Optimal angle of the holomorphic functional calculus for the classical Ornstein-Uhlenbeck operator on $L^p$

TL;DR

This paper proves that the classical Ornstein-Uhlenbeck operator on L^p spaces has an optimal angle for its holomorphic functional calculus, using a simple proof of R-sectoriality and applying abstract calculus theory.

Contribution

It provides a straightforward proof of the optimal angle for the holomorphic functional calculus of the Ornstein-Uhlenbeck operator, improving understanding of its spectral properties.

Findings

The Ornstein-Uhlenbeck operator is R-sectorial of angle arcsin|1-2/p|.

The operator admits a bounded H-infinity functional calculus with this optimal angle.

The proof simplifies previous approaches to the spectral analysis of the operator.

Abstract

We give a simple proof of the fact that the classical Ornstein-Uhlenbeck operator $L$ is R-sectorial of angle $arcsin|1-2/p|$ on $L^{p}(\mathbb{R}^{n},\exp(-|x|^2/2)dx)$ (for $1<p<\infty$). Applying the abstract holomorphic functional calculus theory of Kalton and Weis, this immediately gives a new proof of the fact that $L$ has a bounded $H^{\infty}$ functional calculus with this optimal angle.

Equations

$$T_{t}f(x)={\int }_{{\mathbb{R}}^d}{M}_t(x,y)f(y)dy,\text{ for }x\in {\mathbb{R}}^d$$

$$(x,y)\mapsto \frac{1}{(2\pi {)}^{\frac{d}{2}}}{\left(\frac{1}{1-e^{-2t}}\right)}^{\frac{d}{2}}\exp \left(-\frac{1}{2}\frac{|e^{-t}x-y{|}^2}{(1-e^{-2t})}\right),$$

$$M_t(x,y)=\frac{1}{(2\pi {)}^{\frac{d}{2}}}{\left(\frac{1}{1-e^{-2t}}\right)}^{\frac{d}{2}}\exp \left(-s_t{\left(\frac{x+y}{2\sqrt{2}}\right)}^2-\frac{1}{4s_t}{\left(\frac{x-y}{\sqrt{2}}\right)}^2\right)\exp \left(\frac{1}{2}\left(\varphi (x)-\varphi (y)\right)\right),$$

$$-\frac{1}{2}\frac{|e^{-t}x-y{|}^2}{(1-e^{-2t})}$$

$$=-\frac{1}{2}\frac{|e^{-t}x-y{|}^2}{(1-e^{-2t})}-\frac{1}{4}(x^2-y^2)+\frac{1}{2}\left(\varphi (x)-\varphi (y)\right)$$

$$=-\frac{1}{2(1-e^{-2t})}\left(|e^{-t}x-y{|}^2+\frac{(1-e^{-2t})}{2}(x^2-y^2)\right)+\frac{1}{2}\left(\varphi (x)-\varphi (y)\right)$$

$$=-\frac{1}{2(1-e^{-2t})}\left(e^{-2t}{x}^2-2e^{-t}xy+y^2+\frac{(1-e^{-2t})}{2}(x^2-y^2)\right)+\frac{1}{2}\left(\varphi (x)-\varphi (y)\right)$$

$$=-\frac{1}{2(1-e^{-2t})}\left(\frac{1}{2}\left(1+e^{-2t}\right)x^2-2e^{-t}xy+\frac{1}{2}\left(1+e^{-2t}\right)y^2\right)+\frac{1}{2}\left(\varphi (x)-\varphi (y)\right)$$

$$=-\frac{1}{8(1-e^{-2t})}$$

$$\times \left(\left((1+e^{-t}{)}^2+(1-e^{-t}{)}^2\right)x^2+2\left((1-e^{-t}{)}^2-(1+e^{-t}{)}^2\right)xy+\left((1+e^{-t}{)}^2+(1-e^{-t}{)}^2\right)y^2\right)$$

$$+\frac{1}{2}\left(\varphi (x)-\varphi (y)\right)$$

$$=-\frac{1}{8(1-e^{-2t})}\left((1-e^{-t}{)}^2(x+y{)}^2+(1+e^{-t}{)}^2(x-y{)}^2\right)+\frac{1}{2}\left(\varphi (x)-\varphi (y)\right)$$

