Consider a "weight" function $\rho: \mathbb{R}^{d} \rightarrow \mathbb{R}$ and the following weighted Sobolev norm: \begin{equation} \|h\|_{\dot{H}^{1}(\rho;\mathbb{R}^{d})}:=(\int_{\mathbb{R}^{d}} \|\nabla h(x)\|^{2}\rho(x) d x)^{1/2} \end{equation} where $dx$ stands for the Lesbegue measure on $\mathbb{R}^{d}$. When $\rho(x)=1$ everywhere we recover the "classical" Homogeneous Sobolev norm: \begin{equation} \|h\|_{\dot{H}^{1}(\mathbb{R}^{d})}:=(\int_{\mathbb{R}^{d}} \|\nabla h(x)\|^{2} d x)^{1/2} \end{equation} and in this case we have also a characterization of the dual norm $\dot{H}^{-1}(\mathbb{R}^{d})$ via the Fourier transform, which writes: \begin{equation} \|\nu\|_{\dot{H}^{-1}(\mathbb{R}^{d})}:=\sup_{\|h\|_{\dot{H}^{1}(\mathbb{R}^{d})}\leq 1} |\langle h,\nu\rangle|=(\int \|\omega\|^{-2} |\hat{\nu}(\omega)|^{2} d \omega)^{1/2} \end{equation} for $\nu$ a finite measure and $\hat{\nu}$ denotes its Fourier transform. My question is the following: do we have the same kind of characterization via the Fourier transform for the dual norm of the weighted Sobolev under some conditions on $\rho$ ?
I am looking for examples in simple cases where $\rho$ is the indicator of a compact of $\mathbb{R}^{d}$ or when it is a Gaussian density function.
