= Nonvanishing level-set bound between three Lp norms
{title2=$\mu\{|f|>\varepsilon\}\ge\left(\frac{C_q^q}{2C_r^q}\right)^{r/(r-q)}$}
Suppose $1\le p<q<r<\infty$, $\|f\|_p\le C_p$, $\|f\|_q\ge C_q>0$ and $\|f\|_r\le C_r$. Set $\varepsilon=(C_q^q/(2C_p^p))^{1/(q-p)}$. The part of $\int|f|^q$ where $|f|\le\varepsilon$ is at most $\varepsilon^{q-p}C_p^p=C_q^q/2$. On the complementary <level set>, the <Holder inequality> bounds the integral by $C_r^q\mu\{|f|>\varepsilon\}^{1-q/r}$, 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.
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