Update: See edit below.
I'm working on a homework problem. It states:
For what $s$ does the inclusion $V\in H^1(\mathbb{R})$ imply that the map $u\mapsto Vu$ is continuous as a map from $H^1(\mathbb{R})$ to $H^1(\mathbb{R})$
Thoughts
Naively, I was thinking that we must at least have $s\leq 1$, but this isn't necessarily true since the product of even $L^2$ functions may not belong to $L^2$.
Next, I looked to my favorite theorem for Sobolev spaces to get some direction: the Sobolev embedding theorem. I feel like we must at least have $s>3/2$, because then $H^s\subset C^1_0(\mathbb{R})$ (here $C^1_0$ is the space of continuous functions which, along with their derivatives, vanish at infinity) and so $V$ and $V'$ are uniformly bounded on $\mathbb{R}$. Therefore making some rough estimates I believe I can show the map $u\mapsto Vu$ is bounded in this case. I provide my "proof" below for those who are interested.
Proof. Set $s>3/2$. In this case, the Sobolev embedding theorem shows that $H^s\subset C^1_0$; in particular $V\in C^1_0$. Another application of the Sobolev embedding theorem shows that $H^1\subset C_0$. Therefore \begin{align*} \|Vu\|_{H^1}^2&=\int_{\mathbb{R}}|Vu|^2dx+\int_{\mathbb{R}}|V'u+Vu'|^2du\\ &\leq \sup_{x\in\mathbb{R}}|V(x)|^2\int_{\mathbb{R}}|u|^2dx+\int_{\mathbb{R}}|V'u|^2+2|V'Vu'u|+|Vu'|^2dx\\ &\leq \sup_{x\in\mathbb{R}}|V|^2\int_{\mathbb{R}}|u|^2dx+\sup_{x\in\mathbb{R}}|V'|^2\int_{\mathbb{R}}|u|^2dx+\int_{\mathbb{R}}2|(V'u')(Vu)|dx+\sup_{x\in\mathbb{R}}|V|^2\int_{\mathbb{R}}|u'|^2dx. \end{align*} By applying the inequality $2ab\leq a^2+b^2$ to the third integral and proceeding with the same estimates, we get $$ \|Vu\|_{H^1}\leq C\|u\|_{H^1} $$ for some $C>0$. Therefore the map $u\mapsto Vu$ is continuous from $H^1$ to $H^1$. $\square$
This might hold for a smaller $s$ since I only need uniform boundedness of derivatives (which would then have to only exist in the sense of distribution) for this argument.
Any help is greatly appreciated. Thank you.
I apologize for posting for homework help so much as of late. I am having a rough semester. As usual please only hints as I'm still attempting to hold on to some integrity.
Edit: I've realized that Sobolev space $H^s(\mathbb{R}^n)$ is an algebra with $2s>n$ is extremely close to my question, and with its help I've shown that we have the result when $s\geq 1$.
I'm now wondering if this is the optimal choice of $s$. Does this work for any smaller $s$? If not, then is there a simple counterexample?
