Besov spaces visualization

It is more general: the normal Sobolev spaces H\^s are based on L\^2 meaning that for being the Sobolev space of order k you need that all (weak) partial derivatives up to order k are in L\^2 Now remember that there are also Hölder spaces C\^{k,α} where you want your partial derivatives being Hölder-continuous. Besov space are now family of spaces that include both Hölder and Sobolev spaces as special cases and interpolate in the correct way. There is another scale of spaces called Triebel-Lizorkin spaces, they unify slightly different kinds of Sobolev spaces (in particular the integer order L\^p-based Sobolev spaces W\^{k,p}).

A good reference for these spaces is the series of monographs by Hans Triebel (called Function spaces 1-3 or something like that).

For the motivation of studying such spaces: if you study linear PDEs you are usually totally fine with H\^s. These are great since they are Hilbert spaces (even though you don't use that too often) and if you are on R\^d then you have simple characterization using the Fourier transform. Now if you want to study nonlinear PDEs, then you frequently run into the problem that the linearization of your nonlinear operator is no longer a continuous operator on L\^p for any fixed p, so you need more flexibility in your spaces. Sometimes W\^{k,p} suffices, but not always. What often happens in nonlinear PDEs is that your operator shifts frequencies around so you have to control different frequencies rather precisely and there is a great tool for that called Littlewood-Paley theory (or dyadic decomposition) and Besov spaces have a rather nice description in this dyadic decomposition. I am rather imprecise here, for real theorems look in for instance Tao's book on nonlinear dispersive equations if you want to learn mainly the PDE-side and don't want to be bothered with complicated spaces or Cazenave's book if you want to see Besov spaces in action.
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