My Mathematica 13.2 result is mathematical nonsense
FourierTransform[ HeavisideTheta[t1^2 + t2^2 + t3^2 - a],
{t1, t2, t3},
{\[Omega]1, \[Omega]2, \[Omega]3} ]
2 Sqrt[2] \[Pi]^(3/2) DiracDelta[\[Omega]1] DiracDelta[\[Omega]2] DiracDelta[\[Omega]3]
The problem is the transformation from cartesian to spherical coordinates.
By spherical symmetry, the problem can be simplified to the simple choice of the result vector
\[Omega] = {0 ,0 , \Omega3}
Then you have to calculate the integral over the z-axis with
z=cos theta, dz = sin \theta d\theta
2 Pi \Integrate[ t^2 * Exp[ I \[Omega ] t z ] ,{z,-1,1}, {t,a, Infinity}
Keep in mind, this is a spherical distribution in spherical coordinates, where \CapitalOmega is the absolute value of the cartesian vector \omega, tas a map onto the complex numbers to be applied onto smooth functions with compact support, that projects to the mass outside of a sphere.
In order to get the limits, a cut-off factor of the t-integral has to be introduced
t -> exp[-t^2/s^2]/\sqrt[2 PI s^2]
The
\int_a^Infinity t^2 Exp[ i \omega t ]
does not make any sense.
Perhaps, anywhere the idea is introduced errorneously, that
Fourier[ t^n f[ t ] , t , omega ] = D[ Fourier[ f[ t ] , t, \omega ] , { \omega, n } ]
The formula is not applicable of course, if a non-constant measure of integration is part of the game.