Do abstract harmonic analysts use topological methods to solve problems or deal with Borel hierarchy?

Question

Question: Does there exist a branch in harmonic analysis that uses topological method to solve problems?

For example, in topology, partition of unity allows one to construct collection of bounded continuous functions with their supports form a closed covering. In fact, the collection of functions can be infinitely differentiable. Compactification allows one to apply any compactness techniques, and much more.

Also, do harmonic analysts deal with Borel hierarchy in their work? For example, complexity of Borel sets, $\Sigma^0_\xi$ and $\Pi^0_\xi$ sets for any countable ordinal $\xi.$

My motivation of this question comes from giving a talk at an abstract harmonic analysis seminar. As I will present a mixture of topology and Borel hierarchy stuffs where audience mostly work in abstract harmonic analysis, I would like to find out what interests abstract harmonic analyst the most from the field of general topology and descriptive set theory (in particular, Borel hierachy).

Answer 1

My guess would be that some harmonic analysts may use a little bit of the Borel hierarchy. But probably using the notation $F_\sigma$ and $G_\delta$ rather than $\mathbf\Sigma^0_\xi$ and $\mathbf\Pi^0_\xi$. And probably never going beyond the first few levels.

So when you define $\mathbf\Sigma^0_\xi$ and $\mathbf\Pi^0_\xi$ at first, maybe show where $F_\sigma$ and $G_\delta$ are in there.

Answer 2

Look at certain papers by people such as Etienne Matheron, Randall Dougherty, Alexander S. Kechris, Alain Louveau, Hugh Woodin, etc.

A good place to begin is Trigonometric series and set theory (2012) by Kechris, especially p. 9.

Note that most of this work is by people primarily interested in descriptive set theory, not harmonic analysis, so you'll want to google or otherwise search for papers (citing this work) that are written by people mostly interested in harmonic analysis to find something that might be of particular interest to a harmonic analyst.