What does it tell you about two functions if their $L^p$ norms are the same for all $p\in[1,\infty]$? Certainly they could be related by composition with a diffeomorphism with Jacobian of norm 1, or even one could be a "pulled apart version of the other one" in the sense of
$x^2\chi\_{[0,2]}$ vs $x^2 \chi\_{[0,1]} + (x-1)^2 \chi\_{[2,3]}$. To try to ignore the second type of issue, I'll restrict to smooth functions, and ask the following precise question:


Given $f,g\in C^\infty(\mathbb{R})$ such that $\Vert f \Vert\_{L^p(\mathbb{R})} = \Vert g \Vert\_{L^p(\mathbb{R})} <\infty$ for all $p\in [1,\infty]$ is it necessarily true that $f(x) = g(\pm x + C)$ for some constant $C$ and for a choice of either $+$ or $-$ for all $x \in \mathbb{R}$? 


As Qiaochu Yuan shows in his answer, smoothness doesn't solve the issue of "pulling apart" at all. Thus, I am interested in the following:


What is the "smallest" (in whatever sense) but still nonempty subset of $S \subset [1,\infty]$ such that there is $f,g\in C^\infty(\mathbb{R})$ such that $S$ is the set of $p$ such that
$\Vert f \Vert\_{L^p} \neq \Vert g \Vert\_{L^p}$?



How small can the set of $p$ such that the $L^p$ norms are different for two fixed functions?
The complement of $S$ in $(1,\infty)$ can't contain an interval or even a sequence converging to a point in $(1,\infty)$. Let $f$ and $g$ be two real functions on ${\mathbb{R}}$ all whose moments exist. Assume $\int|f|^p=\int|g|^p$ for all $p\in S^c$. put $h(z)=\int|f|^z-\int|g|^z$. where $z\in \mathbb{C}$. $h$ is an analytic functions in $\{Real z>1\}$ which we assume is not constantly zero (since $S$ is not empty) so can't vanish on a converging sequence.


