# Weighted translation semigroup

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!

• 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? Feb 8, 2021 at 20:29
• 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". Feb 9, 2021 at 10:48
• There are many subdomains there corresponding to different boundary conditions where you get a generator. Feb 9, 2021 at 11:03
• If $a$ and $1/a$ are bounded, then you can use similarity transformation argumets to reduce your problem to the $a=1$ case. Feb 9, 2021 at 11:04
• Take $B=a\cdot$ and $C=\frac{d}{dx}$ and write out $B^{-1}CB$. Feb 9, 2021 at 14:05