Consider the function given by $$ F(x)=\int_0^\infty e^{-\int_0^t\int_{x-vs}^{x+vs}f(y)\,dyds} \,dt $$ Is it possible to invert this and write $f$ in terms of $F$?
Some thoughts: If $F$ was in the form $$ F(x)=\int_0^\infty f(t)e^{-xt} \,dt $$ I could potentially use the inverse of a Laplace transform. If, for example, $f(x)=x$, we get $$ F(x)=\frac12 \sqrt{\frac{\pi}{xv}} $$ but the general case might not be invertible. Any ideas?