A Fourier transform of a function of $x$ is related by the following equation:
$$φ'(p) = \int_{}^{}φ(x) e^{ix·p} \mathrm{d}x.$$
Let's say that $x$ is a contravariant vector. Does it follow that $p$ transforms as a covariant vector?
In other words, is it true that, if we transform $x \to x',$ then $p$ must transform as
$$ p'_\mu = \frac{ \partial x^\nu }{ \partial x'^\mu } p_\nu $$ so, as a result, I will have to replace all $p$ in $φ'(p)$ with $p'$ that is related by the above expression?