Sobolev Embedding Theorems, Besov Spaces, etc.
First we need the definition of Hölder and Sobolev spaces (Wikipedia).
The idea of Hölder spaces is that the k-th derivative of the functions are alpha-Lipschitz. The idea of Sobolev spaces is that the k-th derivate of the functions are in Lp space.
Then we have different Sobolev inequalities. These inequalities basically say which Sobolev space can be embedded in another one. Of course, if two spaces are measured with the same Lp norm, the one with higher degree of smoothness can be embedded in the less smooth one. But if we measure one with Lp and measure the other with Lq (p!=q), then the relation is more complicated and depends on p, q, smoothness degrees and the dimension n of the space R^n.
Then we have interesting Gagliardo-Nirenberg-Sobolev inequality which says for diff. function u: R^n —>R, ||u||_Lp* <= C || D u ||_Lp where p* is the Sobolev conjugate of p, and D is the differentiation operator.
Now, we have Besov spaces. As far as I understand, there are different ways to define it: Fourier (or Wavelets)-based approach and moduli of smoothness approach.
For example, see DeVore, Sharpley, “Besov Spaces on Domains in R^d” (Section 2).
Jan Schneider, Function Spaces with Varying Smoothness, Ph.D. Dissertation.
It starts by Fourier-analytical approach to define Besov spaces in R^n (Definition 2.1), reviews some embedding theorem (Theorem 2.2), and equivalent of some norms (Theorem 2.3) in Besov spaces. Then it defines the same thing for domains \Sigma \subset R^n, and proves similar results. Afterwards, it generalizes the same result to the case where the smoothness measurement changes. The motivation is that this way is better to capture pointwise singularities. Again, similar properties hold.
And finally a few other references:
Frayssee and Jaffard, “The Sobolev Embeddings are Usually Sharp”.
I am not sure if understand the paper correctly, but this is my take from its Introduction: For a Sobolev space W^s(Lp)(R^n), there exists a function belong to that space which has a Holder exponent of s - d/n (the Holder exponent more or less show the maximum pointwise smoothness of the function). Now, the question is whether this function belonging to the aforementioned Sobolev space is an exception or actually most functions in that Sobolev space has this Holder exponent of s - d/n.
