Proposition 14.1.13.label Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, and $0 < p < q < r \le \infty$, then $L^{p}(X; E) \cap L^{r}(X; E) \subset L^{q}(X; E)$, where for each $f \in L^{p}(X; E) \cap L^{r}(X; E)$,
\[\norm{f}_{L^q(X; E)}\le \norm{f}_{L^p(X; E)}^{\lambda}\norm{f}_{L^q(X; E)}^{1 - \lambda}\]
with
\[\frac{1}{q}= \frac{\lambda}{p}+ \frac{(1 - \lambda)}{r}\quad \lambda = \frac{q^{-1}- r^{-1}}{p^{-1}- r^{-1}}\]
Proof, [Proposition 6.10, Fol99]. If $r = \infty$, then $\norm{f}_{E}^{q} \le \norm{f}_{L^\infty(X; E)}^{q - p}\norm{f}_{E}^{p}$ and $\lambda = p/q$, so
\[\norm{f}_{L^q(X; E)}\le \norm{f}_{L^p(X; E)}^{p/q}\cdot \norm{f}_{L^\infty(X; E)}^{1 - p/q}= \norm{f}_{L^p(X; E)}^{\lambda}\norm{f}_{L^q(X; E)}^{1 - \lambda}\]
If $r < \infty$, then by Hölder’s inequality applied to the pair $p/(\lambda q)$ and $r/[(1 - \lambda)q]$,
\begin{align*}\norm{f}_{L^q(X; E)}^{q}&= \int \norm{f}_{E}^{\lambda q}\norm{f}_{E}^{(1 - \lambda)q}d\mu \\&= \norm{\norm{f}_E^{\lambda q}}_{L^{p/(\lambda q)}(X; \real)}\cdot \norm{\norm{f}_E^{(1 - \lambda) q}}_{L^{r/[(1 - \lambda)q]}(X; \real)}\\&= \norm{f}_{L^p(X; E)}^{\lambda q}\norm{f}_{L^q(X; E)}^{(1 - \lambda)q}\\ \norm{f}_{L^q(X; E)}&\le \norm{f}_{L^p(X; E)}^{\lambda}\norm{f}_{L^q(X; E)}^{1 - \lambda}\end{align*}
$\square$
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