# On semigroup maximal operators associated with divergence-form operators
with complex coefficients

Andrea Carbonaro  Andrea Carbonaro, Università degli Studi di Genova,
Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy
<EMAIL_ADDRESS>and  Oliver Dragičević  Oliver Dragičević, Department
of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana,
Jadranska 19, SI-1000 Ljubljana, Slovenia, and Institute of Mathematics,
Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia
<EMAIL_ADDRESS>
(Date: July 20, 2022)

###### Abstract.

Let $L_{A}=-{\rm div}\,(A\nabla)$ be an elliptic divergence form operator with
bounded complex coefficients subject to mixed boundary conditions on an
arbitrary open set $\Omega\subseteq\mathbb{R}^{d}$. We prove that the maximal
operator ${\mathscr{M}}^{A}f=\sup_{t>0}\left|{\exp(-tL_{A})f}\right|$ is
bounded in $L^{p}(\Omega)$, whenever $A$ is $p$-elliptic in the sense of [CD-
DivForm]. The relevance of this result is that, in general, the semigroup
generated by $-L_{A}$ is neither contractive in $L^{\infty}$ nor positive,
therefore neither the Hopf–Dunford–Schwartz maximal ergodic theorem
[DunfordSchwartz1, Chap. VIII] nor Akcoglu’s maximal ergodic theorem [Akc75]
can be used. We also show that if $d\geqslant 3$ and the domain of the
sesquilinear form associated with $L_{A}$ embeds into $L^{2^{*}}(\Omega)$ with
$2^{*}=2d/(d-2)$, then the range of $L^{p}$-boundedness of ${\mathscr{M}}^{A}$
improves into the interval $(rd/((r-1)d+2),rd/(d-2))$, where $r\geqslant 2$ is
such that $A$ is $r$-elliptic. With our method we are also able to study the
boundedness of the two-parameter maximal operator
$\sup_{s,t>0}\left|{T^{A_{1}}_{s}T^{A_{2}}_{t}f}\right|$.

###### 2010 Mathematics Subject Classification:

42B25, 47D06, 35J15, 42B20

##  1\. Principal result of the paper

Let $\Omega\subseteq\mathbb{R}^{d}$ be an arbitrary open set. Denote by
${\mathcal{A}}(\Omega)$ the family of all complex uniformly strictly accretive
(also called elliptic) $d\times d$ matrix functions on $\Omega$ with
$L^{\infty}$ coefficients. That is, ${\mathcal{A}}(\Omega)$ is the set of all
measurable $A:\Omega\rightarrow{\mathbb{C}}^{d\times d}$ for which there exist
$\lambda,\Lambda>0$ such that for almost all $x\in\Omega$ we have

$\displaystyle\operatorname{Re}\left\langle A(x)\xi,\xi\right\rangle$
$\displaystyle\hskip-19.0pt\geqslant\lambda|\xi|^{2}\,,$ $\displaystyle\hskip
20.0pt\forall\xi\in{\mathbb{C}}^{d};$ (1) $\displaystyle\left|{\left\langle
A(x)\xi,\eta\right\rangle}\right|$
$\displaystyle\hskip-6.0pt\leqslant\Lambda\left|{\xi}\right|\left|{\eta}\right|\,,$
$\displaystyle\hskip 20.0pt\forall\xi,\eta\in{\mathbb{C}}^{d}.$ (2)

For any $A\in{\mathcal{A}}(\Omega)$ denote by $\lambda(A)$ the largest
admissible $\lambda$ in (1) and by $\Lambda(A)$ the smallest $\Lambda$ in (2).

Given $A\in{\mathcal{A}}(\Omega)$ and $p\in(1,\infty)$, we say that $A$ is
$p$-elliptic [CD-DivForm] if $\Delta_{p}(A)>0$, where

$\Delta_{p}(A):=\underset{x\in\Omega}{{\rm
ess}\inf}\min_{\begin{subarray}{c}\xi\in{\mathbb{C}}^{d}\\\
|\xi|=1\end{subarray}}\operatorname{Re}\left\langle
A(x)\xi,\xi+|1-2/p|\bar{\xi}\right\rangle_{{\mathbb{C}}^{d}}.$ (3)

It follows straight from (3) that $\Delta_{2}(A)=\lambda(A)$, and that
$\Delta_{p}$ is invariant under conjugation of $p$, meaning that
$\Delta_{p}(A)=\Delta_{p^{\prime}}(A)$, where $1/p+1/p^{\prime}=1$. Denote by
${\mathcal{A}}_{p}(\Omega)$ the class of all $p$-elliptic matrix functions on
$\Omega$. Then ${\mathcal{A}}_{p}(\Omega)={\mathcal{A}}_{p^{\prime}}(\Omega)$.
It is known [CD-DivForm] that $A\in{\mathcal{A}}_{p}(\Omega)$ if and only
$A^{*}\in{\mathcal{A}}_{p}(\Omega)$, where $A^{*}$ denotes the conjugate
transpose of $A$. Moreover, $\\{{\mathcal{A}}_{p}(\Omega):p\in[2,\infty)\\}$
is a decreasing chain of matrix classes such that

$\begin{array}[]{rcc}\\{\text{elliptic matrices on
}\Omega\\}&=&{\mathcal{A}}_{2}(\Omega)\,,\\\ \\{\text{real elliptic matrices
on
}\Omega\\}&=&{\displaystyle\bigcap_{p\in[2,\infty)}{\mathcal{A}}_{p}(\Omega)}\,.\end{array}$

We will refer to $\lambda(A)$, $\Lambda(A)$ and $\Delta_{p}(A)$ collectively
as the $p$-ellipticity constants of $A$.

Suppose that either

  1. (a)

${\mathscr{V}}=W^{1,2}_{0}(\Omega)$,

  2. (b)

${\mathscr{V}}=W^{1,2}(\Omega)$, or

  3. (c)

${\mathscr{V}}$ is the closure in $W^{1,2}(\Omega)$ of the set of restrictions
$\\{f|_{\Omega}:f\in C_{c}^{\infty}(\mathbb{R}^{d}\backslash D)\\}$, where $D$
is a (possibly empty) closed subset of $\partial\Omega$.

For every $A\in{\mathcal{A}}(\Omega)$ we denote by $L_{A}$ the maximal
accretive Kato-sectorial operator on $L^{2}(\Omega)$ associated with the
densely defined, closed and sectorial form

${\text{\got a}}(f,g):=\int_{\Omega}\left\langle A\nabla f,\nabla
g\right\rangle_{{\mathbb{C}}^{d}},\quad f,g\in{\mathscr{V}}.$

We denote by $(T^{A}_{t})_{t>0}$ the associated contractive analytic semigroup
on $L^{2}(\Omega)$; see [Kat, Chapter VI], [AuTc] and [Ouhabook, Chapters I
and IV]. It was proven in [CD-DivForm, Theorem 1.3], [Egert2018, Theorem 2]
and [CD-MIXED, Lemma 17] that $(T^{A}_{t})_{t>0}$ extends to a contractive
analytic semigroup on $L^{p}(\Omega)$ whenever
$A\in{\mathcal{A}}_{p}(\Omega)$.

The maximal operator associated with $(T^{A}_{t})_{t>0}$ is defined by the
rule

${\mathscr{M}}^{A}f(x):=\sup_{t>0}\left|{T^{A}_{t}f(x)}\right|.$

To the best of our knowledge, boundedness of ${\mathscr{M}}^{A}$ for general
complex elliptic $A$ is unknown.

The following is our principal result.

