Threshold decomposition between Lp spaces ($g=f{1}_{\{|f|>a\}},h=f{1}_{\{|f|\le a\}}$)

If $p_0<p<p_1$, the displayed split of an $L^p$ function gives $\Vert g{\Vert }_{p_0}\le a^{1-p/p_0}\Vert f{\Vert }_p^{p/p_0}$ and $\Vert h{\Vert }_{p_1}\le a^{1-p/p_1}\Vert f{\Vert }_p^{p/p_1}$. Weighted multiplication with $1/p=(1-\theta )/p_0+\theta /p_1$ cancels the threshold, yielding $\Vert g{\Vert }_{p_0}^{1-\theta }\Vert h{\Vert }_{p_1}^{\theta }\le \Vert f{\Vert }_p$. Choosing $a=\Vert f{\Vert }_p$ also gives the continuous inclusion $\Vert f{\Vert }_{L^{p_0}+L^{p_1}}\le 2\Vert f{\Vert }_p$. The analogous split works if the larger endpoint is infinity.