# sobolev spaces – differentiation into \$ L ^ p \$ and weak differentiation.

In Weiss and Stein's book on Fourier analysis in Euclidean spaces, they define the notion of & 39; derivative & # 39; in $$L ^ p$$& # 39 ;. To be precise, they define the difference operators

$$( Delta_h f) (x) = frac {f (x + h) – f (x)} {h}$$

and say that $$f$$ has a derivative $$g$$ in the $$L ^ p ( mathbf {R} ^ n)$$ if $$Delta_h f to g$$ in that $$L ^ p$$ Norm as $$h to 0$$, I have not met this definition yet. It seems as if it should correspond to some kind of weak differentiation, or at least be related in some sense? However, I can not find other references that speak about this type of construction in the language used by Stein and Weiss. Is there another name for this term and how is it related to a weak differentiation?