Denote by $(t, x_{1}, \ldots, x_{d-1})$ the coordinates of $x \in \mathbb{R}^{d}$ and set $$\mathbb{R}^{d}_{+}=\left\{t, x_{1}, \ldots, x_{d-1} \in \mathbb{R}^{d}|t > 0\right\}. $$ Write $\theta$ for the reflection $$(t, x_{1}, \ldots, x_{d-1}) \mapsto (-t, x_{1}, \ldots, x_{d-1}).$$ as well as for the induced map on $L^2(\mathbb{R}^{d})$.
The resolvent $C=(1-\Delta)^{-1}$, where $\Delta$ is the Laplacian on $\mathbb{R}^{d}$, is known to be reflection positive on $L^2(\mathbb{R}^{d})$, i.e., one has $$ \langle C f ,\theta f \rangle_{L^2} \geq 0$$ for all $f \in C^{\infty}_{c}(\mathbb{R}^{d}_{+})$, the compactly supported smooth functions on $\mathbb{R}^{d}_{+}$.
Fix $s \in \mathbb{R}$ and set $\rho(x)=(1+\|x\|)^s$ where $x \in \mathbb{R}^{d}$. Is $C$ reflection positive on the weighted $L^2$-space with weight $\rho$ ? In other words, is it true that $$ \langle C f ,\rho \hspace{2pt}\theta f \rangle_{L^2} \geq 0$$ for all $f \in C^{\infty}_{c}(\mathbb{R}^{d}_{+})$ ?
