Suppose we have $u(x)\in H_2^{loc}(\Omega_{\rho})$, where $\Omega_{\rho}=\{x\in \mathbb{R}^n, |x|>\rho\}$, and in $\Omega_{\rho}$, $u$ satisfies the equation $$ \Delta u-V(x)u=0, $$ where $V$ is a bounded function. If we require that $\int_{\Omega_{\rho}}{|u|^2w(x)dx}<\infty$, here $w(x)$ is some weight function, for example, we choose $w(x)=\exp{|x|^{\alpha}}$, $\alpha>0$. Then my question is how to show that $$ \int_{\Omega_{\rho}}{|D^{\beta}u|^2w(x)dx}<\infty $$ is also true for $|\beta|\leq 2$?

I came across this problem when reading a paper, and reference the author offered there was Hormander's The Analysis of Linear Differential operators, the 2nd volume, where I failed to find something similar. I'm very appreciated that if someone can point out the exact theorem or proposition that corresponds to the above result in Hormander's book. Thanks in advance.

Suppose we have $u(x)\in H_2^{loc}(\Omega_{\rho})$, where $\Omega_{\rho}=\{x\in \mathbb{R}^n, |x|>\rho\}$, and in $\Omega_{\rho}$, $u$ satisfies the equation $$ \Delta u-V(x)u=0, $$ where $V$ is a bounded function. If we require that $\int_{\Omega_{\rho}}{|u|^2w(x)dx}<\infty$, here $w(x)$ is some weight function, for example, we choose $w(x)=\exp{|x|^{\alpha}}$, $\alpha>0$. Then my question is how to show that $$ \int_{\Omega_{\rho}}{|D^{\beta}u|^2w(x)dx}<\infty $$ is also true for $|\beta|\leq 2$?

I came across this problem when reading a paper, and the reference the author offered was Hormander's The Analysis of Linear Differential operators, the 2nd volume, where I failed to find something similar. However, I found that in the first section of Chapter 17, he gave the interior Shauder estimates by using fundamental solution and fractional integral. But the result is local and without weight. I don't know if this can modified to the case I concerned here. Another resource that maybe helpful is S Agmon, A Douglis, L Nirenberg's classical paper "Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions.I" , in thm 15.1, they established the interior estimated for general elliptic operators, but I don't know whether their methods can be applied to the case with weight like $w(x)=\exp{|x|^{\alpha}}$. I'm very appreciated that if someone can point out how this is done. Thanks in advance.

Suppose we have $u(x)\in H_2^{loc}(\Omega_{\rho})$, where $\Omega_{\rho}=\{x\in \mathbb{R}^n, |x|>\rho\}$, and in $\Omega_{\rho}$, $u$ satisfies the equation $$ \Delta u-V(x)u=0, $$ where $V$ is a bounded function. If we require that $\int_{\Omega_{\rho}}{|u|^2w(x)dx}<\infty$, here $w(x)$ is some weight function, for example, we choose $w(x)=\exp{|x|^{\alpha}}$, $\alpha>0$. Then my question is how to show that $$ \int_{\Omega_{\rho}}{|D^{\beta}u|^2w(x)dx}<\infty $$ is also true for $|\beta|\leq 2$?

Thanks in advance.

Suppose we have $u(x)\in H_2^{loc}(\Omega_{\rho})$, where $\Omega_{\rho}=\{x\in \mathbb{R}^n, |x|>\rho\}$, and in $\Omega_{\rho}$, $u$ satisfies the equation $$ \Delta u-V(x)u=0, $$ where $V$ is a bounded function. If we require that $\int_{\Omega_{\rho}}{|u|^2w(x)dx}<\infty$, here $w(x)$ is some weight function, for example, we choose $w(x)=\exp{|x|^{\alpha}}$, $\alpha>0$. Then my question is how to show that $$ \int_{\Omega_{\rho}}{|D^{\beta}u|^2w(x)dx}<\infty $$ is also true for $|\beta|\leq 2$?

I came across this problem when reading a paper, and reference the author offered there was Hormander's The Analysis of Linear Differential operators, the 2nd volume, where I failed to find something similar. I'm very appreciated that if someone can point out the exact theorem or proposition that corresponds to the above result in Hormander's book. Thanks in advance.