Lecture 6. The Lp theory of Markovian semigroups

Our goal, in this lecture, is to define, for $1 \le p \le +\infty$ , $P_t$ on $L^p(X,\mu)$ . This may be done in a natural way by using the Riesz-Thorin interpolation theorem that we recall below.

Theorem: [Riesz-Thorin interpolation theorem] Let $1 \le p_0, p_1,q_0,q_1 \le \infty$ , and $\theta \in (0,1)$ . Define $1 \le p,q \le \infty$ by

$\frac{1}{p}=\frac{1-\theta}{p_0} + \frac{\theta}{p_1}, \quad \frac{1}{q}=\frac{1-\theta}{q_0} + \frac{\theta}{q_1}.$
If $T$ is a linear map such that
$T:L^{p_0} \rightarrow L^{q_0}, \quad \| T \|_{ L^{p_0} \rightarrow L^{q_0} } =M_0$
$T:L^{p_1} \rightarrow L^{q_1}, \quad \| T \|_{ L^{p_1} \rightarrow L^{q_1} } =M_1,$
then, for every $f \in L^{p_0} \cap L^{p_1}$ ,
$\| T f \|_q \le M_0^{1-\theta} M_1^{\theta} \| f \|_p.$
Hence $T$ extends uniquely as a bounded map from $L^{p}$ to $L^{q}$ with
$\| T \|_{ L^{p} \rightarrow L^{q} } \le M_0^{1-\theta} M_1^{\theta} .$

The statement that $T$ is a linear map such that
$T:L^{p_0} \rightarrow L^{q_0}, \quad \| T \|_{ L^{p_0} \rightarrow L^{q_0} } =M_0$
$T:L^{p_1} \rightarrow L^{q_1}, \quad \| T \|_{ L^{p_1} \rightarrow L^{q_1} } =M_1,$
means that there exists a map $T: L^{p_0} \cap L^{p_1}\rightarrow L^{q_0} \cap L^{q_1}$ with
$\sup_{ f \in L^{p_0} \cap L^{p_1} , \| f \|_{p_0} \le 1 } \| Tf \|_{q_0} =M_0$
and
$\sup_{ f \in L^{p_0} \cap L^{p_1} , \| f \|_{p_1} \le 1 } \| Tf \|_{q_1} =M_1.$
In such a case, $T$ can be uniquely extended to bounded linear maps $T_0: L^{p_0} \rightarrow L^{q_0}$ , $T_1: L^{p_1} \rightarrow L^{q_1}$ . With a slight abuse of notation, these two maps are both denoted by $T$ in the theorem.

We now are in position to state the following theorem:

Theorem: Let $(P_t)_{t \ge 0}$ be a strongly continuous self-adjoint contraction Markovian semigroup on $L^2(X,\mu)$ . The space $L^1 \cap L^{\infty}$ is invariant under $P_t$ and $P_t$ may be extended from $L^1 \cap L^{\infty}$ to a contraction semigroup $(P^{(p)}_t)_{t \ge 0}$ on $L^{p}$ for all $1 \le p \le \infty$ : For $f \in L^p$ ,
$\| P_t f \|_{ L^p} \le \| f \|_{ L^p}.$
These semigroups are consistent in the sense that for $f \in L^p \cap L^{q}$ ,
$P^{(p)}_t f=P^{(q)}_t f.$

Proof: If $f,g \in L^1 \cap L^{\infty}$ which is a subset of $L^1 \cap L^{\infty}$ , then,
$\left| \int_{X} (P_t f) g d\mu \right| = \left| \int_{X} f(P_t g) d\mu \right |$
$\le \| f \|_{ L^1} \| P_t g \|_{ L^\infty}$
$\le \| f \|_{ L^1} \| g \|_{ L^\infty}.$
This implies
$\| P_t f \|_{ L^1} \le \| f \|_{ L^\infty}.$
The conclusion follows then from the Riesz-Thorin interpolation theorem.

Exercise: Show that if $f \in L^{p}$ and $g \in L^{q}$ with $\frac{1}{p}+\frac{1}{q}=1$ then,
$\int_{\mathbb{R}^n} f P^{(q)}_t g d\mu=\int_{\mathbb{R}^n} g P^{(p)}_t f d\mu.$

Exercise:

1) Show that for each $f \in L^{1}$ , the $L^{1}$ -valued map $t \rightarrow P^{(1)}_t f$ is continuous.
2) Show that for each $f \in L^{p}$ , $1<p<2$ , the $L^{p}$ -valued map $t \rightarrow P^{(p)}_t f$ is continuous.
3) Finally, by using the reflexivity of $L^{p}$ , show that for each $f \in L^{p}$ and every $p \ge 1$ , the $L^{p}$ -valued map $t \rightarrow P^{(p)}_t f$ is continuous.