######  Theorem 1.

Let $p\in(1,\infty)$ and $A\in{\mathcal{A}}_{p}(\Omega)$. There exists $C>0$,
that depends only on the $p$-ellipticity constants of $A$, such that

${\left\|{{\mathscr{M}}^{A}f}\right\|_{p}}\leqslant
C{\left\|{f}\right\|_{p}},\quad\forall f\in L^{p}(\Omega).$ (4)

Unless the form a is given by real coefficients, the semigroup
$(T^{A}_{t})_{t>0}$ is neither bounded in $L^{\infty}(\Omega)$, nor positive
[Ouhabook]. The novelty of Theorem 1 is that the relevant literature known to
us deals with semigroups of sub-positive contraction whose generators have a
bounded holomorphic functional calculus in $L^{p}$ [cowling] and [Stein70,
Chapter III].

The known methods allow to prove (4) only in the case where $A$ is a complex
rotation of a real matrix (see Section 2.2). Namely, [cowling, Stein70]
exploit the Hopf–Dunford–Schwartz maximal ergodic theorem and its
generalizations, that is to say, the boundedness of the ergodic maximal
operator associated with a positive contractive semigroup [DunfordSchwartz1,
Chap. VIII], [CW, Chap. 4], [Akc75] and [Garsia65]; see also [HNVW2, Section
10.7.d] for a summary of the known results in this direction. However, in the
case of complex matrices $A$ there is no general result that replaces the
Hopf–Dunford–Schwartz maximal ergodic theorem. Instead, the boundedness of the
maximal ergodic operator associated with $(T^{A}_{t})_{t>0}$ will be a
consequence of our Theorem 1.

###  1.1. Principal ideas for the proof of Theorem 1

It is known that for real elliptic matrices $B$ the maximal operator
${\mathscr{M}}^{B}$ is bounded in $L^{p}(\Omega)$ for all $p\in(1,\infty]$
(see Corollary 4).

The first idea for proving Theorem 1 comes from afar (for instance, see
[Stein76]): by triangular inequality,

${\mathscr{M}}^{A}f(x)\leqslant{\mathscr{M}}^{B}f(x)+\sup_{t>0}\left|{T^{A}_{t}f(x)-T^{B}_{t}f(x)}\right|,$
(5)

we reduce the problem to proving that the second maximal operator on the
right-hand side is bounded in $L^{p}(\Omega)$. The point is that the maximal
operator

${\mathscr{M}}^{A,B}f(x)=\sup_{t>0}\left|{T^{A}_{t}f(x)-T^{B}_{t}f(x)}\right|$

is expected to behave much better than ${\mathscr{M}}^{A}$. As an example,
consider the case of the two semigroups $\exp(-tzL)$ and $\exp(-tL)$, where
$\operatorname{Re}z>0$ and $L=-\Delta$ is the positive Euclidean Laplacian.
The Fourier multiplier
$m_{z}(\xi):=\exp(-z\left|{\xi}\right|^{2})-\exp(-\left|{\xi}\right|^{2})$
satisfies the estimates $\left|{m_{z}(\xi)}\right|\,\text{\posebni\char
46\relax}\,\min(|\xi|,|\xi|^{-1})$ and $\left|{\xi\cdot\nabla
m_{z}(\xi)}\right|\,\text{\posebni\char 46\relax}\,1$, from which one can
prove [Bou86, Carbery86, BMSW18] that the square function

$f\mapsto\left(\sum_{n\in{\mathbb{Z}}}\left|{\exp(-2^{n}zL)f-\exp(-2^{n}L)f}\right|^{2}\right)^{1/2}$

and, consequently, the maximal operator

$f\mapsto\sup_{t>0}\left|{\exp(-tL)f-\exp(-tzL)f}\right|$

are bounded in $L^{p}(\mathbb{R}^{d})$ for all $p\in(1,\infty)$.

The second idea used in the proof of Theorem 1 comes from [cowling], where the
author developed a subordination method for studying the maximal operator
$\sup_{t>0}\left|{m(t{\mathscr{A}})f}\right|$ associated with, say, a
generator ${\mathscr{A}}$ of a holomorphic semigroup with bounded imaginary
powers on $L^{p}$ and a multiplier $m\in L^{1}(\mathbb{R}_{+},\,\text{\rm
d}\lambda/\lambda)$: at least formally we have

$m(t{\mathscr{A}})f=\frac{1}{2\pi}\int_{\mathbb{R}}t^{iu}[{\mathcal{M}}m](u){\mathscr{A}}^{iu}f\,\text{\rm
d}u,$ (6)

where ${\mathcal{M}}$ denotes the Mellin transform of $m$, which is defined by
rule

$[{\mathcal{M}}m](u)=\int^{\infty}_{0}m(\lambda)\lambda^{-iu}\frac{\,\text{\rm
d}\lambda}{\lambda},\quad u\in\mathbb{R}.$ (7)

It follows from (6) that

${\left\|{\sup_{t>0}\left|{m(t{\mathscr{A}})f}\right|}\right\|_{p}}\leqslant\frac{1}{2\pi}\int_{\mathbb{R}}\left|{[{\mathcal{M}}m](u)}\right|{\left\|{{\mathscr{A}}^{iu}f}\right\|_{p}}\,\text{\rm
d}u.$ (8)

This reduces the estimate of the $L^{p}$-norm of maximal function on the left-
hand side of (6) to finding poinwise estimates of ${\mathcal{M}}m$ and
$L^{p}$-estimates of imaginary powers of ${\mathscr{A}}$.

If we apply this method to the multiplier

$m_{\pm\theta}(\lambda):=\exp(-e^{\pm
i\theta}\lambda)-\exp(-\lambda),\quad\lambda>0,\quad 0<\theta<\pi/2,$ (9)

and the generator $L_{B}$ where $B\in{\mathcal{A}}(\Omega)$ is real, then it
follows from [CD-MIXED, Theorem 3] that ${\mathscr{M}}^{e^{\pm i\theta}B,B}$
is bounded in $L^{p}(\Omega)$ whenever the matrix $A=e^{\pm i\theta}B$ is
$p$-elliptic (see Section 2.2 for details). This argument breaks down when the
complex rotations of $B$ are replaced by a general
$A\in{\mathcal{A}}_{p}(\Omega)$, because the two generators $L_{A}$ and
$L_{B}$, in general, are not related via functional calculus (see also the
comments in Section 3). Let us explain how we get around this.

Let $A,B\in{\mathcal{A}}(\Omega)$, where $B$ is real. The operators
$T^{A}_{t}$ and $T^{B}_{t}$ are related by the Duhamel’s formula:

$T^{A}_{t}f-T^{B}_{t}f=\int^{t}_{0}T^{B}_{s}(L_{B}-L_{A})T^{A}_{t-s}f\,\text{\rm
d}s,\quad\forall f\in(L^{2}\cap L^{p})(\Omega).$ (10)

Duhamel’s formula is typically used for showing that the difference of the two
semigroups is small (with respect to some topology) whenever the matrices $A$
and $B$ are close, for example, with respect to the $L^{\infty}$-norm. So it
would seem futile to use the estimate

$\left|{T^{A}_{t}f-T^{B}_{t}f}\right|\leqslant\left|{\int^{t}_{0}L_{B}T^{B}_{s}T^{A}_{t-s}f\,\text{\rm
d}s}\right|+\left|{\int^{t}_{0}T^{B}_{s}L_{A}T^{A}_{t-s}f\,\text{\rm
d}s}\right|,$ (11)

but actually it works.

