Suppose $f$ is a compactly supported smooth function from $\mathbb{R}^n$ to $\mathbb{C}$ and $A$ is a diffeomorphism on $\mathbb{R}^n$, do we have any theorems relating the $L^p$ norm of $\hat{f}$ and the $L^q$ norm of $\widehat{f\circ A}$? Here hat denotes the Fourier transform. $p,q>1$,
Obviously for $A$ an invertible linear map this follows trivially from a change of variable. What about when $A$ is a more general nonlinear map?
