Let $n \ge 2$. Set $\dot{H}^1(\mathbb{R}^n)$ to be the homogeneous Sobolev space, defined as the Hilbert completion of $C_0^\infty(\mathbb{R}^n)$ with respect to the norm $\| \varphi \|^2_{\dot{H}^1} \equiv \int_{\mathbb{R}^n} |\nabla \varphi(x) |^2dx = \int_{\mathbb{R}^n} |\xi|^2|\hat{\varphi}(x) |^2dx$, where the "hat" denotes Fourier transform.
Let $R_0(\lambda) = (-\Delta - \lambda^2)^{-1}$, $\text{Im }\lambda >0$ denote the resolvent of the free Laplacian. $R_0(\lambda)$ can be defined as a bounded operator $\dot{H}^1 \to \dot{H}^1$ by first giving the usual definition of $R(\lambda) \varphi$ via the Fourier transform of $\varphi \in \mathcal{S}(\mathbb{R}^n)$, and then noticing that
\begin{equation} \label{define free resolv dot H one}
\| R_0(\lambda) \varphi \|^2_{\dot{H}^1} = \int_{\mathbb{R}^n} \frac{|\xi|^2 |\hat{\varphi}(\xi)|^2}{|\xi|^2- \lambda^2}d\xi \le \sup_{r \in \mathbb{R}}{\frac{1}{r - \lambda^2}} \int_{\mathbb{R}^n} |\xi|^2 |\hat{\varphi}(\xi)|^2 = C(\lambda) \| \varphi \|^2_{\dot{H}^1}.
\end{equation}
So then $R_0(\lambda)$ may be extended to all of $\dot{H}^1$ by density.
I would like to know whether the cutoff resolvent $\chi R_0(\lambda) \chi,$ ($\chi \in C_0^\infty$) can be meromorphically continued from $\text{Im } \lambda > 0$ through the real spectrum and into all of $\mathbb{C}$ (or to the Logarithmic cover of $\mathbb{C}$ in the case of even dimensions), and if $\dot{H}^1 \to \dot{H}^1$ estimates can be obtained for the continuation.
For dimension $n \ge 3$, I believe the answer is yes because of the following fact, which is a consequence of combining the Gagliardo-Nirenberg-Sobolev inequalty along with Holder's inequality:
If $n \ge 3$ and $\chi \in C_0^\infty(\mathbb{R}^n)$, then there exists a constant $C_\chi> 0$ such that for all $u \in C_0^\infty(\mathbb{R}^n)$, we have $$\| \chi u \|_{L^2(\mathbb{R}^n)} \le C_\chi \| \nabla u \|_{L^2(\mathbb{R}^n)}.$$
This estimate allows one to take $u \in C_0^\infty(\mathbb{R}^n)$ and compute $$ \nabla \chi R_0(\lambda) \chi u= (\nabla\chi) \tilde{\chi} R_0(\lambda) \tilde{\chi} \chi u + \chi \tilde{\chi} R_0(\lambda) \tilde{\chi} \nabla \chi u + \chi \tilde{\chi} R_0(\lambda) \tilde{\chi} \chi \nabla u, $$
where $\tilde{\chi} \in C_0^\infty(\mathbb{R}^n)$ and $\tilde{\chi} \equiv 1$ on the support of $\chi$. By applying the above estimate when $n \ge 3$, we may lean on the $L^2 \to L^2$ bounds that are known for the continuation of $\tilde{\chi} R_0(\lambda) \tilde{\chi}$ (as shown in Chapter 3 of the book in progress by Dyatlov and Zworski) to achieve the desired bounds $\dot{H}^1 \to \dot{H}^1.$
In dimension two, I think there are counterexamples to the Sobolev-type inequality given above, and so I feel we cannot so easily obtain the continuation from $\text{Im } \lambda > 0$. I am currently stuck, and would like to know if anyone knows of reference where this or something similar is worked out.
Thank you very much for any suggestions.
