Parseval Theorem for Lp Norms

Question

After doing much research online, it seems that Parseval's Theorem is only valid for $L_2$ norms. So for example, for a time-domain signal $f(t)$ and with it's frequency response $F(j\omega)$, then Parseval's theorem states the following $$\|f(t) \|_2 = \|F(j\omega) \|_2$$

However, is there a similar representation for the $L_1$ norm, or the $L_\infty$ norm? It seems that since $L_p$ for $p \neq 2$ are not Hilbert spaces, then the frequency-domain equivalent of such spaces do not exist. Is this true?

Thanks for your help!

Answer 1

There is the Hausdorff-Young inequality: $$\|F\|_q \leq \|f\|_p$$ for $1 < p \leq 2$, $1/p + 1/q = 1$.

https://en.wikipedia.org/wiki/Hausdorff%E2%80%93Young_inequality