Extension of a Bounded Operator on $L^p$ to $L^r$

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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 $$\|Tf\|_{L^q}\le C\|f\|_{L^q}.$$

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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.