Function form of Lorentz invariant functions In QFT, Green function of gauge field is Lorentz invariant(i.e. $\forall \Lambda \in SO(3,1), f(\Lambda p)=\Lambda f(p)$).And according to the textbook I'm reading, The form of such functions is restricted as
$$
f_{\mu}(p)=\alpha(p^2)p_{\mu}\\
F_{\mu\nu}(p)=\beta(p^2)g_{\mu\nu}+\gamma(p^2)p_{\mu}p_{\nu}
$$
where $F_{\mu\nu}$ is also assumed to be symmetric in Lorentz indices.
My problem is that I can't understand how this relation is proved.When $f$ and $F$ is linear, it is Schur's lemma but how is it applied to $C^{\infty}$functions?
 A: I sketch a proof for the equivariant function $f$ which is vector-valued and for causal vectors $p$, assuming only the continuity of $f$ in the last step to extend the result from timelike vectors to lightlike vectors.
I believe the proof can be completed and the equivariant symmetric tensor case case could be treated similarly.
Consider $f(\Lambda p) = \Lambda f(p)$ where  $f$ is vector valued as I said.
Define $$G^\mu(p) := f^\mu(p) - \frac{p_\nu f^\nu(p)}{p^2} p^\mu\:.$$
Evidently
$$p_\mu G^\mu(p) = 0 \quad \forall p\:. \tag{1}$$
On the other hand, by construction
$$G(\Lambda p) = \Lambda G(p)\tag{2}$$ 
Now suppose that $k= (c,0,0,0)$ with $c \neq 0$ and let $R \in O(3)$ be any spatial $3$-rotation leaving $k$ fixed. It must be
$$RG(k) = G(Rk)= G(k)$$
as a consequence the vector $G(k)$ is parallel to $k$ (since $O(3)$ admits only $0$ as a fixed point) so that, for some real $a_k$,
$$G(k) = a_k k\:.$$
If $p$ is in the future or past light-cone, there is $\Lambda \in SO(1,3)$ with $p= \Lambda k$ for some $c$. Thus
$$G(p) = G(\Lambda k)= \Lambda G(k) = \Lambda a_k k = a_k p$$
and, since $p^2 \neq 0$,  (1) implies $a_k=0$ so that $G(p)=0$ if $p$ is a timelike vector.  We finally have that
$$f^\mu(p) = \frac{p_\nu f^\nu(p)}{p^2} p^\mu\:.$$
We can define $$\alpha(p^2) := \frac{p_\nu f^\nu(p)}{p^2}$$
since the right-hand side depends only on $p^2$ in view of $\Lambda f(p)= f(\Lambda p)$. Summing up, at least for timelike vectors $p$ and including the light-like vectors assuming $f$ continuous,
$$f_\mu(p) =\alpha(p^2) p_\mu \:.$$
The crucial points are that (a) $O(3) \subset O(1,3)$, that (b)  $O(3)$ has not non-zero fixed points, and that (c) $O(1,3)$ acts transitively on a set of timelike vectors with fixed length.
