Reference on periodic Besov spaces

Question

I am looking for a reference for the construction and the study of the main properties of Besov spaces on the torus (i.e. $\mathrm{B}_{p,q}^s(\mathbb{T}^d))$.

Indeed, I know the classical $\mathrm{B}_{p,q}^s(\mathbb{R}^d)$ on the whole space and I would like to know if the theory is the same when we have periodicity. This is mainly a bibliographic question since I think the construction is quite the same, replacing Fourier transform by Fourier series (but I wonder if we could easily 'deduce' the periodic case from the euclidean case).

Thank you !

Answer 1

I found the first two references below on MSE or maybe MO somewhere but where exactly eludes me. Regardless: first, there is a set of notes by Gubinelli and Perkowski,

We will use Littlewood–Paley blocks to obtain a decomposition of distributions into an infinite series of smooth functions. Of course, we have already such a decomposition at our disposal: $f = \sum_k \hat f(k)e^*_k$. But it turns out to be convenient not to consider each Fourier coefficient separately, but to work with projections on dyadic Fourier blocks.

Secondly, there is Chapter 3 'Periodic Spaces' of Schmeisser and Triebel's book [ST]. Schmeisser and Triebel do not deduce the periodic case from the euclidean case, but reprove each theorem.

More recently I have stumbled upon a paper [DHWX] of Dai, Hu, Wu, and Xiao that (re?)develops the theory they need and then applies it to a certain PDE called the Boussinesq equation.

I have read none of these carefully, but I hope they are useful to you.

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[ST] Schmeisser, Hans-Jürgen; Triebel, Hans, Topics in Fourier analysis and function spaces, Chichester: John Wiley & Sons. 300 p.; \textsterling 23.95 (1987). ZBL0661.46025.

[DHWX] Dai, Yichen; Hu, Weiwei; Wu, Jiahong; Xiao, Bei, The Littlewood-Paley decomposition for periodic functions and applications to the Boussinesq equations, Anal. Appl., Singap. 18, No. 4, 639-682 (2020). ZBL1444.42023.