## Lecture 7. Diffusion semigroups in Lp

In the previous lectures, we have seen that if L is an essentially self-adjoint diffusion operator with respect to a measure, then by using the spectral theorem one can define a self-adjoint strongly continuous contraction semigroup on L2 with generator L. This semigroup is moreover positivity preserving and a contraction on the space of bounded square integrable functions. Our goal, in this lecture, is to define, for $1 \le p \le +\infty$, the semigroup on $\mathbf{L}_{\mu}^p (\mathbb{R}^n,\mathbb{R})$. This can be done by using the RieszThorin interpolation theorem that we remind below. In this subsection, in order to simplify the notations we simply denote $\mathbf{L}_{\mu}^p (\mathbb{R}^n,\mathbb{R})$ by $\mathbf{L}_\mu^p$. We first start with general comments about semigroups in Banach spaces.

Let $(B,\| \cdot \|)$ be a Banach space (which for us will be $\mathbf{L}_{\mu}^p$, $1 \le p \le +\infty$).

We first have the following basic definition.

Definition: A family of bounded operators $(T_t)_{t \ge 0}$ on $B$ is called a contraction semigroup if:

• $T_0 =\mathbf{Id}$ and for $s,t \ge 0$, $T_{s+t}=T_s T_t$;
• For each $x \in B$ and $t \ge 0$, $\| T_t x \| \le \|x \|$.

A contraction semigroup $(T_t)_{t \ge 0}$ on $B$ is moreover said to be strongly continuous if for each $x \in B$, the map $t \to T_t x$ is continuous.

In this Lecture, we will prove the following result:

Theorem: let $L$ be an essentially self-adjoint diffusion operator. Denote by $(\mathbf{P}_t)_{t \ge 0}$ the self-adjoint strongly continuous semigroup associated to $L$ and constructed on $\mathbf{L}_{\mu}^2$ thanks to the spectral theorem. Let $1 \le p \le +\infty$. On $\mathbf{L}_\mu^p$, there exists a unique contraction semigroup $(\mathbf{P}^{(p)}_t)_{t \ge 0}$ such that for $f \in \mathbf{L}_\mu^p \cap \mathbf{L}_\mu^2$, $\mathbf{P}^{(p)}_t f=\mathbf{P}_t f$. The semigroup $(\mathbf{P}^{(p)}_t)_{t \ge 0}$ is moreover strongly continuous for $1 \le p < +\infty$.

This theorem can be proved by using two sets of methods: Hille-Yosida theory on one hand and interpolation theory on the other hand. We shall use interpolation theory in $L^p$ space which takes advantage of the fact that $\mathbf{P}_t$ only needs to be constructed on $L^1$ and $L^\infty$. The Hille-Yosida method starts from the generator and we sketch it below.

Definition:
Let $(T_t)_{t \ge 0}$ be a strongly continuous contraction semigroup on a Banach space $B$. There exists a closed and densely defined operator $A: \mathcal{D}(A) \subset B \rightarrow B$ where $\mathcal{D}(A)=\left\{ x \in B,\quad \lim_{t \to 0} \frac{T_t x -x}{t} \text{ exists} \right\},$ such that for $x \in \mathcal{D}(A)$, $\lim_{t \to 0} \left\| \frac{T_t x -x}{t} -Ax \right\|=0.$ The operator $A$ is called the generator of the semigroup $(T_t)_{t \ge 0}$. We also say that $A$ generates $(T_t)_{t \ge 0}$.

The following important theorem is due to Hille and Yosida and provides, through spectral properties, a characterization of closed operators that are generators of contraction semigroups.

Let $A: \mathcal{D}(A) \subset B \rightarrow B$ be a densely defined closed operator. A constant $\lambda \in \mathbb{R}$ is said to be in the spectrum of $A$ if the the operator $\lambda \mathbf{Id}-A$ is not bijective. In that case, it is a consequence of the closed graph theorem that if $\lambda$ is not in the spectrum of $A$ , then the operator $\lambda \mathbf{Id}-A$ has a bounded inverse. The spectrum of an operator $A$ shall be denoted $\rho(A)$.

