Consider the following equation $$ g(x)=\int_0^\infty t \frac{\partial f(x,t)}{\partial t}\,dt $$ Is it possible to write $f(x,t)$ in terms of $g(x)$?
My attempt: I proceeded by differentiating both sides with respect to $t$. However, the term $g(x)$ is independent of $t$, leading to a vanishing derivative on the left-hand side. On the right, differentiation under the integral sign results in $$ 0 = \int_0^\infty t \frac{\partial^2 f(x,t)}{\partial t^2}\,dt. $$ This suggests that $f(x,t)$ could be a function whose second partial derivative with respect to $t$, when multiplied by $t$, integrates to zero over the given domain. Yet, this condition alone does not seem sufficient to uniquely determine $f(x,t)$ from $g(x)$.
Given the integral's insensitivity to the form of $f(x,t)$ for a fixed $g(x)$, additional constraints or information about $f(x,t)$ seem necessary to make further progress.