For simplicity consider the first term on the right-hand side (11). Our idea
is to transfer a fractional power of $L_{B}$ from the first semigroup to the
second one as follows:

$L_{B}T^{B}_{s}T^{A}_{t-s}=L^{1-\alpha}_{B}T^{B}_{s}L^{\alpha}_{B}T^{A}_{t-s},\quad\alpha\in(0,1).$

Now we would like to transform the fractional power of $L_{B}$ into a
fractional power of $L_{A}$, more precisely we aim at the decomposition

$L^{\alpha}_{B}=U^{B,A}_{\alpha}L^{\alpha}_{A},$

where $U^{B,A}_{\alpha}$ is a bounded operator on $L^{p}(\Omega)$.

This is not possible for all $A,B,p,\alpha$, because for, say,
$1/2\leqslant\alpha<1$ and $p=2$ the domains of $L^{\alpha}_{A}$ and
$L^{\alpha}_{B}$ are either unrelated or unknown. This procedure with
$\alpha=1/2$ and $p=2$ would require a solution to the Kato problem for the
square root of $L_{A}$ which is still an open problem in the general setting
we consider in this paper.

However, by combining a classic result of Kato [Kato61] with [CD-MIXED,
Theorem 3] and a classic complex interpolation argument, we are able to make
the procedure described above work for a small positive power
$\alpha=\alpha(p)<1/2$ whenever $A\in{\mathcal{A}}_{p}(\Omega)$ (see
Proposition 8). This gives

$\int^{t}_{0}L_{B}T^{B}_{s}T^{A}_{t-s}f\,\text{\rm
d}s=\int^{t}_{0}\psi_{1-\alpha}(sL_{B})U^{B,A}_{\alpha}\psi_{\alpha}((t-s)L_{A})f\,\text{\rm
d}\mu^{t}_{\alpha}(s)$

with $U^{B,A}_{\alpha}$ as above,

$\,\text{\rm
d}\mu^{t}_{\alpha}(s)=s^{\alpha-1}(t-s)^{-\alpha}\mathbbm{1}_{[0,t]}(s)\,\text{\rm
d}s,\quad t>0,$ (12)

and

$\psi_{\beta}(\lambda)=\lambda^{\beta}e^{-\lambda},\quad\lambda>0,\quad\beta>0.$
(13)

The advantage of this decomposition is that $\mu^{t}_{\alpha}(\mathbb{R})$ is
a finite measure on $\mathbb{R}$ with total mass independent of $t$. More
precisely,

$\mu^{t}_{\alpha}(\mathbb{R})=B(\alpha,1-\alpha)=\frac{\pi}{\sin(\alpha\pi)},\quad\forall
t>0.$ (14)

Then, by using Cowling’s subordination [cowling] twice in combination with our
functional calculus result in [CD-MIXED, Proposition 3], we obtain that the
two-variable maximal operator

$\sup_{0<s\leqslant
t}\left|{\psi_{1-\alpha}(sL_{B})U^{B,A}_{\alpha}\psi_{\alpha}((t-s)L_{A})f}\right|$

is bounded in $L^{p}(\Omega)$, whenever $A\in{\mathcal{A}}_{p}(\Omega)$ (see
Section 3.1). In this way we can prove that ${\mathscr{M}}^{A,B}$ — and thus
${\mathscr{M}}^{A}$ — is bounded in $L^{p}(\Omega)$ whenever
$A\in{\mathcal{A}}_{p}(\Omega)$.

##  2\. More notation and preliminaries

Let $1<p<2$ and $A\in{\mathcal{A}}_{p}(\Omega)$. With a slight abuse of
notation, we maintain the symbol $L_{A}$ for denoting the negative generator
of $(T^{A}_{t})_{t>0}$ in $L^{q}(\Omega)$, $p\leqslant q\leqslant p^{\prime}$.
We have $L^{q}(\Omega)={\rm
N}_{q}(L_{A})\oplus\overline{\operatorname{R}_{q}(L_{A})}$ and
$Q_{q}f=\lim_{t\rightarrow+\infty}T^{A}_{t}f$ is a consistent family of
contractive projections on ${\rm N}_{q}(L_{A})$ for $p\leqslant q\leqslant
p^{\prime}$ [CDMY]. A simple calculation based on the definition of $L_{A}$ by
means of the sesquilinear form a shows that ${\rm N}_{2}(L_{A})={\rm
N}_{2}(L_{I})$ consists of the vector space of all functions which are
constant on each connected component of $\Omega$ of finite measure whose
boundary does not intersect $D$ in a set of positive relative $2$-capacity
[AreWarm03], and are $0$ on all other connected components of $\Omega$. In
particular, $N_{2}(L_{A})\subseteq L^{q}(\Omega)$, which implies ${\rm
N}_{q}(L_{A})={\rm N}_{2}(L_{A})={\rm N}_{2}(L_{I})$, whenever $A$ is
$p$-elliptic and $p\leqslant q\leqslant p^{\prime}$. In order to simplify our
proofs, in this paper we shall always assume that ${\rm
N}_{2}(L_{I})=\\{0\\}$.

Under this assumption $L_{A}$ is an injective sectorial operator of angle
$<\pi/2$ on $L^{q}(\Omega)$, whenever elliptic and $p\leqslant q\leqslant
p^{\prime}$. In this range of $q$’s the complex powers $L^{z}_{A}$ are well
defined on $L^{q}(\Omega)$ for every $z\in{\mathbb{C}}$. Also, for every
$z\in{\mathbb{C}}$, the closed densely defined operators $L^{z}_{A}$,
$p\leqslant\chi\leqslant p^{\prime}$, are consistent [Haase, Yagi2010].

######  Proposition 2 ([CD-MIXED]).

Let $p>1$ and $A\in{\mathcal{A}}_{p}(\Omega)$. Then $L_{A}$ has bounded
$H^{\infty}$-calculus of angle $<\pi/2$ on $L^{p}(\Omega)$ in the sense of
[CDMY]. It particular, $L_{A}$ has bounded imaginary powers on $L^{p}(\Omega)$
and there exist $\theta_{p}\in(0,\pi/2)$ and $C>0$ such that

${\left\|{L^{iu}_{A}}\right\|_{p}}\leqslant Ce^{\theta_{p}|u|},\quad\forall
u\in\mathbb{R}.$ (15)

The angle $\theta_{p}$ and the constant $C$ only depend on $p$ and the
$p$-ellipticity constants of $A$ [CD-DivForm, Corollary 5.17 and Proposition
5.23].

###  2.1. Cowling’s subordination technique

It is by now a classic result in semigroup theory that for a semigroup
$(S_{t})_{t>0}$ of linear transformations on some Lebesgue spaces $L^{p}$, the
boundedness of the associated maximal ergodic operator in $L^{p}$ together
with a $H^{\infty}$-calculus of angle $<\pi/2$ in $L^{p}$ for the negative
generator implies the boundedness of the associated maximal operator in
$L^{p}$. One way to prove this result is to use Cowling’s subordination
technique [cowling] (see also [Meda, CD-mult]) which we briefly describe below
in the specific case we are interested in here (see (16), Proposition 3 and
Corollary 4). Note that Cowling originally formulated his argument only for
symmetric semigroups. Note also that as an alternative to Cowling’s
subordination method, we can slightly modify an argument of Stein [Stein70,
Chapter III] that exploits complex interpolation and fractional integration.

