Consider a function $F$ on the half space $\{(x,y,z)|z>0\}$. If $F$ is analytic, it is straightforward to show that 


A) The integral of $F$ over the hemisphere $(x-x\_0)^2 + (y-y\_0)^2 + z^2 = R^2$ vanishes for all $x\_0$, $y\_0$, and $R$.


implies


B) $F = 0$ everywhere


My question is whether this is known to be true for any continuous function (or under some less restrictive conditions).



Vanishing of integral on hemispheres implies vanishing of function?
As @alvarezpaiva notes, this is a question about hyperbolic hyperplane (Radon) transform. This has been studied. See, e.g., Kurusa, the Radon transform in hyperbolic space [Geometriae Dedicata, 1991] (available for free, thanks to Springer). and references therein (Kurusa inverts it on fairly natural subspaces of $L^2,$ I am not sure if his results are optimal.


