Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 5 / 4 / i / Solution

The interpolated exponent is strictly between $p_0$ and $p_1$. Apply Hölder's inequality to $|f{|}^p=|f{|}^{p(1-\theta )}|f{|}^{p\theta }$ with conjugate exponents $p_0/[p(1-\theta )]$ and $p_1/(p\theta )$. Their reciprocals sum to one, soThis gives $L^{p_0}\cap L^{p_1}\subseteq L^p$.

For the other inclusion take any $a>0$ and use the threshold decomposition between Lp spacesBecause $p_0-p<0<p_1-p$,Thus $g\in L^{p_0}$, $h\in L^{p_1}$ and $f=g+h$, proving $L^p\subseteq L^{p_0}+L^{p_1}$. Raising the two norm bounds to powers $1-\theta$ and $\theta$ gives a power of $a$ equal toand a power of $\Vert f{\Vert }_p$ equal to one. HenceFor later use, choose $a=\Vert f{\Vert }_p$ when $f\ne 0$. Each endpoint norm is then at most $\Vert f{\Vert }_p$, so $\Vert f{\Vert }_{L^{p_0}+L^{p_1}}\le 2\Vert f{\Vert }_p$ for the sum of Lp spaces norm. If $f=0$, take $g=h=0$.