I am interested in the regularity of the solutions to the following partial differential equation on $\mathbb{R}^2$: $$ \left( a \frac{\partial}{\partial x} + \frac{\partial^2}{\partial y^2} \right) u = f, $$ where $a\neq 0$ is real and $f \in L_2\left(\mathbb{R}^2\right)$.
If we assume that $u\in L_2$, then I suspect that the solution $u$ sits in the Sobolev space $H^1(\mathbb{R}^2)$, but I have not been able to prove this. I have tried to look at this equation in the Fourier domain, but this did not lead to anything. Note that this equation is not elliptic, so we cannot use the known theory for elliptic differential equations to proceed. Can anyone provide me some insight in how to analyze this?