Is Fourier transform method suitable for to solve the following equation \begin{align*} \int g(x-t)e^{-t^2/2} dt = e^{-a|x|} \end{align*}
Suppose we take the Fourier transform of the above equation \begin{align*} &G(\omega) \sqrt{2\pi} e^{-\omega^2/2}=\frac{2}{1+\omega^2}\\ &G(\omega)=\frac{2e^{\omega^2/2}}{\sqrt{2\pi}(1+\omega^2)} \end{align*}
But the function on the right is not integrable. What can we do in case like this? Is there a strategy to solving general cases like this?
The Fourier transform method is useful in showing that there is no solution.
Since $G$ is unbounded, what your argument shows is that there is no solution in $L^1$. It is clear also that $G$ is not in $L^2$, so that the is no solution in $L^2$. Moreover, $G$ is not a tempered distribution, so that there is no solution in the space of tempered distributions.