Is Fourier transform method suitable for solving equation $\int g(x-t)e^{-t^2} dt = e^{-a|x|}$ 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.
