I have trouble finding the regularity of the solutions to a particular equation. I define $$\mathcal{L}f(x)=f''(x)+x^2f'(x)+ \operatorname{p.\!v.\!\!}\int_{-\infty}^{+\infty} \dfrac{f'(t)e^{-t^2}}{t-x}dt$$
I showed that the operator is invertible on a subspace of $\big\{f\in H^1(e^{-t^2}dt), \int_\mathbb{R} f(t) e^{-t^2}dt=0\big\}$ in itself so there exists a unique solution in that space.
What am i looking for ? I want to characterize the image $$ \mathcal{L}\big(H^3(\mathbb{R})\cap\mathcal{C}^3(\mathbb{R})\big)\triangleq F. $$ I was able to show that $$ F\subset H^1\bigg(\dfrac{dt}{t^2+1}\bigg)\cap\mathcal{C}^1(\mathbb{R})\cap\big\{g,g(x)=o(x^2) \text{ at }\infty\big\}. $$ If it's too difficult to precisely characterize $F$, I'd like to know if there is some general function space contained in $F$. Thank you in advance.
