The basic idea of the Littlewood-Paley Theory consists in a localization procedure in the frequency space. The interest of this method is that the derivatives (or more generally the Fourier multipliers) act in a very special way on distributions which Fourier transform is supported in a ball or a ring.
To explain this, I will first present and prove the Bernstein's Lemma. From this lemma we will define a specific dyadic partition of unity that will be the main tool to construct the Besov spaces. I will then give you multiple properties of those spaces and especially about the paradifferential calculus that turns out to be very useful in the study of PDE.
I will then give some examples of why those spaces are useful in the theory of PDEs.
