Observe that given a non negative function $\omega: \mathbb{R^2} \rightarrow [0, \infty)$, we can define the weighted $L^{p}(\mathbb{R}^2, \omega) $ spaces. They are measurable functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ such that their weighted $L^p$ norm is finite i.e., $$ \int |f|^p \omega d \mu < \infty. $$ Similarly, we can define the weighted Sobolev spaces $W^{k,p}(\mathbb{R}^2, \omega)$. They are the completion with respect to the weighted $L^p$ norm of the spaces of $k$ times differentiable functions that have finite weighted Sobolev $p$-norm. Now consider the linear operator $$ T: W^{k,p}(\mathbb{R}^2, \omega) \rightarrow W^{k-1,p}(\mathbb{R}^2, \omega) $$ given by $$ T(v(x,y)) := \frac{\partial v(x,y)}{ \partial x}.$$ My question is as follows: does there exist some weight $\omega(x,y)$ and integers $k$ and $p$ such that the following are true:

1. The functions $(x,y) \rightarrow x$, $(x,y) \rightarrow e^x$, $(x,y) \rightarrow y$ and $(x,y) \rightarrow e^y$ all belong to $W^{k,p}(\mathbb{R}^2, \omega)$ and $T$ maps into $W^{k-1,p}(\mathbb{R}^2, \omega)$.

2. The map $T$ is a compact operator i.e., it takes bounded sequences to convergent subsequences.

3. The map $T$ is surjective.

My guess is that $$ \omega(x,y) := e^{-(x^2+y^2)} $$ should do the trick, but I am not certain. The motivation for this question is to show that a certain pde has a solution, via the Implicit Function Theorem. The map $T$ arises as the linearization of some non linear operator.