Fix $A\in{\mathcal{A}}(\Omega)$. Define the associated maximal ergodic
operator by the rule

${\mathscr{E}}^{A}f(x)=\sup_{t>0}\left|{\frac{1}{t}\int^{t}_{0}T^{A}_{s}f(x)\,\text{\rm
d}s}\right|,\quad f\in L^{2}(\Omega).$

A rapid calculation shows that for all $f\in(L^{2}\cap L^{p})(\Omega)$ and
$t>0$ we have

$T^{A}_{t}f=\frac{1}{t}\int^{t}_{0}T^{A}_{s}f\,\text{\rm
d}s-\frac{1}{t}\int^{t}_{0}\psi_{1}(sL_{A})f\,\text{\rm d}s,$

where $\psi_{\alpha}$, $\alpha>0$, is defined by (13) and the bounded operator
$\psi_{1}(tL_{A})$ on $L^{2}(\Omega)$ is defined by means of McIntosh’s
$L^{2}$-functional calculus [Mc]. Therefore,

$\sup_{t>0}\left|{T^{A}_{t}f(x)}\right|\leqslant{\mathscr{E}}^{A}f(x)+\sup_{t>0}\left|{\psi_{1}(tL_{A})f(x)}\right|,\quad\forall
f\in(L^{2}\cap L^{p})(\Omega).$ (16)

######  Proposition 3.

Let $p>1$. Suppose that $A\in{\mathcal{A}}_{p}(\Omega)$. For every $\alpha>0$
there exists $C>0$, which depends only on $\alpha$, $p$ and the
$p$-ellipticity constants of $A$, such that

${\left\|{\sup_{t>0}\left|{\psi_{\alpha}(tL_{A})f}\right|}\right\|_{p}}\leqslant
C{\left\|{f}\right\|_{p}},\quad\forall f\in L^{p}(\Omega).$

###### Proof.

We use Cowling’s subordination (8). The Mellin transform (7) of
$\psi_{\alpha}$ is $[{\mathcal{M}}\psi_{\alpha}](u)=\Gamma(\alpha-iu)$,
$u\in\mathbb{R}$, where $\Gamma$ denotes the Euler gamma function. Stirling’s
formula gives

$\left|{\Gamma(\alpha-
iu)}\right|\sim\sqrt{2\pi}\,|u|^{\alpha-1/2}e^{-\frac{\pi}{2}|u|},\quad\textrm{for
}|u|\rightarrow\infty.$ (17)

Therefore,

${\left\|{\sup_{t>0}\left|{\psi_{\alpha}(tL_{A})f}\right|}\right\|_{p}}\,\text{\posebni\char
46\relax}\,\int_{\mathbb{R}}(1+|u|)^{\alpha-1/2}e^{-\frac{\pi}{2}|u|}{\left\|{L^{iu}_{A}f}\right\|_{p}}\,\text{\rm
d}u.$

Now the desired result follows from Proposition 2. ∎

######  Corollary 4.

Let $B\in{\mathcal{A}}(\Omega)$ be a real matrix. Then ${\mathscr{M}}^{B}$ is
bounded in $L^{p}(\Omega)$, for every $p\in(1,+\infty]$.

###### Proof.

As remarked before, when $B$ is real elliptic $(T^{B}_{t})_{t>0}$ is positive
and contractive on $L^{\infty}(\Omega)$ and on $L^{1}(\Omega)$ by the duality
$(T^{B}_{t})^{*}=T^{B^{*}}_{t}$ [Ouhabook]. Hence by the Hopf–Dunford–Schwartz
theorem ${\mathscr{E}}^{B}$ is bounded in $L^{p}(\Omega)$ for all $p>1$.
Moreover, $B\in{\mathcal{A}}_{p}(\Omega)$ for all $p>1$, therefore by
Proposition 3 the maximal operator
$f\mapsto\sup_{t>0}\left|{\psi_{1}(tL_{B})f}\right|$ is bounded in
$L^{p}(\Omega)$ for all $p>1$. The desired result for $p\neq\infty$ now
follows from (16), while the boundedness of the maximal operator in
$L^{\infty}(\Omega)$ trivially follows from the contractivity of
$(T^{B}_{t})_{t>0}$ on $L^{\infty}(\Omega)$. ∎

###  2.2. Complex rotations of real elliptic matrices

In the case of a symmetric contraction semigroup $(S_{t})_{t>0}$ on some
$\sigma$-finite measure space, Cowling [cowling] observed that the
subordination technique explained in Section 2.1 can be used for proving the
boundedness on $L^{p}(\Omega)$, $1<p<\infty$, of the maximal operator

$f\mapsto\underset{|\arg(z)|<\omega_{p}-\varepsilon}{\sup}\left|{S_{z}f}\right|,$

where $\omega_{p}=\pi/2-\omega^{*}_{p}$ and $\omega^{*}_{p}$ denotes the angle
of the $H^{\infty}$-calculus in $L^{p}$ of the negative generator of
$(S_{t})_{t>0}$. Recall that by [CD-mult], for every symmetric contraction
semigroup we have
$\omega^{*}_{p}\leqslant\phi^{*}_{p}:=\arcsin\left|{1-2/p}\right|$. Despite
the fact that for a real elliptic matrix $B$, in general, the semigroup
$(T^{B}_{t})_{t>0}$ is not symmetric, Cowling’s argument can be easily adapted
and in combination with Proposition 2 gives the following extension of
Corollary 4.

######  Proposition 5.

Let $B\in{\mathcal{A}}(\Omega)$ be a real matrix. Suppose that $1<p<\infty$
and $0\leqslant\theta<\pi/2$ are such that
$e^{i\theta}B\in{\mathcal{A}}_{p}(\Omega)$. Then ${\mathscr{M}}^{e^{\pm
i\theta}B}$ is bounded on $L^{p}(\Omega)$.

###### Proof.

Recall the definition (9) of $m_{\pm\theta}$. Then,

${\mathscr{M}}^{e^{\pm
i\theta}B}f\leqslant{\mathscr{M}}^{B}f+\sup_{t>0}\left|{m_{\pm\theta}(tL_{B})f}\right|.$

By Corollary 4 the maximal operator ${\mathscr{M}}^{B}$ is bounded on
$L^{p}(\Omega)$.

As for the maximal operator associated with $m_{\pm\theta}$, its Mellin
transform (7) is

$[{\mathcal{M}}m_{\pm\theta}](u)=i\theta\left(\frac{e^{\mp\theta\,u}-1}{\theta
u}\right)\Gamma(1-iu),\quad u\in\mathbb{R}.$

Hence by (17),

$\left|{[{\mathcal{M}}m_{\pm\theta}](u)}\right|\leqslant
C(1+|u|)^{1/2}\exp((\theta-\pi/2)|u|),\quad\forall u\in\mathbb{R}.$

Now use the assumptions, the fact that ${\mathcal{A}}_{p}(\Omega)$ is
invariant by Hermitian transposition, the identity $L_{e^{\pm
i\theta}B}=e^{\pm i\theta}L_{B}$, Proposition 2 (applied with $A=e^{\pm
i\theta}B$) and Cowling’s subordination (8) with ${\mathscr{A}}=L_{B}$ and
$m=m_{\pm\theta}$. ∎

######  Remark 6.

Let $B$ a real elliptic matrix. By [CD-DivForm, Propositions 5.13 and 5.21]
$e^{\pm i\theta}B$ is $p$-elliptic precisely when
$\theta<\pi/2-\vartheta^{*}_{p}$, where
$\vartheta^{*}_{p}=\vartheta^{*}_{p}(B)$ is given by [CD-OU, (3.7)]. This is
consistent with the general result for symmetric contractions [CD-mult]
explained above, because $T^{e^{\pm i\theta}B}_{t}=T^{B}_{te^{\pm i\theta}}$
and when the real matrix $B$ is symmetric we have
$\vartheta^{*}_{p}=\phi^{*}_{p}=\arcsin\left|{1-2/p}\right|$.