Theorem: A necessary and sufficient condition that a densely defined closed operator $A$ generates a strongly continuous contraction semigroup is that:

• $\rho (A) \subset (-\infty,0]$ ;
• $\| (\lambda \mathbf{Id} -A)^{-1} \| \le \frac{1}{\lambda}$ for all $\lambda > 0$.

These two conditions are unfortunately difficult to directly check for diffusion operators.
We can bypass the study of the closure in $L^p$ of a diffusion operator by using interpolation theory.

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

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

The proof of the theorem can be found in this post by Tao.

One of the (numerous) beautiful applications of the Riesz-Thorin theorem is to construct diffusion semigroups on $L^p$ by interpolation. More precisely, let $L$ be an essentially self-adjoint diffusion operator. We denote by $(\mathbf{P}_t)_{t \ge 0}$ the self-adjoint strongly continuous semigroup associated to $L$ constructed on $\mathbf{L}_{\mu}^2$ thanks to the spectral theorem. We recall that $(\mathbf{P}_t)_{t \ge 0}$ satisfies the submarkov property: That is, if $0 \le f \le 1$ is a function in $\mathbf{L}_{\mu}^2$, then $0 \le \mathbf{P}_t f \le 1$.

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

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

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

Exercise:

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

We mention, that in general, the $\mathbf{L}_{\mu}^{\infty}$ valued map $t \rightarrow \mathbf{P}^{(\infty)}_t f$ is not continuous.

Due to the consistency property, we always remove the subscript $p$ from $\mathbf{P}^{(p)}_t$ and only use the notation $\mathbf{P}_t$.

To finish this Lecture, we finally connect the heat semigroup in $L^p$ to $L^p$ solutions of the heat equation.

Proposition: Let $f \in L^p_\mu(\mathbb{R}^n)$, $1 \le p \le \infty$, and let $u (t,x)= P_t f (x), \quad t \ge 0, x\in \mathbb{R}^n.$ Then, if $L$ is elliptic with smooth coefficients, $u$ is smooth on $(0,+\infty)\times \mathbb{R}^n$ and is a strong solution of the Cauchy problem $\frac{\partial u}{\partial t}= L u,\quad u (0,x)=f(x).$

Proof: The proof is identical to the $L^2$ case. For $\phi \in C^\infty_0 ((0,+\infty) \times \mathbb{R}^n)$, we have
$\int_{\mathbb{R}^n \times \mathbb{R}} \left( \left( -\frac{\partial}{\partial t} -L \right) \phi (t,x) \right) u(t,x) d\mu(x) dt$
$=\int_{\mathbb{R}} \int_{\mathbb{R}^n} \left( \left( -\frac{\partial}{\partial t} -L \right) \phi (t,x) \right) P_t f (x) dx dt$
$= \int_{\mathbb{R}} \int_{\mathbb{R}^n} P_t \left( \left( -\frac{\partial}{\partial t} -L \right) \phi (t,x) \right) f (x) dx dt$
$= \int_{\mathbb{R}} \int_{\mathbb{R}^n} -\frac{\partial}{\partial t} \left( P_t \phi (t,x) f(x) \right) dx dt =0.$
Therefore $u$ is a weak solution of the equation $\frac{\partial u}{\partial t}= L u$. Since $u$ is smooth it is also a strong solution $\square$.

We now address the uniqueness of solutions. As in the $L^2$ case, we assume that $L$ is elliptic with smooth coefficients and that there is a sequence $h_n\in C_0^\infty(\mathbb{R}^n)$, $0 \le h_n \le 1$, such that $h_n\nearrow 1$ on $\mathbb{R}^n$, and $||\Gamma (h_n,h_n)||_{\infty} \to 0$, as $n\to \infty$.

