Suppose that a tempered distribution $f(x)$ on 4-dimensional Minkowski space with signature $+---$ vanishes for $x²<0$ and its Fourier transform $\tilde f(p)$ vanishes for $p²<0$.

Are then $f$ and $\tilde f$ necessarily Lorentz invariant? I am interested in a proof or a counterexample, and in the latter case ideally in a description of all possible $f$ with the above property.