Description of Bessel potential spaces

Question

Hi, let $1 < p <\infty $, $ 0 < \alpha < 1$, and $ \mathscr{L}^p_\alpha(R^n) $ be the usual Bessel potential space defined by $$ \mathscr{L}^p_\alpha = (1-\triangle)^{-\alpha/2}L^p(R^n). $$ Let $ h \in R^n $, $ f \in L^p $ we denote $$ \omega_p(h) = \| f(x+h) - f(x) \|_{L^p}. $$

This quantity can be used to describe the space $ \mathscr{L}^p_\alpha $. In fact, for $ p = 2 $ the following are equivalent:

$$ f \in \mathscr{L}^2_\alpha $$

$$ f \in L^2 , and \int_{R^n}\frac{(\omega_2(h))^2}{|h|^{n+2\alpha}}dh < \infty . $$

For general $ p \in (1,\infty) $, only partial results of this kind are available in a Chinese book. Could someone give me some references for recent progress on this topic?

Answer 1

For integer $\alpha$, one can prove using Calderon-Zygmund theory that the Bessel potential spaces coincide with the Sobolev spaces. A standard reference for this would be Stein's harmonic analysis books.

There are some online lecture notes by Alessandra Lunardi which contain the following assertions: $$B^s_{p,p}(R^n) \subset H^{s,p}(R^n) \subset B^s_{p,2} (R^n) \text{ for } 1 < p\leq 2,$$ $$B^s_{p,2}(R^n) \subset H^{s,p}(R^n) \subset B^s_{p,p}(R^n) \text{ for } 2\leq p < \infty,$$ and the inclusions are strict if $p \neq 2$. The Besov spaces $B^s_{p,q}$ correspond to your second norm, while the $H^{s,p}$ spaces are your Bessel potential spaces.

There are several references you might want to look at to learn more about this subject. Bergh and Lofstrom is a classic. Luc Tartar has an entertaining introduction to the subject which is available online. The tome by Hans Triebel (referenced in A. Lunardi's lecture notes) probably contains the answer to your question, but it might be hard to locate.

Answer 2

You can find more about Bessel potential spaces in the book Sobolev spaces, Academic, 1975, by Adams. You will find there some Sobolev-type embeddings in Bessel potential spaces.

Best regards,