Consider a function $f \in L^2$ and $f \notin L^{p}$ (for $p>2$). Now multiple the Fourier Coefficients by random signs. Almost surely, the new function, $g$, will be in $L^{p}$ (by Khinchin's inequality and Fubini's theorem). Thus we have two functions, $f$ and $g$, both of whose Fourier coefficients have the same absolute values, one of which has finite $L^{p}$ norm and the other of which has infinite $L^{p}$ norm. Thus, no expression involving only the absolute values of the Fourier coefficients can compute (or even bound!) the $L^p$ norm of a function.

Consider a function $f \in L^2$ and $f \notin L^{p}$ (for $p>2$). Now multiple the Fourier coefficients by random signs. Almost surely, the new function, $g$, will be in $L^{p}$ (by Khinchin's inequality and Fubini's theorem). Thus we have two functions, $f$ and $g$, both of whose Fourier coefficients have the same absolute values, one of which has finite $L^{p}$ norm and the other of which has infinite $L^{p}$ norm. Thus, no expression involving only the absolute values of the Fourier coefficients can compute (or even bound!) the $L^p$ norm of a function.