Singular Integral Operators and Convergence of Fourier Series

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.

A Resolution of a Tensor Algebra

    This post is about projective resolutions of algebras, thought of as a bimodules over themselves.  As long as is an associative, unital algebra (which it always will be in this post), there is a canonical projective resolution of , called the bar resolution, which is sufficient for most purposes.  However, this resolution is of infinite length, and so it isn’t useful in bounding projective dimensions of modules.  For those purposes, it is natural to look for finite projective resolutions of .

    I came across such a problem in my research, and came up with limited and ultimately unhelpful results.  My interest was in the case that is a tensor algebra of an algebra over a bimodule .  Under what conditions on and would a nice, finite resolution of exist?  My result is as follows:

Let be an algebra, be a bimodule over , and let denote the kernel of the multiplication map .  If is projective as an bimodule, and , then there is a projective resolution of of length 3.

This is kinda neat, but its not super useful unless it can be used to produce projective resolutions of modules.  Hence, the second result:

Let , , and be defined as above.  If is projective and is flat as a right module, then any left  module has a projective resolution of length 3.

Sadly, I wanted the assumptions of the former to prove the latter, which my techniques don’t.

Dunkl Operators

This post is basically a write-up of notes for a talk I gave for the Olivetti club, the weekly Cornell grad student talk (of course, the post has pretty much everything I wanted to say, while my talk unsurprisingly did not). In 1989 Dunkl introduced a commuting family of operators which is a deformation of the family of directional derivative operators. More specifically, given a finite reflection group W acting on a vector space V, the choice of one complex parameter for each conjugacy class in W determines a family of commuting linear operators on , and if all the parameters are chosen to be 0 then this family is just the family of directional derivative operators. For almost all parameter choices, this family is surprisingly well-behaved, and many constructions involving directional derivatives can be extended, including the Fourier transform, the heat equation, and the wave equation. As far as I can tell, these operators are pretty mysterious. From looking at the formula, it’s not even obvious they should commute, and further properties are even more surprising. In this post I’ll tell you some of the surprising things about them, but I unfortunately won’t be able to say much about why they exist.