$$=-\left(\frac{1-e^{-t}}{1+e^{-t}}{\left(\frac{x+y}{2\sqrt{2}}\right)}^2+\frac{1}{4}\frac{1+e^{-t}}{1-e^{-t}}{\left(\frac{x-y}{\sqrt{2}}\right)}^2\right)+\frac{1}{2}\left(\varphi (x)-\varphi (y)\right)$$

$$=-\left(s_t{\left(\frac{x+y}{2\sqrt{2}}\right)}^2+\frac{1}{4s_t}{\left(\frac{x-y}{\sqrt{2}}\right)}^2\right)+\frac{1}{2}\left(\varphi (x)-\varphi (y)\right).$$

$$U_{p}f(x)=f(x)\exp \left(-\frac{\varphi (x)}{p}\right),\text{ for }x\in {\mathbb{R}}^d.$$

$$E:=\left\{z\in \mathbb{C};s_z\in {\Sigma }_{\frac{\pi }{2}-{\theta }_p};z\notin i\pi \mathbb{Z}\right\}$$

$$E_{ϵ,\delta }=\left\{z\in \mathbb{C};|\Re (s_z){|}^2/|s_z{|}^2={\cos }^2(\arg (s_z))>M_p^2+ϵ;\text{dist}\left(z,i\pi (2\mathbb{Z}+1)\right)>\delta ;z\notin 2i\pi \mathbb{Z}\right\}$$

$$U_p\exp (-tL)U_p^{-1}f=\left(x\mapsto {\int }_{{\mathbb{R}}^d}{k}_t(x,y)f(y)dy\right),$$

$$k_t(x,y)=\frac{1}{(2\pi {)}^{\frac{d}{2}}}{\left(\frac{1}{1-e^{-2t}}\right)}^{\frac{d}{2}}\exp \left(-s_t{\left(\frac{x+y}{2\sqrt{2}}\right)}^2-\frac{1}{4s_t}{\left(\frac{x-y}{\sqrt{2}}\right)}^2\right)\exp \left(\left(\frac{1}{2}-\frac{1}{p}\right)\left(\varphi (x)-\varphi (y)\right)\right)$$

$$k_z(x,y)=\frac{1}{(2\pi {)}^{\frac{d}{2}}}{\left(\frac{1}{1-e^{-2z}}\right)}^{\frac{d}{2}}\exp \left(-s_z{\left(\frac{x+y}{2\sqrt{2}}\right)}^2-\frac{1}{4s_z}{\left(\frac{x-y}{\sqrt{2}}\right)}^2\right)\exp \left(\left(\frac{1}{2}-\frac{1}{p}\right)\left(\varphi (x)-\varphi (y)\right)\right),$$

$$|k_z(x,y)|$$

$$\le \frac{1}{(2\pi {)}^{\frac{d}{2}}}{\left|\frac{1}{1-e^{-2z}}\right|}^{\frac{d}{2}}\exp \left(-\Re (s_z){\left(\frac{x+y}{2\sqrt{2}}\right)}^2+M_p\frac{1}{4}\left|x^2-y^2\right|-\frac{1}{4}\Re \left(\frac{1}{s_z}\right){\left(\frac{x-y}{\sqrt{2}}\right)}^2\right)$$

$$=\frac{1}{(2\pi {)}^{\frac{d}{2}}}{\left|\frac{1}{1-e^{-2z}}\right|}^{\frac{d}{2}}\exp \left(-\Re (s_z){\left(\frac{x+y}{2\sqrt{2}}\right)}^2+M_p\left|\frac{x+y}{2\sqrt{2}}\right|\left|\frac{x-y}{\sqrt{2}}\right|-\frac{1}{4}\Re \left(\frac{1}{s_z}\right){\left(\frac{x-y}{\sqrt{2}}\right)}^2\right)$$

$$|k_z(x,y)|$$

$$=\frac{1}{(2\pi {)}^{\frac{d}{2}}}{\left|\frac{1}{1-e^{-2z}}\right|}^{\frac{d}{2}}\exp \left(-{\left(\sqrt{\Re (s_z)}u-\frac{M_p}{2\sqrt{\Re (s_z)}}k\right)}^2-\frac{1}{4}\left(\Re \left(\frac{1}{s_z}\right)-\frac{M_p^2}{\Re (s_z)}\right)k^2\right).$$

