Given a strictly positive function $a:[0,1]\longrightarrow \mathbb{R}$, I have a question about the generation of a $C_0$-semigroup on $L^p([0,1])$ ($1\le p<\infty$) by the following maximal operator $$A_mf=- \frac{d(af)}{dx}\quad\text{ for } f\in D(A_m)=\{f\in L^{p}([0,1]),\quad af\in W^{1,p}([0,1])\}.$$ It is well-known that for $a=1$, the semigroup in question is the one of nilpotent-translation on a subdomain of $D(A_m)$. However, for general $a$ I have no idea, since here I took $a$ as a general function and $A_m$ with maximal domain.

Another thing: It is clear that if $a$ is token to be in $W^{1,p}([0,1])\}$ the result can be deduced easily.

It would be great if you could provide me with some ideas and discussions in the general case: **Minimal assumptions on $a$ and $p$, in order to obtain a generator on a subdomain of the maximal domain $D(A_m)$**.

Thank you!

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