Recently I have been reading Singular Integrals and Differentiability Properties of Functions by E. M. Stein, where Chapter IV is devoted to Littlewood–Paley theory. Whereas Stein explains clearly the main tools and features of the theory (e.g., the use of $g$ and $g^*_\lambda$ functions, dyadic decompositions, multipliers), after finishing the whole chapter I got totally lost. All the results look technical, and I have no idea what this theory is trying to accomplish...
Questions
- How would you explain the idea and the goal of the theory?
- What are some typical applications of the theory, which you think can convince the beginner that it is really powerful and indispensable?