I am taking a basic course in harmonic analysis right now. Going in, I thought it was about generalizing Fourier transform / series: finding an alternative representation of some function where something works out nicer than it did before.

Now, having taken the first few weeks of this, it is not at all about Fourier analysis but about the Hardy-Littlewood-Maximal-operator, interpolation theorems, Stein's theorem/lemma and a lot of constants which we try to improve constantly in some bounds. We are following Stein's book on singular integrals, I guess.

Can anyone tell me where this is leading? Why are we concerned with this kind of operators and in which other areas are the results helping?