I have trouble finding the regularity of the solutions to a particular equation. I define $$\mathcal{L}f(x)=f''(x)+x^2f'(x)+p.v\int_{-\infty}^{+\infty} \dfrac{f'(t)e^{-t^2}}{t-x}dt$$

I showed that the operator is invertible on a subspace of $\{f\in H^1(e^{-t^2}dt), \int_\mathbb{R} f(t) e^{-t^2}dt=0\}$ in itself so there exists a unique solution in that space.

What am i looking for ? I want to know the image of $\mathcal{L}\Big(H^3(\mathbb{R})\cap\mathcal{C}^3(\mathbb{R})\Big)=:F$. I was able to show that $F\subset H^1\Big(\dfrac{dt}{t^2+1}\Big)\cap\mathcal{C}^1(\mathbb{R})\cap\{g,g(x)=o(x^2) \text{ at }\infty\}$. If it's too difficiult to precisely know $F$, I'd like to know if there is some general functionnal space contained in $F$. Thank you in advance.