##  3\. Proof of Theorem 1

The proof of Proposition 5 is based on the fact that $T^{e^{\pm
i\theta}B}_{t}-T^{B}_{t}$ coincides with the multiplier
$m_{\pm\theta}(tL_{B})$ of the generator $L_{B}$: this is clearly false when
we replace the complex rotations of the real matrix $B$ by a general complex
$p$-elliptic matrix $A$, since $T^{A}_{t}$ and $T^{B}_{t}$ may not commute and
the generators $L_{A}$ and $L_{B}$ may not have a joint functional calculus.
As we already explained in Section 1.1, despite this difficulty, we prove
Theorem 1 by combining Corollary 4 with Proposition 2 and using Cowling’s
subordination (6) appropriately. To do that, another ingredient is needed: a
classic result of Kato [Kato61] concerning the domain of the fractional powers
of Kato-sectorial operators, which we state below in the particular case we
will use in this paper.

Recall that ${\mathcal{V}}$ denotes the domain of the sesquilinear form
associated with $L_{A}$, $A\in{\mathcal{A}}(\Omega)$.

######  Lemma 7 (Kato [Kato61] and Lions [Lions62]).

Let $A,B\in{\mathcal{A}}(\Omega)$ and $0<\alpha<1/2$. Then

$\operatorname{D}_{2}(L^{\alpha}_{A})=\operatorname{D}_{2}(L^{\alpha}_{B})=\left[L^{2}(\Omega),{\mathscr{V}}\right]_{2\alpha}=:{\mathcal{D}}_{\alpha},$

where ${\mathcal{D}}_{\alpha}$ denotes the complex interpolation space of
index $2\alpha$ between $L^{2}(\Omega)$ and ${\mathscr{V}}$. Moreover there
exist $C_{\alpha},C^{\prime}_{\alpha}>0$ depending only on $\alpha$ and the
ellipticity constants of $A$ and $B$ such that

$C_{\alpha}{\left\|{L^{\alpha}_{A}f}\right\|_{2}}\leqslant{\left\|{L^{\alpha}_{B}f}\right\|_{2}}\leqslant
C^{\prime}_{\alpha}{\left\|{L^{\alpha}_{A}f}\right\|_{2}},\quad\forall
f\in{\mathcal{D}}_{\alpha}.$ (18)

Moreover, for every $\alpha_{0}\in[0,1/2)$ we have
$\sup_{0\leqslant\alpha\leqslant\alpha_{0}}\left(\left|{C_{\alpha}}\right|^{-1}+\left|{C^{\prime}_{\alpha}}\right|\right)<+\infty$.

Recall that for every $A\in{\mathcal{A}}(\Omega)$ we have ${\rm
N}_{2}(L_{A})={\rm N}_{2}(L_{I})$ and that for simplicity in this paper we
always assume that $\Omega$ and ${\mathscr{V}}$ are such that ${\rm
N}_{2}(L_{I})=\\{0\\}$. Therefore $\overline{{\rm
R}}_{2}(L_{A})=L^{2}(\Omega)$ and (18) is equivalent to the fact that

$U^{B,A}_{\alpha}:=L^{\alpha}_{B}L^{-\alpha}_{A}$

extends to a bounded (invertible) operator in $L^{2}(\Omega)$, for all
$\alpha\in(0,1/2)$ and all $A,B\in{\mathcal{A}}(\Omega)$. This together with a
standard complex interpolation argument based on (15) gives the next
proposition which extends the $L^{2}$-boundedness of $U^{B,A}_{\alpha}$ to the
$L^{p}$-boundedness in the range of $p$-ellipticity of $A$ and $B$.

######  Proposition 8.

Let $p>1$. Suppose that $A,B\in{\mathcal{A}}_{p}(\Omega)$. Then there exists
$\alpha=\alpha(p)\in(0,1/2)$ such that $U^{B,A}_{\alpha}$ extends to a bounded
operator in $L^{p}(\Omega)$.

###### Proof.

For simplicity, suppose that $1<p<2$. For every $\alpha>0$ define the strip

$\Sigma_{\alpha}=\\{z\in{\mathbb{C}}:\ 0<\operatorname{Re}z<\alpha\\}.$

Fix $\alpha_{0}\in(0,1/2)$ and consider the family of operators

$S_{z}=e^{z^{2}}L^{i\operatorname{Im}z}_{B}U^{B,A}_{\operatorname{Re}z}L^{-i\operatorname{Im}z}_{A},\quad
z\in\Sigma_{\alpha_{0}}.$

The imaginary powers of $L_{A}$ and $L_{B}$ are bounded in $L^{2}(\Omega)$
(this is a consequence of [CrDe], but it also follows from (15) applied with
$p=2$) and, by Lemma 7, the operators $U^{B,A}_{\operatorname{Re}z}$ are
bounded in $L^{2}(\Omega)$ uniformly in $z\in\overline{\Sigma}_{\alpha_{0}}$.
Hence,

${\left\|{S_{z}}\right\|_{2}}\leqslant C(\alpha_{0}),\quad\forall
z\in\overline{\Sigma}_{\alpha_{0}}.$ (19)

Moreover, by (15) we have

${\left\|{S_{i\sigma}}\right\|_{p}}\leqslant
C_{p},\quad\forall\sigma\in\mathbb{R}.$

Suppose we have already proved that for all $f\in L^{2}(\Omega)$ the function

$z\mapsto S_{z}f\in L^{2}(\Omega)$

is continuous and bounded on $\overline{\Sigma}_{\alpha_{0}}$, and holomorphic
in the interior of the strip. Then, by Stein’s complex interpolation theorem,
for every $q\in(p,2)$ there exist $\alpha(q)\in(0,\alpha_{0})$ and $C_{q}>0$
such that

${\left\|{U^{B,A}_{\alpha(q)}}\right\|_{q}}\leqslant C_{q}.$

This proves the proposition, because the interval of $p$-ellipticity of $A$
and $B$ is open [CD-DivForm].

It remains to show that $z\mapsto S_{z}f\in L^{2}(\Omega)$ is continuous and
bounded on $\overline{\Sigma}_{\alpha_{0}}$, and holomorphic in the interior
of the strip. We first show that

$S_{z}f=e^{-z^{2}}L^{z}_{B}L^{-z}_{A}f,\quad\forall
z\in\Sigma_{1/2},\quad\forall f\in\operatorname{R}_{2}(L_{A}).$ (20)

Indeed, since the imaginary powers of $L_{B}$ are bounded in $L^{2}(\Omega)$,
we have
$\operatorname{D}(L^{z}_{B})=\operatorname{D}(L^{\operatorname{Re}z}_{B})$ and
$L^{z}_{B}=L^{i\operatorname{Im}z}_{B}L^{\operatorname{Re}z}_{B}$; see, for
example, [Haase, Proposition 3.2.1 (c)] applied with
$\alpha=i\operatorname{Im}z$ and $\beta=\operatorname{Re}z$. Therefore, by
Lemma 7 we have

$L^{z}_{B}=L^{i\operatorname{Im}z}_{B}L^{\operatorname{Re}z}_{B}L^{-\operatorname{Re}z}_{A}L^{\operatorname{Re}z}_{A}=L^{i\operatorname{Im}z}_{B}U^{B,A}_{\operatorname{Re}z}L^{\operatorname{Re}z}_{A}.$
(21)

Using once again [Haase, Proposition 3.2.1 (c)] we deduce that

$L^{\operatorname{Re}z}_{A}L^{-z}_{A}\subset\overline{L^{\operatorname{Re}z}_{A}L^{-z}_{A}}=L^{\operatorname{Re}z-z}_{A}=L^{-i\operatorname{Im}z}_{A}.$
(22)

By [Haase, Proposition 3.1.1 (c)] we have
$L_{A}=L^{z+(1-z)}_{A}=L^{1-z}_{A}L^{z}_{A}=L^{z}_{A}L^{1-z}_{A}$, which
implies

$\operatorname{R}_{2}(L_{A})\subseteq\operatorname{R}_{2}(L^{z}_{A}),\quad\forall
z\in\overline{\Sigma}_{1}.$ (23)

Now (20) follows by combining (21) with (22) and (23).

