Looking for references to study $U^p$ and $V^p$ spaces

Question

I am studying some papers in the analysis of nonlinear PDEs and I am encountering the $U^p$ and $V^p$ spaces for the first time. Where can I find references more detailed than papers?

Edited

The spaces I mentioned above are defined in section two of this paper:

https://arxiv.org/pdf/0708.2011.pdf.

Answer 1

You can take a look at Herbert Koch's contribution in

Koch, Herbert; Tataru, Daniel; Vişan, Monica, Dispersive equations and nonlinear waves. Generalized Korteweg-de Vries, nonlinear Schrödinger, wave and Schrödinger maps, Oberwolfach Seminars 45. Basel: Birkhäuser/Springer (ISBN 978-3-0348-0735-7/pbk; 978-3-0348-0736-4/ebook). xii, 312 p. (2014). ZBL1304.35003.

Chapter 4 of Koch's notes there is specifically concerning the $U^p$ and $V^p$ spaces.

You should note that these notes are from 2014, and so there are a few newer developments covered in the PhD thesis mentioned in Raffaele Scandone's comment that are not included in this discussion. Though for reaching the paper you linked to certainly the material in the book should be more than sufficient.

Answer 2

Here is one fairly recent paper of Sebastian Herr (the second author in the paper you linked) and Timothy Candy in which they use $U^p$ and $V^p$ spaces. During a winter school in Obergurgl in 2022, Herr said that paper is a good reference for the $U^p-V^p$ spaces. It is probably worth having a look, although in that case they use the spaces for a nonlinear wave equation, which makes things not that simple as it is of second order in time (unlike the KP-II equation). That is why you will find two distinct adapted function spaces, $U^2_{\pm}$.

The book mentioned by Willie Wong was very helpul to me as it takes the time to explain things in detail; I almost didn't look at the original paper by Hadac, Herr, Koch as the main argument is all contained in that book. I didn't know of this PhD thesis from Scandone's comment, it seems it is quite a valid reference which confronts all the previous ones (it also mention the paper of Candy and Herr).

Edit:

A list of all the references I know:

• Koch, Tataru - 2005 (The first published paper which uses these spaces; very technical)
• Koch, Tataru - 2007 (A priori bounds for the 1D cubic NLS in negative Sobolev spaces; I have seen there are nice heuristics in the explanation of the spaces)
• Hadac, Herr, Koch - 2009 (Well... the paper on the KP-II equation; historically, it's the first nonlinear problem in which it seems that the $X^{s,b}$ spaces by Bourgain do not work and the use of $U^p-V^p$ seems somewhat necessary)
• Koch, Tataru, Visan - 2014 (The book; a very nice introductory reference which compares several models; see the first part (Koch's notes); contains most of the arguments from the paper on the KP-II equation in a simplified way)
• Koch, Tataru - 2018 (Very detailed (see appendix B), maybe a bit technical; a bit more recent than the book)
• Candy, Herr - 2018 (On the division problem for the Wave maps equation; very well-written and understandable)
• PhD thesis of Andreas Geyer-Schulz - 2019 (It seems a good and detailed reference which compares the previous ones)

There are surely many other papers that make use of these spaces. I think it is worth having a look at most of the above ones.