Theorem 12.4.3 ( That rises from the book Analytic Inequalities and Their Applications in PDEs By Yuming Qin but no proof is given.)
Let $\Omega $ be a smooth open set in $\mathbb {R^{n}}$ and $f,g\in L^{p}(\Omega)$ with $1\leq p\leq +\infty.$ Then we have $$f+g\in L^{p}(\Omega)$$ and $$\left \| f+g \right \|_{ L^{p}(\Omega)}\leq\left \| f \right \|_{ L^{p}(\Omega)}+\left \| g \right \|_{ L^{p}(\Omega)}.\quad (12.4.9)$$
If $p=1$,then the equality in $(12.4.9)$ holds if and only if $\arg f(x)=\arg g(x)$ a.e.in $\Omega.$
If $p>1$,then the equality in $(12.4.9)$ holds if and only if there exist constants $C_{1}$ and $C_{2}$ which are not all zero such that $C_{1}f(x)=C_{2}g(x)$ a.e. in $ \Omega $ or there exists a non-negative measueable function $h$ such that $fh=g$ a.e. in the set $A=\{x \in\Omega|f(x)g(x)\ne0\}.$
But @Daniel Fischer♦ had proved that "For $1 < p < \infty$,$$\lVert f+g\rVert_p = \lVert f\rVert_p + \lVert g\rVert_p$$
holds if and only if there are non-negative real constants $\alpha,\beta$, not both zero, such that $\alpha f(x) = \beta g(x)$ almost everywhere."see here
Which one is right ? Is the condition "non-negative" necessary for the equality holds when $p>1$? What does the symbol $\arg f(x)=\arg g(x)$ mean in Theorem 12.4.3 $p=1$?