\section{Interpolations}
\label{sec:interpolation}

\begin{theorem}[Riesz-Thorin]
    \label{thm:rt}
    Let $1 \le p_0, p_q, q_0, q_1 \le \infty$ and $\theta \in (0, 1)$. Let $1 \le p, q \le \infty$ with
    \[
    \frac{1}{p} = \frac{1 - \theta}{p_0} + \frac{\theta}{p_1} \quad \frac{1}{q} = \frac{1 - \theta}{q_0} + \frac{\theta}{q_1}
    \]
    Let $T: (L^{p_0} + L^{q_0}) \to(L^{p_1} + L^{q_1})$ with
    \[
    M_0 = \norm{T}_{L(L^{p_0}; L^{q_0})} \quad 
    M_1 = \norm{T}_{L(L^{p_1}; L^{q_1})}
    \]
    then $T$ extends uniquely as a bounded map $L^p \to L^q$ with
    \[
    \norm{T}_{L(L^p; L^q)} \le M_0^{1 - \theta}M_1^{\theta}
    \]
\end{theorem}

\begin{theorem}[{{\cite[Theorem 4.31]{Baudoin}}}]
    \label{thm:heat-rt}
    Let $L$ be an essentially self-adjoint diffusion operator with symmetrising measure $\mu$, and $\bracs{\bp_t|t \ge 0}$ be its heat semigroup. For each $p \in [1, \infty]$ and $t \ge 0$, $\bp_t$ extends uniquely to a mapping $L^p(\mu) \to L^p(\mu)$ with
    \[
    \norm{\bp_t f}_{L^p(\mu)} \le \norm{f}_{L^p(\mu)}
    \]
\end{theorem}
\begin{proof}
    Let $f, g \in L^1(\mu) \cap L^\infty(\mu)$, then
    \begin{align*}
        \dpn{\bp_tf, g}{L^2(\mu)} &= \dpn{f, \bp_tg}{L^2(\mu)} \le \norm{f}_{L^1(\mu)}\norm{\bp_tg}_{L^\infty(\mu)} \\
        &\le \norm{f}_{L^1(\mu)}\norm{g}_{L^\infty(\mu)}
    \end{align*}
    so $\norm{\bp_t f}_{L^1(\mu)} \le \norm{f}_{L^1(\mu)}$.
\end{proof}