Extension of a Bounded Operator on $L^p$ to $L^r$ Let $1<p\leq \infty$, and let $p^{-1} + q^{-1} = 1$. Let $T$ be a bounded operator on $L^p$ such that $\int(Tf)g = \int f(Tg)$ for all $f,g \in L^p \cap L^q.$ Show that $T$ uniquely extends to a bounded operator on $L^r$ for all $r \in [p,q]$ (if $p< q$) or $r\in [q,p]$ (if $q < p$).
This is exercise 6.41 in Folland. Per his list of errata, for the case $p=\infty$ we need the measure to be semifinite.
First, I suppose we should recall that $(L^p \cap L^q) \subset L^r \subset (L^p + L^q).$ The question is in the section showing the Riesz-Thorin and the Marcinkewicz interpolation theorems so I think we want to use one of those, but its not obvious to me how to proceed. Any hints?
EDIT (using the hint given by user75064):
Since $T$ is bounded on $L^p$, for any $g\in L^p$ we have $$\|Tg\|_{L^p} \le C\|g\|_{L^p}.$$
I want to be able to say the following, but we need $f,g \in L^p \cap L^q$:
\begin{eqnarray*}
\|Tf\|_{L^q}&=&\sup\left\{ \left|\int (Tf)g\right| : \|g\|_{L^p} = 1\right\}\\
&=& \sup\left\{ \left|\int f(Tg)\right| : \|g\|_{L^p} = 1\right\}.\\
\end{eqnarray*}
Now, for any $g$ with $\|g\|_{L^p} = 1$, we see that $$\left|\int f(Tg)\right| \le \int \left|f(Tg)\right| = \|f(Tg)\|_{L^1} \le \|f\|_{L^q}\|Tg\|_{L^p} \le C\|f\|_{L^q},$$
where the second to last inequality follows from Holder's inequality.  Thus $$\|Tf\|_{L^q}\le C\|f\|_{L^q}.$$
 A: The first, and main, step is to show that $T$ is bounded on $L^q$. This is done "by duality": the starting point is the identity 
$$\|f\|_{L^q}=\sup\left\{ \int fg : \|g\|_{L^p}\le 1\right\}$$
Apply it to $ Tf $: 
 $$\|Tf\|_{L^q}=\sup\left\{ \int (Tf)g : \|g\|_{L^p}\le 1\right\}$$
Now it's time to re-read the assumptions of the problem and decide what to do next.
A: Before discussing boundness of $T$ on $L_q$, we must define $T$ on $L_q$, because $T$ is defined only on $L_p$. So the strategy is the following:


*

*Define $T$ on $L_q$ and prove its well-definedness.

*Prove $T$ is bounded on $L_q$ so that $T$ maps $L_q$ to $L_q$.

*Extend $T$ into $T:L_p + L_q \to L_p + L_q$ as $T(f+g) = Tf + Tg$ for $f \in L_p$ and $g\in L_q$. Also, have to check well-definedness. Then, $T:L_r \to L_r$ can be defined in the sense that $L_r \subset L_p + L_q$.

*Use the Riesz-Thorin interpolation theorem to prove $T$ is bounded on $L_r$.

*Prove the uniqueness of all extensions.



STEP $1$
Fix $g \in L_q$. Let $\|T\|_p$ be the operator norm of $T$ on $L_p$. Note that \begin{equation}
\left|\int (Tf)g\right| \le \int \left|(Tf)g\right| \le \|Tf\|_p \|g\|_q
\le \|T\|_p \|f\|_p \|g\|_q
\end{equation}
which is bounded for any $f \in L_p$. Now define $\phi_g:L_p \to \mathbb{C}$ as \begin{equation}\phi_g (f) = \int (Tf)g. \end{equation} Since $\phi_g$ is a bounded linear functional, $\phi_g \in \left(L_p\right)^*$ ($L_p$ dual space) and there uniquely exists $g' \in L_q$ such that $\displaystyle{\phi_g(f) = \int fg'}$. This is because the mapping $\displaystyle{(\cdot)\mapsto\int f(\cdot)}$ is an isometrical isomorphism from $L_q$ to $\left(L_p\right)^*$. It is natural to define $T$ on $L_q$ as $Tg = g'$. We should check well-definedness and linearity of $T$, but just skip them here. Before the next step, please notice that now we have $\displaystyle{\phi_g (f) = \int (Tf)g = \int f(Tg)}$ for all $f \in L_p, g \in L_q$.

STEP $2$
As Bohring and user75064 mentioned, by Hölder's inequality, \begin{align*}
\|Tg\|_q
&= \sup\left\{ \left|\int f(Tg)\right| : \|f\|_{L^p} = 1\right\}\\
&= \sup\left\{ \left|\int (Tf)g\right| : \|f\|_{L^p} = 1\right\}\\
&\le \|Tf\|_p\|g\|_q \le \|T\|_p \|g\|_q.
\end{align*}
Thus $T$ is bounded on $L_q$.

STEP $3,4,5$ will be not so hard to prove!
