Constant for Sobolev multiplier We denote the set of functions $q : \mathbb R\to\mathbb C$ for which
$$
\|q\|_{L^2_u} := \left(\sup_{n\in\mathbb N}\int_n^{n+1}|q(x)|^2\,dx\right)^{1/2} < \infty
$$
by $L^2_u(\mathbb R)$ (uniformly locally $L^2$). It is well-known that $L^2_u(\mathbb R)$ is exactly the class of functions $q$ for which $qf\in L^2(\mathbb R)$ for all $f\in H^1(\mathbb R)$. Each such function induces a bounded operator from $H^1(\mathbb R)$ to $L^2(\mathbb R)$. Hence, there exists $C = C_q> 0$ such that
$$
\|qf\|_2\,\le\,C_q\|f\|_{H^1}.
$$
The constant $C_q$ can be bounded by $\|q\|_{L^2_u}$, that is, we have
$$
\|qf\|_2\,\le\,C\|q\|_{L^2_u}\|f\|_{H^1}
$$
for all $q\in L^2_u(\mathbb R)$ and all $f\in H^1(\mathbb R)$.
I would like to know the best constant $C$ for this. I have derived one, but it is very bad. I got it in the following way. First, for $q\in L^2$ one has
$$
\|qf\|_2 = \|\hat q * \hat f\|_2\,\le\,\|q\|_2\|\hat f\|_1\,\le\,\sqrt\pi\|q\|_2\|f\|_{H^1}.
$$
Now, I let $q\in L^2_u$ and pick a smooth partition of unity $(\phi_n)$ with $\operatorname{supp}\phi_n\subset [n-1,n+1]$. Then
$$
\|qf\|_2^2 = \sum_n\|q(\sqrt{\phi_n}f)\|_2^2\,dx\,\le\,\pi\sum_n\|q\|_{L^2(n-1,n+1)}^2\|\sqrt{\phi_n}f\|_{H^1}^2\,\le\,2\pi\|q\|_{L^2_u}^2\sum_n\|\sqrt{\phi_n}f\|_{H^1}^2.
$$
Now, it remains to estimate the last sum:
\begin{align*}
\sum_n\|\sqrt{\phi_n}f\|_{H^1}^2
&= \sum_n\int\phi_n|f|^2 + \Big|\frac{\phi_n'f}{\sqrt{\phi_n}} + \sqrt{\phi_n}f'\Big|^2\,dx\\
&\le 2\|f\|_2^2 + 2\sum_n\int\Big|\frac{\phi_n'f}{\sqrt{\phi_n}}\Big|^2 + \phi_n|f'|^2\,dx\\
&\le 2\|f\|_2^2 + 4\|f'\|_2^2 + 2\sum_n\int_{n-1}^{n+1}\frac{|\phi_n'|^2}{\phi_n}|f|^2\,dx\\
&\le (2+4S)\|f\|_2^2 + 4\|f'\|_2^2,
\end{align*}
where $S = \sup(|\phi_0'|^2/\phi_0)$, because the $\phi_n$ are just translated versions of each other. Now, my $S$ is about $144$ or so. It would be much nicer to have it close to one.
So, I am either looking for a partition of unity $(\phi_n)$ for which $|\phi_n'|^2/\phi_n$ is small or for just another way of treating the whole thing. Maybe someone here has a reference?
 A: I have been able to reduce the constant to a minimum (at least with respect to the above technique). I define
$$
\phi_0(x) :=
\begin{cases}
2(1+x)^2 &\text{if }x\in [-1,-\tfrac 1 2],\\
1-2x^2 &\text{if }x\in[-\tfrac 12,\tfrac 12],\\
2(1-x)^2 &\text{if }x\in [\tfrac 12,1]
\end{cases}.
$$
Then $\phi_0\in C^1(\mathbb R)$, its support is $[-1,1]$ and $|\phi_0'|^2/\phi_0\le 8$. Moreover, the functions $\phi_n(x) = \phi_0(x-n)$ are a partition of unity as above. Since also $|\phi_0'|\le 2$, we get
\begin{align*}
\sum_{n\in\mathbb Z}\big\|\sqrt{\phi_n}f\big\|_{H^1}^2
&= \sum_{n\in\mathbb Z}\int_{n-1}^{n+1}\left(\phi_n|f|^2 + \Big|\frac{\phi_n'}{2\sqrt{\phi_n}}f + \sqrt{\phi_n}f'\Big|^2\right)\,dx\\
&= \sum_{n\in\mathbb Z}\int_{n-1}^{n+1}\left(\phi_n|f|^2 + \frac{|\phi_n'|^2}{4\phi_n}|f|^2 + \phi_n|f'|^2 + \phi_n'\operatorname{Re}(\overline{f}f')\right)\,dx\\
&\le \sum_{n\in\mathbb Z}\int_{n-1}^{n+1}\left(3|f|^2 + |f'|^2 + 2|f'f|\right)\,dx\,\le\,8\|f\|_{H^1}^2.
\end{align*}
Thus,
$$
\|qf\|_2\,\le\,4\sqrt\pi\|q\|_{L^2_u}\|f\|_{H^1}.
$$
