Let $f \in L^p(R)$, if exists $h \in L^p(R)$ such that $$\displaystyle{lim_{y\to 0} \int_R \left|\frac{f(x+y)-f(x)}{y} -h(x) \right|^p dx = 0} $$ The function $h$ is called the derivate $L^p$ of $f$. Prove that, if $g \in L^q(R)$, and the derivate $L^p$ of $f$ is $h$, then the point derivate of $f*g$ is $h*g$
And show if $f \in L^p(R)$ and the derivate $L^p$ of $f$ exists, then $f$ is absolutely continuous in any bounded interval its point derivate in $L^p$.
Update. So far I have already managed to rpove thw first part:
$$ \left| \frac{f*g(x+y)-f*g(x)}{y} -h*g(x) \right| \to 0 $$ For this I used Hölder's inequality. For the next part I have as a hint to use aproximation of identity, but I don't know how to use it.