$$|k_z(x,y)|$$

$$=\frac{1}{(2\pi {)}^{\frac{d}{2}}}{\left|\frac{1}{1-e^{-2z}}\right|}^{\frac{d}{2}}\exp \left(-\frac{1}{4}\left(\Re \left(\frac{1}{s_z}\right)-\frac{M_p^2}{\Re (s_z)}\right){\left(\frac{x-y}{\sqrt{2}}\right)}^2\right).$$

$$=\frac{1}{(2\pi {)}^{\frac{d}{2}}}{\left|\frac{1}{1-e^{-2z}}\right|}^{\frac{d}{2}}\exp \left(-\frac{1}{8}\left(\Re \left(\frac{1}{s_z}\right)-\frac{M_p^2}{\Re (s_z)}\right){\left(x-y\right)}^2\right).$$

$$x\mapsto \frac{1}{(2\pi {)}^{\frac{d}{2}}}{\left|\frac{1}{1-e^{-2z}}\right|}^{\frac{d}{2}}\exp \left(-\frac{1}{8}\left(\Re \left(\frac{1}{s_z}\right)-\frac{M_p^2}{\Re (s_z)}\right)x^2\right).$$

$$\left|\left(U_p\exp (-zL)U_p^{-1}f\right)(x)\right|\le (g_z\ast |f|)(x).$$

$$\Re \left(\frac{1}{s_z}\right)-\frac{M_p^2}{\Re (s_z)}=\frac{\Re (s_z)}{|s_z{|}^2}-\frac{M_p^2}{\Re (s_z)}=\frac{1}{\Re (s_z)}\left(\frac{|\Re (s_z){|}^2}{|s_z{|}^2}-M_p^2\right)>0,$$

$$\underset{z\in E_{ϵ,\delta }}{\sup }{\int }_{{\mathbb{R}}^d}\underset{|y|>|x|}{\sup }|g_z(y)|dx<\infty ,$$

$$\underset{z\in E_{ϵ,\delta }}{\sup }{\int }_{{\mathbb{R}}^d}\underset{|y|>|x|}{\sup }|g_z(y)|dx$$

$$=\underset{z\in E_{ϵ,\delta }}{\sup }{\int }_{{\mathbb{R}}^d}\frac{1}{(2\pi {)}^{\frac{d}{2}}}{\left|\frac{1}{1-e^{-2z}}\right|}^{\frac{d}{2}}\exp \left(-\frac{1}{8}\left(\Re \left(\frac{1}{s_z}\right)-\frac{M_p^2}{\Re (s_z)}\right)x^2\right)dx$$

$$=\underset{z\in E_{ϵ,\delta }}{\sup }\frac{1}{(2{)}^{\frac{d}{2}}}{\left|\frac{1}{1-e^{-2z}}\right|}^{\frac{d}{2}}{\left(\frac{1}{8}\left(\Re \left(\frac{1}{s_z}\right)-\frac{M_p^2}{\Re (s_z)}\right)\right)}^{-\frac{d}{2}}$$

$$\le \underset{z\in E_{ϵ,\delta }}{\sup }{2}^d{\left|\frac{1}{1-e^{-2z}}\right|}^{\frac{d}{2}}{\left(\frac{ϵ}{\Re (s_z)}\right)}^{-\frac{d}{2}}$$

$$=\underset{z\in E_{ϵ,\delta }}{\sup }{ϵ}^{-\frac{d}{2}}{2}^d{\left(\left|\frac{\Re (s_z)}{1-e^{-2z}}\right|\right)}^{\frac{d}{2}}$$

$$\le \underset{z\in E_{ϵ,\delta }}{\sup }{ϵ}^{-\frac{d}{2}}{2}^d{\left(\frac{|s_z|}{|1-e^{-z}||1+e^{-z}|}\right)}^{\frac{d}{2}}$$

$$=\underset{z\in E_{ϵ,\delta }}{\sup }{ϵ}^{-\frac{d}{2}}{2}^d{\left(\frac{\left|\frac{1-e^{-z}}{1+e^{-z}}\right|}{|1-e^{-z}||1+e^{-z}|}\right)}^{\frac{d}{2}}$$

$$=\underset{z\in E_{ϵ,\delta }}{\sup }{ϵ}^{-\frac{d}{2}}{2}^d{\left(\frac{1}{|1+e^{-z}|}\right)}^d$$

$$<\infty$$

Full text

The author gratefully acknowledges financial support by the discovery Grant DP160100941 of the Australian Research Council. This research is also supported by an Australian Government Research Training Program (RTP) Scholarship.

