# Description of Bessel potential spaces

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?

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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.