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>.



Looking for references to study $U^p$ and $V^p$ spaces
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**](http://dx.doi.org/10.1007/978-3-0348-0736-4), 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](https://zbmath.org/?q=an:1304.35003).


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](https://mathoverflow.net/questions/419075/looking-for-references-to-study-up-and-vp-spaces#comment1076145_419075) 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.


