Let $B\_t(0)$ denote the $n$ dimensional ball of radius $t$ centered at the origin. Does there exist a $\phi\in C(\mathbb{R}^n)$ function with the properties:


$
\phi (x) =
\begin{cases}
1&x\in B\_r(0)
\\\
0&x\not\in B\_{r+3}(0)
\end{cases}
$


and for any real-valued function $f\in \mathcal{H}^\tau(\mathbb{R^n})$ ($\tau\in\mathbb{R}$, $\tau>d/2$) we have 


$
\left\|\phi(\cdot) f(\cdot)\right\|\_{\mathcal{H}^\tau(A\_1)}\leq C\left\|f\right\|\_{\mathcal{H}^\tau(A\_2)}
$


where $C$ is a constant independent of $f$, $A\_1 = B\_{r+2}(0)\setminus B\_{r+1}(0)$ and $A\_2 = B\_{r+3}(0)\setminus B\_{r}(0)$. 


This has a similar feel to extension theory results if we think of $\phi$ as an extension operator which preserves $f$ on $B\_r(0)$.



Extension theory with bump function
**Short answer: yes.**


Let $\psi\_\epsilon(x):=\frac{1}{\epsilon^n}\exp{\epsilon^2/(\epsilon^2-|x|^2)}$ for $|x|<\epsilon$, and $\psi\_{\epsilon}(x)=0$ for $|x|\geq \epsilon$. Set $\epsilon=2$, and define
$\phi$ is the convolution of $C\phi\_{\epsilon}$ with the characteristic function of $B\_{r+3/2}(0)$, that is, 


**$\phi(x):= C\psi\_\epsilon(x)\* \chi\_{B\_{r+3/2}(0)}$.** Here $C$ is a normalizing constant (this may not be needed, but I haven't checked). 


This yields a smooth cut-off function which is 1 in the ball $B\_{r+1}(0)$, and zero outside $B\_{r+2}(x)$. 


To see this does the trick, one can use a localization theorem, for example, Theorem 3.20 in 'Strongly Elliptic Systems and Boundary Integral Equations' by W. McLean. This theorem states:


'Suppose that $\phi \in C^r\_{comp}(\mathbb{R}^n)$ for some integer $r\geq 1$, and let $|s|\leq r$. If $u\in H^s(\Omega)$ then $\phi u \in H^s(\Omega)$, and $||\phi u||\_{H^s(\Omega)} \leq C\_r||\phi||\_{W^{r,\infty}(\mathbb{R}^n)}Q\_u$ where $Q\_u=||u||\_{H^s(\Omega).
}$ *(Apologies, I encountered trouble while trying to typeset the LaTeX here).*


The same result holds with $H^s(\Omega)$ replaced with $\tilde{H}^s(\Omega)$.'


The proof proceeds using $\Omega = A\_2$, and then 


either by
(a) considering the situation for $s=r$, using duality to see it holds for $s=-r$, and the intermediate $s$ by interpolation. This is suggested by Yakov above.


or (b) by examining $\hat{\phi u}$ and using Peetre's inequality.


Since the constructed $\phi \in C^\infty$ and has compact support, it will satisfy the inequality you seek. In my comment I asked whether you wanted a $\phi$ of minimal regularity (relative to $\tau$); my construction works but may be overkill.


