research was that I now have more background in some special parts of real analysis like singular integral theory and interpolation theorems. The paper proved to be too difficult for me to deal with in the time I had. So in this report I’ll talk about the background which I got from this experience. Fefferman’s paper [2] gives another proof of the Carleson-Hunt theorem. Carleson-Hunt theorem. Let f ∈ Lp[0,2π] for 1 < p < ∞. Then except on a set with Lebesgue measure zero, the partial sums fn of the Fourier series, defined as follows,