Are there any interactions between abstract harmonic analysis and harmonic analysis of real variables?

Question

I am a student mainly interested in mathematical analysis, and recently I had a course on harmonic analysis of real variables. In the course, I studied something like Paley-Littlewood decompositions, Calderon-Zygmund theory, Fourier multiplier/ pseudodifferential operators, and its multilinear analogues(paraproducts and the Coifman-Meyer theory).

I also heard that there is a field called "abstract harmonic analysis", which does some Fourier analysis on locally compact abelian groups (with some extension to nonabelian ones) or Lie groups. So I picked up Folland's book on abstract harmonic analysis. I anticipated the standard Euclidean theories (Littlewood-Paley or Calderon-Zygmund) being extended to topological groups, but the contents of the book were quite different from my expectations.

So here is my question: Are there any interactions between abstract harmonic analysis and harmonic analysis of real variables? (Of course, they are both motivated by Fourier series/Fourier transforms, but I want some more modern connections between these two fields of mathematics.) To be more specific, I want to know whether the real variable theory has direct analogues in the abstract setting and whether there exist applications of the abstract side to some concrete problems (especially PDEs).

Answer 1

In my opinion, the difference between the fields is less the domain you are working on, but more on the types of techniques you use. In the 'real-variable harmonic analysis' you discuss, the primary techniques are quantitative, i.e. you are often trying to bound some quantity in terms of some other quantity, most often to show some operator is bounded. On the other hand, in 'abstract harmonic analysis', the techniques are much more qualitative, relying more on topological techniques, weak compactness, etc. Both approaches are necessary to the general development of the field. Of course, the topology of $\mathbf{R}^n$ is very necessary to developing Littlewood-Paley / Calderon-Zygmund type techniques, so you shouldn't expect these approaches to generalize to all locally compact groups. But there are certainly groups 'like' $\mathbf{R}^n$, i.e. Lie groups, that do have similar theories. When it comes to applications of soft-type analysis in PDEs, I'm not too knowledgeable about those that could be directly considered as 'abstract harmonic analysis', but results like the Atiyah-Singer index theorem could be considered as soft-analytical results that have major importance in PDEs.