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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    $\begingroup$ I am sorry but I do not completely understand. Why do you get for $a=1$ the nilpotent shift? I do not see a boundary condition. What am I missing? $\endgroup$ Feb 8, 2021 at 20:29
  • $\begingroup$ I am sorry maybe I should be more specified. In the case $a=1$, the operator $A=A_m$ with domain $D(A)=\{f\in D(A_m):\, f(0)=0\}$ is the generator in question with the semigroup $(T(t)f)(s):=\chi_{[0,1]}(s-t)f(s-t)$. "I wrote it is a generator on a subdomain, $D(A)$ in this case". $\endgroup$ Feb 9, 2021 at 10:48
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    $\begingroup$ There are many subdomains there corresponding to different boundary conditions where you get a generator. $\endgroup$ Feb 9, 2021 at 11:03
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    $\begingroup$ If $a$ and $1/a$ are bounded, then you can use similarity transformation argumets to reduce your problem to the $a=1$ case. $\endgroup$ Feb 9, 2021 at 11:04
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    $\begingroup$ Take $B=a\cdot$ and $C=\frac{d}{dx}$ and write out $B^{-1}CB$. $\endgroup$ Feb 9, 2021 at 14:05


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