We are now ready to study the regularity of $z\mapsto S_{z}f$. Fix
$f\in\operatorname{D}_{2}(L_{A})\cap\operatorname{R}_{2}(L_{A})$ and
$g\in\operatorname{D}_{2}(L_{B^{*}})\cap\operatorname{R}_{2}(L_{B^{*}})$. Let
$z\in{\mathbb{C}}$ be such that $0\leqslant\operatorname{Re}z<1/2$. By Lemma 7
we have $\operatorname{D}_{2}(L^{z}_{A})=\operatorname{D}_{2}(L^{z}_{B})$,
which together with (23) gives

$\left\langle L^{z}_{B}L^{-z}_{A}f,g\right\rangle_{L^{2}}=\left\langle
L^{-z}_{A}f,L^{\bar{z}}_{B^{*}}g\right\rangle_{L^{2}}.$

By [Haase, Proposition 3.2.1 (f)] (applied with $\alpha_{0}=\alpha_{1}=1/2$)
the maps $z\mapsto L^{-z}_{A}f$ and $z\mapsto\overline{L^{\bar{z}}_{B^{*}}g}$
are analytic in the strip $-1/2<\operatorname{Re}z<1/2$.

We now observe that, by assumption,
$L^{2}(\Omega)=\overline{\operatorname{R}}_{2}(L_{B^{*}})$. Hence
$\operatorname{D}_{2}(L_{B_{*}})\cap\operatorname{R}_{2}(L_{B^{*}})$ is dense
in $L^{2}(\Omega)$. It follows form (20) and the previous considerations that
for every $f\in\operatorname{D}_{2}(L_{A})\cap\operatorname{R}_{2}(L_{A})$ the
map $z\mapsto S_{z}f\in L^{2}(\Omega)$ is continuous in
$\overline{\Sigma}_{\alpha_{0}}$ and holomorphic in the interior of the strip.
By the uniform boundedness of the family
$\\{S_{z}:z\in\overline{\Sigma}_{\alpha_{0}}\\}$ (see (19)) and the density of
$\operatorname{D}_{2}(L_{A})\cap\operatorname{R}_{2}(L_{A})$ in
$L^{2}(\Omega)$, we deduce that the same is true for every $f\in
L^{2}(\Omega)$. ∎

###  3.1. Proof of Theorem 1

Fix a real elliptic matrix $B$ (for example, $B=I$). By Corollary 4 the
maximal operator associated with $(T^{B}_{t})_{t>0}$ is bounded in
$L^{p}(\Omega)$. Hence by (5) it suffices to prove that the maximal operator
${\mathscr{M}}^{A,B}$ is bounded in $L^{p}(\Omega)$. Fix $f\in(L^{2}\cap
L^{p})(\Omega)$. By Duhamel’s formula (11) for every $t>0$ we have

$\displaystyle\left|{T^{A}_{t}f-T^{B}_{t}f}\right|$
$\displaystyle\leqslant\left|{\int^{t}_{0}L_{B}T^{B}_{s}T^{A}_{t-s}f\,\text{\rm
d}s}\right|+\left|{\int^{t}_{0}T^{B}_{s}L_{A}T^{A}_{t-s}f\,\text{\rm
d}s}\right|$ $\displaystyle=\left|{I_{t}(f)}\right|+\left|{II_{t}(f)}\right|.$

####  3.1.1. The maximal operator $\sup_{t>0}\left|{I_{t}(f)}\right|$

Let $\alpha=\alpha(p)\in(0,1/2)$ be as in Proposition 8. Set $\beta=1-\alpha$.
Then we have

$I_{t}(f)=\int^{t}_{0}\psi_{\beta}(sL_{B})U^{B,A}_{\alpha}\psi_{\alpha}((t-s)L_{A})f\,\text{\rm
d}\mu^{t}_{\alpha}(s),$ (24)

where the finite measures $\mu^{t}_{\alpha}$, $t>0$, are defined by (12).
Hence by (14) we have

$\sup_{t}\left|{I_{t}(f)}\right|\leqslant\frac{\pi}{\sin(\alpha\pi)}\sup_{0<s\leqslant
t}\left|{\psi_{\beta}(sL_{B})U^{B,A}_{\alpha}\psi_{\alpha}((t-s)L_{A})f}\right|.$

We now subordinate the operators $\psi_{\beta}(sL_{B})$ and
$\psi_{\alpha}((t-s)L_{A})$ to imaginary powers by means of (6) and obtain

$\sup_{0<s\leqslant
t}\left|{\psi_{\beta}(sL_{B})U^{B,A}_{\alpha}\psi_{\alpha}((t-s)L_{A})f}\right|\leqslant\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}\left|{\Gamma(\beta-
iu)}\right|\left|{\Gamma(\alpha-
iv)}\right|\left|{L^{iu}_{B}U^{B,A}_{\alpha}L^{iv}_{A}f}\right|\,\text{\rm
d}u\,\text{\rm d}v.$

The boundedness of $f\mapsto\sup_{t}\left|{I_{t}(f)}\right|$ now follows by
combining the Stirling’s formula (17) with Propositions 8 and 2.

####  3.1.2. The maximal operator $\sup_{t>0}\left|{II_{t}(f)}\right|$

For every $\alpha\in(0,1/2)$ set

$V^{B,A}=\left(U^{A^{*},B^{*}}_{\alpha}\right)^{*}.$

A rapid calculation shows that

$V^{B,A}{\mathcal{D}}_{\alpha}\subseteq{\mathcal{D}}_{\alpha}\quad\textrm{and}\quad
L^{\alpha}_{B}V^{B,A}f=L^{\alpha}_{A}f,\quad\forall
f\in{\mathcal{D}}_{\alpha}.$

Fix $\alpha=\alpha(p^{\prime})$ as in Proposition 8, but for the pair
$(A^{*},B^{*})$ instead of the pair $(B,A)$. Set $\beta=1-\alpha$. Then
$V^{B,A}$ extends to a bounded operator on $L^{p}(\Omega)$ and

$II_{t}(f)=\int^{t}_{0}L^{\alpha}_{B}T^{B}_{s}V^{B,A}L^{\beta}_{A}T^{A}_{t-s}f\,\text{\rm
d}s=\int^{t}_{0}\psi_{\alpha}(sL_{B})V^{B,A}_{\alpha}\psi_{\beta}((t-s)L_{A})f\,\text{\rm
d}\mu^{t}_{\beta}(s)$

Proceeding as we did in the previous section, we obtain the boundedness on
$L^{p}(\Omega)$ of the maximal operator
$f\mapsto\sup_{t>0}\left|{II_{t}(f)}\right|$ that completes the proof of
Theorem 1.

##  4\. Range improvement in Theorem 1

In this section we consider an open set $\Omega\subseteq\mathbb{R}^{d}$,
$d\geqslant 3$. We fix a subspace ${\mathscr{V}}$ of the type described on
page 1 and we assume that it satisfies the homogeneous embedding property:

${\left\|{u}\right\|_{2^{*}}}\,\text{\posebni\char 46\relax}\,{\left\|{\nabla
u}\right\|_{2}},\quad\forall u\in{\mathscr{V}},$ (25)

where $2^{*}=2d/(d-2)$ is the usual Sobolev upper exponent. For every
$p\geqslant 2$ we define the two exponents

$p^{\circ}=\frac{pd}{d-2},\quad p_{\circ}=\frac{p^{\circ}}{p^{\circ}-1}.$

Note that $2^{\circ}=2^{*}$.

