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Let $1 \leq p \leq 2$. I am looking for a characterization of the couples $(f,g)$ of functions $f,g \in L_p(\mathbb{R})$ such that $$ \lVert f + g \rVert_p^p = \lVert f \rVert_p^p + \lVert g \rVert_p^p.$$

For $p = 2$, this relation is satisfied if and only if $\langle f, g \rangle = 0$. For $p = 1$, it has been shown in this post that the condition is equivalent to $f g \geq 0$ almost everywhere. For a general $p$, the relation is clearly satisfied as soon as the product $fg=0$ almost everywhere. Is this latter sufficient condition also necessary?

If not, then what if we reinforce the condition with $$ \lVert \alpha f + \beta g \rVert_p^p = \lVert \alpha f \rVert_p^p + \lVert \beta g \rVert_p^p$$ for any $\alpha, \beta \in \mathbb{R}$?

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    $\begingroup$ Doesn't the rest of the linked question answer yours? $\endgroup$
    – Kevin Casto
    May 29, 2021 at 3:17
  • $\begingroup$ It does not because it considers the equality case in the Minkowski inequality, which corresponds to $\lVert f + g \rVert_p = \lVert f \rVert_p + \lVert g \rVert_p$, without the $p$th powers. $\endgroup$
    – Goulifet
    May 29, 2021 at 13:58
  • $\begingroup$ An important distinction with the Minkowski case is that there is no inequality between $\lVert f + g \rVert^p_p$ and $\lVert f \rVert_p^p + \lVert g \rVert_p^p$: both can be greater than the other for adequate $f,g$. $\endgroup$
    – Goulifet
    May 29, 2021 at 13:59

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