1. Introduction

The Ornstein-Uhlenbeck operator appears in many areas of mathematics: as the number operator of quantum field theory, the analogue of the Laplacian in the Malliavin calculus, the generator of the transition semigroup associated with the simplest mean-reverting stochastic process (the Ornstein-Uhlenbeck process), or as the operator associated with the classical Dirichlet form on $\mathbb{R}^{d}$ equipped with the Gaussian measure $d\mu=(2\pi)^{-\frac{d}{2}}e^{-|x|^{2}/2}dx$. For the sake of this paper, the Ornstein-Uhlenbeck operator will be defined via the Ornstein-Uhlenbeck semigroup $\{T_{t}\}_{t>0}$ whose action on $f\in L^{p}(\mu)$ is

[TABLE]

where $M_{t}:\mathbb{R}^{2d}\to\mathbb{R}$ is given by

[TABLE]

the Mehler kernel.

Let us recall the basic properties of the Ornstein-Uhlenbeck semigroup used in this article. For each $p\in[1,\infty]$ and each $t>0$, the map $f\mapsto T_{t}f$ is bounded $L^{p}(\mu)\to L^{p}(\mu)$, with operator norm at most $1$, and is a positive operator. For $p\in[1,\infty)$, $T_{t}:L^{p}(\mu)\to L^{p}(\mu)$ is a $C_{0}$ semigroup, i.e. as $t\to 0$, $T_{t}\to I$ strongly and $T_{t}T_{s}=T_{t+s}$ for all $t,s>0$. For a proof of these preliminary facts, see for example Theorem 2.5 of [6]. It should be noted that although the Ornstein-Uhlenbeck semigroup arises in many different areas of mathematics, these basic properties can be proven solely with use of the explicit kernel and elementary techniques. It is a simple calculation to show that $T_{t}$ is bounded with norm $1$ on both $L^{\infty}(\mu)$ and $L^{1}(\mu)$, from which interpolation can be used to deduce boundedness with norm $1$ on $L^{p}(\mu)$ for $p\in[1,\infty]$. Positivity follows from non-negativity of the Mehler kernel. Strong continuity of the semigroup follows as in typical proofs of the strong continuity of the classical heat semigroup, and the semigroup property follows from a somewhat tedious exercise in integrating Gaussian functions. It should be noted that by using other representations of the Ornstein-Uhlenbeck semigroup, such as a spectral multiplier for the multivariate Hermite ONB of $L^{2}(\mu)$ or through a different representation via an integral kernel, one may prove some of these results even more simply, however the difficulty then becomes showing that all these representations for the Ornstein-Uhlenbeck semigroup are equivalent (for example, see [5]). We consider the generator of the Ornstein-Uhlenbeck semigroup on $L^{p}(\mu)$, $p\in[1,\infty)$, whose negative we shall call the Ornstein-Uhlenbeck operator and denote by $L$. This operator is a closed densely-defined unbounded operator on $L^{p}(\mu)$, $p\in[1,\infty)$, which uniquely determines $T_{t}$. Thus from here on, we will use the notation $\exp(-tL)$ for the operator $T_{t}$, on any of these spaces.

This paper presents a new proof of the following theorem.

Theorem 1**.**

For $p\in(1,\infty)$, the Ornstein-Uhlenbeck operator has a bounded $H^{\infty}(\Sigma_{\theta_{p}})$ functional calculus on $L^{p}(\mu)$, where $\sin(\theta_{p})=\left|1-\frac{2}{p}\right|$.

See [5] for the theory of the $H^{\infty}$ functional calculus. That $L$ has a bounded $H^{\infty}$ functional calculus (of some angle $\theta<\pi$) follows from general results in the theory of the $H^{\infty}$ functional calculus (for example, Theorem 10.7.13 of [4] states that any generator of an analytic semigroup on an $L^{p}$ space for $p\in(1,\infty)$ which is a positive contraction semigroup for real time has a bounded $H^{\infty}$ functional calculus of some angle less than $\frac{\pi}{2}$). The difficulty in Theorem 1 is to prove the boundedness of the calculus with precisely the optimal angle $\theta_{p}$.

