= Threshold decomposition between Lp spaces
{title2=$g=f\mathbf1_{\{|f|>a\}},\quad h=f\mathbf1_{\{|f|\leq a\}}$}
If $p_0<p<p_1$, the displayed split of an $L^p$ function gives $\|g\|_{p_0}\leq a^{1-p/p_0}\|f\|_p^{p/p_0}$ and $\|h\|_{p_1}\leq a^{1-p/p_1}\|f\|_p^{p/p_1}$. Weighted multiplication with $1/p=(1-\theta)/p_0+\theta/p_1$ cancels the threshold, yielding $\|g\|_{p_0}^{1-\theta}\|h\|_{p_1}^{\theta}\leq\|f\|_p$. Choosing $a=\|f\|_p$ also gives the continuous inclusion $\|f\|_{L^{p_0}+L^{p_1}}\leq2\|f\|_p$. The analogous split works if the larger endpoint is infinity.
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