Sum of functions in Lp spaces for $p<1$

Question

For $p \in (0,1)$, I know that the Minkowski inequality does not hold. But do we still have that $f+g \in L^p$ for $f,g \in L^p$ if $p \in (0,1)$?

Answer 1

Yes. If $p\in(0,1)$ and $s,t\ge0$ then $(s+t)^p\le s^p+t^p$, hence $||f+g||_p^p\le||f||_p^p+||g||_p^p$.

One can give a proof of that inequality that's so simple it seems like cheating: Fix $s\ge0$ and define $f:[0,\infty)\to\Bbb R$ by$$f(t)=s^p+t^p-(s+t)^p.$$Then $f(0)=0$, and for $t>0$ we have $$f'(t)=p(t^{p-1}-(s+t)^{p-1})\ge0,$$ since $s+t\ge t>0$ and $p-1<0$; hence $f(t)\ge0$.