 Nonvanishing level-set bound between three Lp norms

Suppose $1\le p<q<r<\infty$, $\Vert f{\Vert }_p\le C_p$, $\Vert f{\Vert }_q\ge C_q>0$ and $\Vert f{\Vert }_r\le C_r$. Set $\epsilon =(C_q^q/(2C_p^p){)}^{1/(q-p)}$. The part of $\int |f{|}^q$ where $|f|\le \epsilon$ is at most ${\epsilon }^{q-p}{C}_p^p=C_q^q/2$. On the complementary level set, the Holder inequality bounds the integral by $C_r^q\mu \{|f|>\epsilon {\}}^{1-q/r}$, proving the bound. For $r=\infty$ the bound becomes $C_q^q/(2C_r^q)$. The lower-exponent bound prevents spreading into low amplitudes, while the higher-exponent bound prevents concentration into arbitrarily narrow spikes.