######  Lemma 9 ([Egert2018] and [Elst2019]).

Assume that ${\mathscr{V}}$ satisfies (25). Let $p>2$ and
$A\in{\mathcal{A}}_{p}(\Omega)$. Then for all $r\in[p_{\circ},p^{\circ}]$ the
semigroup $(T^{A}_{t})_{t>0}$ extends to a uniformly bounded analytic
semigroup in $L^{r}(\Omega)$. Moreover, the analyticity angle does not depend
on $r\in[p_{\circ},p^{\circ}]$.

######  Remark 10.

The homogeneous Sobolev embedding (25) implies ${\rm N}_{2}(L_{I})=\\{0\\}$
and the very same considerations of Section 2 show that, under the assumptions
of Lemma 9, we have

$\\{0\\}={\rm N}_{2}(L_{I})={\rm N}_{2}(L_{A})={\rm N}_{r}(L_{A}),\quad\forall
r\in[p_{\circ},p^{\circ}].$

In the next result $\vartheta_{2}$ denotes the sectoriality angle of $L_{A}$
in $L^{2}(\Omega)$. We always have
$0\leqslant\vartheta_{2}\leqslant\arccos(\lambda/\Lambda)<\pi/2$.

######  Lemma 11 ([Egert2018, Egert2]).

Under the assumptions of Lemma 9, for every $r\in[p_{\circ},p^{\circ}]$ the
generator $L_{A}$ has a bounded $H^{\infty}(\mathbf{S}_{\theta})$-calculus in
$L^{r}(\Omega)$, whenever $\theta>\vartheta_{2}$. In particular, there exists
$\theta_{0}\in(0,\pi/2)$ such that

${\left\|{L^{iu}_{A}}\right\|_{r}}\leqslant
C(r,d)e^{\theta_{0}|u|},\quad\forall u\in\mathbb{R},$ (26)

for all $r\in[p_{\circ},p^{\circ}]$.

######  Theorem 12.

Assume that ${\mathscr{V}}$ satisfies (25). Let $p>2$ and
$A\in{\mathcal{A}}_{p}(\Omega)$. Then the maximal operator ${\mathscr{M}}^{A}$
is bounded in $L^{r}(\Omega)$ for all $r\in[p_{\circ},p^{\circ}]$.

###### Proof.

Exactly as we did in the proof of Theorem 1, we fix a real matrix $B$ and
reduce the proof to showing that the maximal operator ${\mathscr{M}}^{A,B}$ is
bounded in $L^{r}(\Omega)$. By using (26) it is easy to see that Proposition 8
extends to the range $[p_{\circ},p^{\circ}]$ and, using again (26), we can now
repeat the arguments in Sections 3.1.1 and 3.1.2. ∎

##  5\. Two-parameter maximal operator

Suppose that $A_{1},A_{2}\in{\mathcal{A}}_{p}(\Omega)$. Define the associated
two-parameter maximal operator by the rule

${\text{\got
M}}^{A_{1},A_{2}}f=\sup_{w,t>0}\left|{T^{A_{1}}_{w}T^{A_{2}}_{t}f}\right|.$

######  Theorem 13.

Let $p\in(1,\infty)$ and $A_{1},A_{2}\in{\mathcal{A}}_{p}(\Omega)$. Then
${\text{\got M}}^{A_{1},A_{2}}$ is bounded on $L^{p}(\Omega)$.

###### Proof.

In the special case when $A_{1}$ is real-valued, the associated semigroup is
nonnegative and, therefore, we have

$\left|{T^{A_{1}}_{w}T^{A_{2}}_{t}f}\right|\leqslant
T^{A_{1}}_{w}{\mathscr{M}}^{A_{2}}f\leqslant{\mathscr{M}}^{A_{1}}{\mathscr{M}}^{A_{2}}f.$
(27)

It follows from Theorem 1 that ${\text{\got M}}^{A_{1},A_{2}}$ is bounded on
$L^{p}(\Omega)$.

When $A_{1}$ is not real-valued, the domination in (27) does not necessarily
hold and we need another argument for proving the boundedness of ${\text{\got
M}}^{A_{1},A_{2}}$. It is a variant of the proof of Theorem 1.

Fix $f\in L^{p}(\Omega)$ and write

$T^{A_{1}}_{w}T^{A_{2}}_{t}f=T^{A_{1}}_{t+w}f+T^{A_{1}}_{w}(T^{A_{2}}_{t}f-T^{A_{1}}_{t}f).$

We clearly have

${\text{\got
M}}^{A_{1},A_{2}}f\leqslant{\mathscr{M}}^{A_{1}}f+\sup_{w,t>0}\left|{T^{A_{1}}_{w}(T^{A_{2}}_{t}f-T^{A_{1}}_{t}f)}\right|$

By Theorem 1 the maximal operator ${\mathscr{M}}^{A_{1}}$ is bounded in
$L^{p}(\Omega)$. As for the second term in the right hand side, we use
Duhamel’s (10) formula and write

$T^{A_{1}}_{w}(T^{A_{2}}_{t}f-T^{A_{1}}_{t}f)=\int^{t}_{0}T^{A_{1}}_{w+s}(L_{A_{1}}-L_{A_{2}})T^{A_{2}}_{t-s}f\,\text{\rm
d}s=I_{w,t}(f)-J_{w,t}(f),$

where

$I_{t,w}(f)=\int^{t}_{0}L_{A_{1}}T^{A_{1}}_{w+s}T^{A_{2}}_{t-s}f\,\text{\rm
d}s;\quad
J_{t,w}(f)=\int^{t}_{0}T^{A_{1}}_{w+s}L_{A_{2}}T^{A_{2}}_{t-s}f\,\text{\rm
d}s.$

We now proceed as we did in Section 3.1.1. We write below the details only for
the first integral.

Let $\alpha=\alpha(p)\in(0,1/2)$ be as in Proposition 8. Then,

$I_{t,w}(f)=\int^{t}_{0}\psi_{1-\alpha}((w+s)L_{A_{1}})U^{A_{1},A_{2}}_{\alpha}\psi_{\alpha}((t-s)L_{A_{2}})f\,\text{\rm
d}\mu^{w,t}_{\alpha}(s),$

where

$\,\text{\rm
d}\mu^{w,t}_{\alpha}(s)=(w+s)^{\alpha-1}(t-s)^{-\alpha}\mathbbm{1}_{[0,t]}(s)\,\text{\rm
d}s.$

By (14) we have

$\mu^{w,t}_{\alpha}([0,t])\leqslant\mu^{t}_{\alpha}([0,t])=\frac{\pi}{\sin(\alpha\pi)},\quad\forall
w,t>0.$

Therefore,

$\sup_{w,t>0}\left|{I_{t,w}(f)}\right|\leqslant\frac{\pi}{\sin(\alpha\pi)}\sup_{s,t>0}\left|{\psi_{1-\alpha}(sL_{A_{1}})U^{A_{1},A_{2}}_{\alpha}\psi_{\alpha}(tL_{A_{2}})f}\right|.$

By Cowling’s subordination, the maximal operator on the right-hand side is
dominated by the sublinear operator

$\frac{1}{\pi^{2}}\int_{\mathbb{R}}\int_{\mathbb{R}}\left|{\Gamma(1-\alpha-
iu)}\right|\left|{\Gamma(\alpha-
iv)}\right|\left|{L^{iu}_{A_{1}}U^{A_{1},A_{2}}_{\alpha}L^{iv}_{A_{2}}f}\right|\,\text{\rm
d}u\,\text{\rm d}v.$

It follows from Stirling’s formula (17) in combination with Propositions 8 and
2 that the sub-linear operator above is bounded in $L^{p}(\Omega)$. ∎

##  Appendix A Proof of Kato’s theorem (Lemma 7) via square functions

In this appendix we give a proof of Lemma 7 which somewhat differs from the
original one by Kato [Kato61]. We find it likely that this proof may already
be known, yet we where not able to find references that would confirm this.

