We are a small group of people that would like to start a reading course with the topic 'Function Spaces'. So far, we have all attended some graduate courses in Functional Analysis, Measure Theory, PDEs and some Harmonic Analysis (about the first four chapters of Grafakos' first book). Especially, we enjoyed the part about Sobolev Spaces a lot and we would like to deepen our knowledge in that area. Furthermore, we would like to enter that huge zoo of function spaces that are important in the modern theory of PDEs (like Besov, Morrey-Capanato, Triebel-Lizorkin, Hardy, BMO, etc.), especially in the theory of regularity of solutions.
However, we are a bit overwhelmed by the vastness of this field and don't know where to start and what to read.
We intend to start with the paper Hitchhiker’s guide to the fractional Sobolev spaces first since we have no background knowledge in fractional Sobolev spaces yet and it seems to be a good resource. But what is next?
Sure, we thought about the classical books like Triebel's 'Theory of Function Spaces' but in this book, a lot of proofs are missing. Also, we found the book 'Morrey and Campanato Meet Besov, Lizorkin and Triebel' but it turned out to be very hard to read for a beginner.
Could anybody give us some advice? We would be very grateful for a step-by-step listing of topics that we should study, together with some good resources (possibly with some exercises!)
Thank you very much! =)