Theorem 1 was originally proven by García-Cuerva, Mauceri, Meda, Sjögren and Torrea in [3], also proving that $\theta_{p}$ is optimal. They use Mauceri’s abstract multiplier theorem to reduce the problem to precisely estimating $u\mapsto||L^{iu}||$. To do so, they express $L^{iu}$ as an integral of the semigroup, using a carefully chosen contour of integration. They then consider the kernels of operators corresponding to different parts of the contour, and decompose them into a local and global part. To treat the global parts they then use a range of subtle kernel estimates.

In [1], Carbonaro and Dragičević reproved and extended the result of Theorem 1 to treat arbitrary generators of symmetric contraction semigroups on an $L^{p}$ space over a $\sigma$-finite measure space. Note that as they work on abstract $L^{p}$ spaces, their result gives dimension independent estimates working over $\mathbb{R}^{d}$. For their proof, they first reduce the problem to proving a bilinear embedding for the semigroup, with constants depending optimally on the angle $\theta_{p}$. They then use the Bellman function method, controlling the bilinear form by an optimally (depending on $p$) chosen function. This function turns out to be a known Bellman function introduced by Nazarov and Treil, but just proving that it has the right properties is a highly non-trivial task.

In contrast, the proof presented in this paper is based on the well-known result that in $L^{p}$ spaces, the optimal angle of the $H^{\infty}$ functional calculus of an operator is equal to its optimal angle of R-sectoriality (see [4] for the theory of R-sectoriality, and its Theorem 10.7.13 for a proof of the stated result). Our proof that the latter is equal to $\theta_{p}$ uses Theorem 10.3.3 of [4], which states an equivalence between an operator $A$ being R-sectorial of angle $\theta<\frac{\pi}{2}$ and $-A$ being the generator of an analytic semigroup of angle $\frac{\pi}{2}-\theta$ which is R-bounded on each smaller sector. To deduce R-boundedness of the Ornstein-Uhlenbeck semigroup on such sectors, a standard result on R-boundedness of integral operators with radially decaying kernels is employed (Proposition 8.2.3 of [4]). This key step only requires simple manipulations of the kernel for the Ornstein-Uhlenbeck semigroup. It is based on an approach designed by van Neerven and Portal in [7], where they recover classical results about the Ornstein-Uhlenbeck semigroup in a very direct manner. Their idea is to separate algebraic difficulties from analytic difficulties by considering a non-commutative functional calculus of the Gaussian position and momentum operators (the Weyl calculus). Using this calculus, one sees how to modify the kernels in a way that makes their analysis straightforward. A posteriori, the use of the Weyl calculus can be removed, and the proof can be read as a simple computation exploiting the change of time parameter $t\mapsto\frac{1-e^{-t}}{1+e^{-t}}$ (which has been used by many authors before).

Throughout the paper, we make use of the following notation. The function $\phi:\mathbb{R}^{d}\to\mathbb{R}$ will have action $x\mapsto\frac{x^{2}}{2}$. The standard Gaussian measure $\mu$ on $\mathbb{R}^{d}$ will thus be written with density $d\mu=(2\pi)^{-\frac{d}{2}}e^{-\phi(x)}dx$. The Lebesgue measure on $\mathbb{R}^{d}$ will be denoted by $\lambda$. As we only ever work over $\mathbb{R}^{d}$ with Borel $\sigma$-algebra, the measurable space over which we consider Lebesgue spaces will be dropped from the notation. For $\theta\in[0,\pi]$, we will write $\Sigma_{\theta}$ for the open sector $\{z\in\mathbb{C}\backslash\{0\};|\arg(z)|<\theta\}$.

2. R-Sectoriality of L

To simplify things, for the rest of the article we will assume that $p\in(1,\infty)$ is fixed. Similarly, all concepts of boundedness and R-boundedness will be on either $L^{p}(\mu)$ or $L^{p}(\lambda)$ without explicit mention of the space, the measure being clear from context.

Lemma 2**.**

$M_{t}$* has the alternate form for $t>0$ and $x,y\in\mathbb{R}^{d}$,*

[TABLE]

where $s_{t}=\frac{1-e^{-t}}{1+e^{-t}}$.

