In quantum mechanics, there a lot of emphasis on $L^2$ spaces since Hilbert spaces describe states in quantum mechanics, so we have
$$ \langle \psi | \psi \rangle = \int |\psi^2(x)|\, dx$$
Even though technically the position operator $\mathbf{x}$ and momentum operator $\mathbf{p} = -i\hbar \frac{d}{dx}$ are not bounded, so maybe wave functions should just be a Banach space or something... However at least we can write $\langle x | p \rangle = e^{ipx}$ and things mostly work out.
Do physicist ever use the other $L_p$ norms? Especially $p \neq 0,1,2,\infty$... we could have:
$$ ||\psi||^p = \int |\psi|^p \, dx $$
Maybe these types of averages are not as natural as the mathematicians would think.
