On a measure space $(\Omega,\Sigma,\mu)$ with $1\le p<r<q\le\infty$ and $f \in L^r(\mu)$, is it possible to find a concrete example with $g \in L^p(\mu)$ and $h \in L^q(\mu)$ with $g+h=f$ and $\vert \vert g \vert \vert _{L^p}+\vert \vert h \vert \vert _{L^q} \le 2 \vert \vert f \vert \vert _{L^r}$ ?
Thanks in advance!