Extension theory with bump function

Question

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 $B_r(0)$.

Answer 1

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.

Answer 2

I believe that the fractional Sobolev spaces can be defined as a complex interpolation space between the integer Sobolev spaces (see this http://www.scribd.com/doc/45316527/Sobolev-Spaces-2ed-Robert-a-Adams-John-J-F-Fournier for example). As noted in the comments, your question is easily seen to hold on the integer valued spaces. Then we can interpolate to get the result for the fractional spaces.