I’m Peter “Viking” Luthy, a journeyman graduate student at Cornell. I’m an analyst, and my current research goals are in harmonic analysis with applications to and from ergodic theory. To avoid being called a hypocrite, Greg asked me to post on occasion and spread my analytic gospel — this isn’t the Everything-but-Analysis Seminar, after all.
My goal in this post is to go through the initial setup of a deep theorem of Carleson dealing with the convergence of Fourier series on . This theorem is almost universally interesting in and of itself. Additionally, it will give ample reason as to why people — myself included — care about objects called singular integral operators. This will also provide some impetus for some future posts as well, particularly one which will outline a famous construction of Fefferman and give some reasons why harmonic analysis in higher dimensions is distinctly harder than in dimension 1.