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).