Since the 1980s, Fourier analysis methods have become of ever greater interest in the study of linear and nonlinear partial differential equations. In particular, techniques based Littlewood-Paley decomposition have proven to be very efficient in the study of evolution and other equations. Littlewood-Paley decomposition was originally developed by Littlewood and Paley’s works in the early 1930s. It provides an elementary device for splitting a function into a sequence of spectrally well localized smooth functions and hence, differentiation acts almost as a multiplication on each term of the sequence. In particular, its systematic use for nonlinear partial differential equations is rather recent. The main breakthrough was archived after Bony introduced the paraproduct calculus. The first aim of these series talks is to describe that the rough frequency splitting supplied by Littlewood-Paley decomposition may still provide elementary and elegant proofs of some classical inequalities, such as Sobolev embedding and Hardy inequalities. The second is to devote to Besov spaces based on Littlewood-Paley decomposition and discuss some of their properties, such as topological properties, characterizations in terms of heat flow or finite differences, embedding in Lebesgue spaces, and Gagliardo-Nirenberg-type inequalities.
