Littlewood-Paley theory in PDEs

The aim of this seminar is to learn basics of a fundamental tool in harmonic analysis, namely the Littlewood-Paley decomposition, keeping in mind its applications to nonlinear PDEs.

The Littlewood-Paley decomposition boils down to representing a function \(f\) by a sum of functions having only certain frequencies, i.e. \[ f = \sum_{k \in \mathbb{Z}} P_kf \] where \(P_kf\) have frequencies localised to \(\sim 2^k\). In other words, it is 'localisation on the Fourier side' or 'a pointwise discrete approximation of the Plancharel theorem'. Despite simplicity of this classical idea (having its roots in 1930s) it is widely applicable to PDEs, since it allows a careful control of derivatives (via Bernstein inequalities) and nonlinearities (via Bony's paraproduct). Therefore, after a short introduction of harmonic analysis preliminaries, we intend to focus on Littlewood-Paley methods in PDEs, including

• multiplier theorems
• useful definition of Besov spaces and the related solvability of nonlinear evolutionary problems for scaling invariant initial data
• commutator estimates with applications to turbulence