Proof.

We will rearrange the exponent from Equation (1) and show that it is equal to the exponent given above for all $x,y\in\mathbb{R}^{d}$ and $t>0$, as that is all that has changed between the two representations. For each $t>0,x,y\in\mathbb{R}^{d}$ we have

[TABLE]

∎

The next definition, albeit a simple one, forms the backbone of the rest of our arguments.

Definition 3**.**

Define the (multiple of an) isometry $U_{p}:L^{p}(\mu)\to L^{p}(\lambda)$ by

[TABLE]

As explained previously, we need only show that the Ornstein-Uhlenbeck semigroup has an analytic extension to a sector of the correct angle, and that it is R-bounded on each smaller sector. We will in fact show a lot more with no more effort. We shall work with the reparametrisation of the kernel of the semigroup in terms of $s_{t}$ from Lemma 2. The function $t\mapsto s_{t}$ is analytic and can clearly be analytically extended to the domain $\mathbb{C}\backslash i\pi(2\mathbb{Z}+1)$. We will consider the analytic extension $z\mapsto s_{z}$ on domains of the form

[TABLE]

where $\sin(\theta_{p})=M_{p}:=\left|1-\frac{2}{p}\right|$. We will show the Ornstein-Uhlenbeck semigroup extends to an analytic semigroup on the domain $E$. Moreover, we will simultaneously show that the Ornstein-Uhlenbeck semigroup is R-bounded on sets of the form

[TABLE]

for all $\epsilon,\delta>0$. Note that, in terms of the reparametrisation $s_{z}$, these sets are open sectors of angle $\frac{\pi}{2}-\theta_{p}$ or less, with certain points removed. We claim that $\Sigma_{\frac{\pi}{2}-\theta_{p}}\subset E$, and that for all $\epsilon^{\prime}>0$ there exists $\epsilon,\delta>0$ such that $\Sigma_{\frac{\pi}{2}-\theta_{p}-\epsilon^{\prime}}\subset E_{\epsilon,\delta}$ (see [7] for details of this calculation). These results combined will imply that the maximal domain of analyticity of the Ornstein-Uhlenbeck semigroup contains the sector $\Sigma_{\frac{\pi}{2}-\theta_{p}}$, and that it is R-bounded on each smaller sector, which combined with the procedure outlined in the introduction will show at least that the Ornstein-Uhlenbeck operator is R-sectorial of the desired angle.

Theorem 4**.**

For $p\in(1,\infty)$, the Ornstein-Uhlenbeck operator on $L^{p}(\mu)$ is R-sectorial of angle $\theta_{p}$, where $\sin(\theta_{p})=M_{p}:=\left|1-\frac{2}{p}\right|$.

Proof.

To determine (R-)boundedness of the analytic extension of $\exp(-tL)$ on $L^{p}(\mu)$ we conjugate by the (multiple of an) isometry $U_{p}:L^{p}(\mu)\to L^{p}(\lambda)$, and work with $U_{p}\exp(-tL)U_{p}^{-1}$ on $L^{p}(\lambda)$. As (multiples of) isometries preserve (R-)boundedness, $\exp(-tL)$ has an analytic extension to $z\in\mathbb{C}$ if and only if $U_{p}\exp(-tL)U_{p}^{-1}$ does, and both families of operators will be R-bounded on the same subdomains of the domain of analyticity. Using the integral kernel of Lemma 2 and the explicit form of the isometry $U_{p}$ from Definition 3, we find the integral representation for $f\in L^{p}(\lambda)$:

[TABLE]

with

[TABLE]

and $s_{t}=\frac{1-e^{-t}}{1+e^{-t}}$. If $U_{p}\exp(-tL)U_{p}^{-1}$ were to have an analytic extension $U_{p}\exp(-zL)U_{p}^{-1}$ for $z$ in some domain containing $[0,\infty)$, uniqueness theory of analytic functions implies that $U_{p}\exp(-zL)U_{p}^{-1}$ would also have an integral representation, with kernel

