Convergence pw if converges in Lp space

Question

Let $p\in[1,\infty]$ be given.

If $f$ and $g$ are non-negative analytic functions such that the following holds: \begin{equation} \int_{\mathbb{R}}|\frac{\partial^ig(t)}{\partial^it}|^pdt=\int_{\mathbb{R}}|\frac{\partial^if(t)}{\partial^it}|^pdt \end{equation}

then does this imply that $f=g$ pointwise?

Answer 1

No. Here is a counterexample: $g(x)=f(x+a)$ for $a\in\mathbb{R}$, $a\ne0$.