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?