Proposition: Let $v(x,t)$ be a non negative function such that $\frac{\partial v}{\partial t} \le L v,\quad v(x,0)=0,$ and such that for every $t > 0$, $\| v ( \cdot,t) \|_{L^p_\mu(\mathbb{R}^n)} <+\infty$, where $1 < p < +\infty$. Then $v(x,t)=0$.

Proof: Let $x_0 \in \mathbb{R}^n$ and $h \in C_0^\infty(\mathbb{R}^n)$. Since $u$ is a subsolution with the zero initial data, for any $\tau\in (0,T)$,
$\int_0^\tau \int_{\mathbb{R}^n} h^2(x) v^{p-1}(x,t) L v(x,t) d\mu(x) dt$
$\geq \int_0^\tau \int_{\mathbb{R}^n} h^2(x) v^{p-1} \frac{\partial v}{\partial t} d\mu(x) dt$
$= \frac{1}{p} \int_0^\tau \frac{\partial }{\partial t} \left( \int_{\mathbb{R}^n} h^2(x) v^{p} d\mu(x)\right) dt$
$= \frac{1}{p}\int_{\mathbb{R}^n} h^2(x) v^{p}(x,\tau) d\mu(x).$
On the other hand, integrating by parts yields
$\int_0^\tau \int_{\mathbb{R}^n} h^2(x) v^{p-1}(x,t) L v(x,t) d\mu(x) dt = - \int_0^\tau \int_{\mathbb{R}^n} 2h v^{p-1} \Gamma(h,v) d\mu dt - \int_0^\tau \int_{\mathbb{R}^n} h^2 (p-1) v^{p-2} \Gamma(v) d\mu dt .$

Observing that
$0\leq \left(\sqrt{\frac{2}{p-1}\Gamma(h)}v - \sqrt{\frac{p-1}{2}\Gamma(v)}h \right)^2 \leq \frac{2}{p-1}\Gamma(h)v^2 + 2 \Gamma(h,v) h v +\frac{p-1}{2}\Gamma(v)h^2 ,$
we obtain the following estimate.
$\int_0^\tau \int_{\mathbb{R}^n} h^2(x) v^{p-1}(x,t) L v(x,t) d\mu(x) dt$
$\leq \int_0^\tau \int_{\mathbb{R}^n} \frac{2}{p-1} \Gamma(h) v^p d\mu dt - \int_0^\tau \int_{\mathbb{R}^n} \frac{p-1}{2}h^2 v^{p-2} \Gamma(v) d\mu dt$
$= \int_0^\tau \int_{\mathbb{R}^n} \frac{2}{p-1} \Gamma(h) v^p d\mu dt - \frac{2(p-1)}{p^2} \int_0^\tau \int_{\mathbb{R}^n} h^2 \Gamma(v^{p/2}) d\mu dt$

Combining with the previous conclusion we obtain ,
$\int_{\mathbb{R}^n} h^2(x) v^{p}(x,\tau) d\mu(x) + \frac{2(p-1)}{p} \int_0^\tau \int_{\mathbb{R}^n} h^2 \Gamma(v^{p/2}) d\mu dt \leq \frac{2 p}{(p-1) } \| \Gamma(h) \|_\infty^2 \int_0^\tau \int_{\mathbb{R}^n} v^p d\mu dt.$
By using the previous inequality with an increasing sequence $h_n\in C_0^\infty(\mathbb{R}^n)$, $0 \le h_n \le 1$, such that $h_n\nearrow 1$ on $\mathbb{R}^n$, and $||\Gamma (h_n,h_n)||_{\infty} \to 0$, as $n\to \infty$, and letting $n \to +\infty$, we obtain
$\int_{\mathbb{R}^n} v^{p}(x,\tau) d\mu(x)=0$ thus $v=0$ $\square$.

As a consequence of this result, any solution in $L^p_\mu(\mathbb{R}^n)$, $1 < p < +\infty$ of the heat equation $\frac{\partial u}{\partial t}= L u$ is uniquely determined by its initial condition, and is therefore of the form $u(t,x)=P_tf(x)$. We stress that without further conditions, this result fails when $p=1$ or $p=+\infty$.

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