I am studying the book Wong, introduction pseudo differential operators and I have a question regarding the Sobolev space and how domain can be used to solve a partial differential equation.
For each $f\in L^2(\mathbb{R})$, the equation \begin{align} (I-\partial_x^2)u(x)=f(x) \end{align} has a solution $u$ in $H^{2,2}(\mathbb{R})$ given that \begin{align} u=\mathcal{F}^{-1}\left(\frac{1}{1+\xi^2}\widehat{u}(\xi)\right) \end{align} Here, $H^{2,2}(\mathbb{R})$ is \begin{align} H^{2,2}(\mathbb{R})=\left\{u \text{ tempered distribution }: \mathcal{F}^{-1}\left((1+\xi^2)^{1/2}\widehat{u}(\xi)\right)\in L^2(\mathbb{R})\right\} \end{align} and \begin{align} (I-\partial_x^2)u(x)=\mathcal{F}^{-1}((1+\xi^2)\widehat{u}(\xi))](x) \end{align}
My question is:
If $f\in L^2(\mathbb{R})$, the equation \begin{align} (1+x^2)(I-\partial_x^2)u(x)=f(x) \end{align} where \begin{align} (1+x^2)(I-\partial_x^2)u(x)=(1+x^2)\mathcal{F}^{-1}((1+\xi^2)\widehat{u}(\xi))(x) \end{align} has a unique solution $u\in H_{1+x^2}^{2,2}(\mathbb{R})$ where
\begin{align}H_{1+x^2}^{2,2}(\mathbb{R}):=\left\{u\text{ tempered distribution }: (1+x^2)\mathcal{F}^{-1}((1+\xi^2)\widehat{u}(\xi))(x) \text{ in } L^2(\mathbb{R})\right\}\end{align} given that
\begin{align} u(x)=\mathcal{F}^{-1}\left(\frac{1}{1+x^2}\mathcal{F}\left(\frac{f(x)}{1+x^2}\right)(\xi)\right)(x) \end{align}?