2.5 Interpolations
Theorem 2.5.1 (Riesz-Thorin).label 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
\[\norm{T}_{L(L^p; L^q)}\le M_{0}^{1 - \theta}M_{1}^{\theta}\]
Theorem 2.5.2 ([Theorem 4.31, Bau14]).label 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
\[\norm{\bp_t f}_{L^p(\mu)}\le \norm{f}_{L^p(\mu)}\]
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 $\norm{\bp_t f}_{L^1(\mu)}\le \norm{f}_{L^1(\mu)}$.$\square$