[TABLE]

where $s_{z}=\frac{1-e^{-z}}{1+e^{-z}}$. To understand why this must be the case, we can act $U_{p}\exp(-tL)U_{p}^{-1}$ on some $f\in L^{p}(\mu)$, and then pair with some $g\in\left(L^{p}(\mu)\right)^{*}=L^{p^{\prime}}(\mu)$ to obtain a function $\mathbb{R}^{+}\to\mathbb{C}$, $t\mapsto\left\langle U_{p}\exp(-tL)U_{p}^{-1}f,g\right\rangle$. This function will have an analytic extension to the set of $z$ for which the operator with integral kernel $k_{z}(x,y)$ is bounded on $L^{p}(\mu)$, and standard uniqueness results for $\mathbb{C}$-valued analytic functions implies that the analytic extension will be given by the operator with integral kernel $k_{z}(x,y)$ applied to $f$ and paired with $g$. Thus $U_{p}\exp(-tL)U_{p}^{-1}$ would have as weak-analytic extension the operator with integral kernel $k_{z}(x,y)$, to the set of $z$ for which this is bounded on $L^{p}(\mu)$. By the equivalence of strong-analytic and weak-analytic Banach space valued functions (see, for example, Chapter VII §3, Exercise 4 of [2]), the claim follows. (There is a slight notational issue here, in that the definition of an analytic semigroup on a Banach space $X$ is only ever analytic in the strong operator topology, such that the functions $z\mapsto\exp(-zL)f$ are $X$-valued norm-analytic functions, for each $f\in X$).

We will now work on bounding $k_{z}(x,y)$. We start by assuming that $z\in E$ (see Equation (2)). Note that this implies $Re(s_{z})>0$ and $1-e^{-2z}\neq 0$. Then we have:

[TABLE]

For notational simplicity, let $u=\left|\frac{x+y}{2\sqrt{2}}\right|$ and $k=\left|\frac{x-y}{\sqrt{2}}\right|$. Then rewriting in terms of $u$ and $k$ and completing the square in $u$ gives

[TABLE]

So

[TABLE]

For each $z\in E$, let $g_{z}:\mathbb{R}^{d}\to\mathbb{R}$ be the function

[TABLE]

Then we have that for all $z\in E$, $f\in L^{p}(\lambda)$ and a.e. $x\in\mathbb{R}^{d}$

[TABLE]

Therefore, provided the family of convolution operators $f\in L^{p}(\lambda)\mapsto g_{z}*f$ is (R-)bounded for $z$ in (a subset of) $E$, we will have proven, by domination and isometry, that $\exp(-zL)$ is (R-)bounded on (the same subset of) $E$ (to see that domination implies R-boundedness, see Proposition 8.1.10 of [4], and note that in the proof of said proposition the fixed positive operator can be replaced by an R-bounded family of positive operators). For $z\in E$, we find

[TABLE]

since $\Re(s_{z})>0$ and $|\Re(s_{z})|^{2}/|s_{z}|^{2}=\cos^{2}(\arg(s_{z}))>M_{p}^{2}$ by definition of $E$ (since $\cos\left(\frac{\pi}{2}-\theta_{p}\right)=\sin\left(\theta_{p}\right)=M_{p}$). So for $z\in E$, $g_{z}\in L^{1}(\lambda)$ and so by Young’s convolution inequality, convolution by $g_{z}$ is a bounded operator on $L^{p}(\lambda)$ with operator norm at most $||g_{z}||_{L^{1}(\lambda)}$. Now we will focus on sets of the form $E_{\epsilon,\delta}$ for some fixed $\epsilon,\delta>0$ (see Equation (3)). We will show that

[TABLE]

from which we can apply Proposition 8.2.3 of [4] to find that the family of convolution operators $\{g_{z}*\}_{z\in E_{\epsilon,\delta}}$ is R-bounded on $L^{p}(\lambda)$. Noting that each $g_{z}$ is radially decaying and positive, the quantity to bound is

[TABLE]

since $z$ is bounded away from $(2\mathbb{Z}+1)i\pi$. So the family of convolution operators $\{g_{z}*\}_{z\in E_{\epsilon,\delta}}$ is R-bounded. By pointwise domination, $U_{p}\exp(-zL)U_{p}^{-1}$ is bounded for $z\in E$, and is R-bounded on subsets $E_{\epsilon,\delta}\subset E$ of the form (3). Hence by isometric equivalence, $\exp(-zL)$ shares the same properties. Hence the claim follows from the discussion preceding this proof. ∎