Let $H$ be a complex Hilbert space. Let ${\text{\got a}}_{1},{\text{\got
a}}_{2}$ be two densely defined, closed and sectorial (thus continuous)
sesquilinear forms on $H$ [Kat, Ouhabook, Haase]. Denote by
${\mathscr{L}}_{{\text{\got a}}_{1}}$ and ${\mathscr{L}}_{{\text{\got
a}}_{2}}$, respectively, the two associated Kato-sectorial operators on $H$.
Suppose further that

$\operatorname{D}({\text{\got a}}_{1})=\operatorname{D}({\text{\got
a}}_{2})={\mathcal{V}}\quad\textrm{and}\quad\operatorname{Re}{\text{\got
a}}_{1}(h)\sim\operatorname{Re}{\text{\got a}}_{2}(h),$

uniformly in $h\in{\mathcal{V}}$. This is equivalent to
${\left\|{h}\right\|_{{\text{\got
a}}_{1}}}\sim{\left\|{h}\right\|_{{\text{\got a}}_{2}}}$ uniformly in
$h\in{\mathcal{V}}$, where

$\|h\|^{2}_{{\text{\got a}}_{j}}=\|h\|^{2}_{H}+\operatorname{Re}{\text{\got
a}}_{j}(h),\quad h\in H,\ j=1,2.$

We also assume that ${\mathscr{L}}_{{\text{\got a}}_{1}}$, and so also
${\mathscr{L}}_{{\text{\got a}}_{2}}$, is injective.

Kato’s theorem on fractional powers [Kato61], which is a part of Lemma 7, can
be rephrased by saying that, for every $\beta\in[0,1/2)$ we have
$\operatorname{D}({\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}})=\operatorname{D}({\mathscr{L}}^{\beta}_{{\text{\got
a}}_{2}})=:{\mathcal{V}}_{\beta}$ and

${\left\|{{\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}}h}\right\|}\sim{\left\|{{\mathscr{L}}^{\beta}_{{\text{\got
a}}_{2}}h}\right\|},\quad\forall h\in{\mathcal{V}}_{\beta}.$ (28)

Once we have this result, Lions’s improvement
${\mathcal{V}}_{\beta}=\left[H,{\mathcal{V}}\right]_{2\beta}$, for all
$\beta\in(0,1/2)$ follows from the modern theory of complex interpolation of
domains of operators (see, for example [MaCaSa01, Theorem 11.6.1]). Indeed, in
(28) fix ${\text{\got a}}_{1}={\text{\got a}}$ and take ${\text{\got
a}}_{2}={\text{\got b}}=({\text{\got a}}_{1}+{\text{\got a}}^{*}_{1})/2$ (the
real part of a). The operator ${\mathscr{H}}$ associated with b is nonnegative
and self adjoint. Therefore,
$\operatorname{D}({\mathscr{H}}^{1/2})=\operatorname{D}({\text{\got
b}})={\mathcal{V}}$ and, by (28),
${\mathcal{V}}_{\alpha/2}=\operatorname{D}({\mathscr{H}}^{\alpha/2})=\left[H,{\mathcal{V}}\right]_{\alpha}$,
$0\leqslant\alpha\leqslant 1$.

We now prove (28) by means of McIntosh’s square functions [Mc]. Recall the
following well-known result.

######  Lemma 14.

Let a be a closed and densely defined sectorial form on a complex Hilbert
space $H$. Let ${\mathscr{L}}_{{\text{\got a}}}$ denote the associated
operator. Then for every $\gamma>0$ the square function

$G_{{\text{\got
a}},\gamma}(x):=\left(\int^{\infty}_{0}\big{\|}\psi_{\gamma}(t{\mathscr{L}}_{{\text{\got
a}}})x\big{\|}^{2}\frac{\,\text{\rm d}t}{t}\right)^{1/2},\quad x\in H$

is bounded on $H$.

###### Proof.

By [CrDe] the operator ${\mathscr{L}}_{{\text{\got a}}}$ has a bounded
$H^{\infty}$-calculus on $H$, thus the result follows from [Mc, Section 8].
See also [CDMY] and [Haase, Corollary 7.1.17]. ∎

Define $T^{{\text{\got a}}_{j}}_{t}=\exp(-t{\mathscr{L}}_{{\text{\got
a}}_{j}})$, $j=1,2$. Fix $\beta\in(0,1/2)$,
$x\in\operatorname{D}({\mathscr{L}}^{\beta}_{{\text{\got a}}_{1}})$ and
$y\in\operatorname{D}({\mathscr{L}}^{\beta}_{{\text{\got a}}^{*}_{2}})$. We
have

$\displaystyle\left|{\left\langle x,{\mathscr{L}}^{\beta}_{{\text{\got
a}}^{*}_{2}}y\right\rangle}\right|$
$\displaystyle=\left|{\int^{\infty}_{0}-\frac{\,\text{\rm d}}{\,\text{\rm
d}t}\left\langle T^{{\text{\got a}}_{1}}_{t}x,T^{{\text{\got
a}}^{*}_{2}}_{t}{\mathscr{L}}^{\beta}_{{\text{\got
a}}^{*}_{2}}y\right\rangle\,\text{\rm d}t}\right|$
$\displaystyle\leqslant\int^{\infty}_{0}\left|{\left\langle{\mathscr{L}}_{{\text{\got
a}}_{1}}T^{{\text{\got a}}_{1}}_{t}x,T^{{\text{\got
a}}^{*}_{2}}_{t}{\mathscr{L}}^{\beta}_{{\text{\got
a}}^{*}_{2}}y\right\rangle}\right|\,\text{\rm
d}t+\int^{\infty}_{0}\left|{\left\langle T^{{\text{\got
a}}_{1}}_{t}x,{\mathscr{L}}^{\beta+1}_{{\text{\got a}}^{*}_{2}}T^{{\text{\got
a}}^{*}_{2}}_{t}y\right\rangle}\right|\,\text{\rm d}t$ $\displaystyle=I+J.$

It follows from Lemma 14 that

$\displaystyle I$
$\displaystyle=\int^{\infty}_{0}\left|{\left\langle\psi_{1-\beta}(t{\mathscr{L}}_{{\text{\got
a}}_{1}}){\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}}x,\psi_{\beta}(t{\mathscr{L}}_{{\text{\got
a}}^{*}_{2}})y\right\rangle}\right|\frac{\,\text{\rm d}t}{t}$
$\displaystyle\leqslant\left(\int^{\infty}_{0}\big{\|}\psi_{1-\beta}(t{\mathscr{L}}_{{\text{\got
a}}_{1}}){\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}}x\big{\|}^{2}\frac{\,\text{\rm
d}t}{t}\right)^{1/2}\left(\int^{\infty}_{0}\big{\|}\psi_{\beta}(t{\mathscr{L}}_{{\text{\got
a}}^{*}_{2}})y\big{\|}^{2}\frac{\,\text{\rm d}t}{t}\right)^{1/2}$
$\displaystyle\,\text{\posebni\char
46\relax}\,_{\beta}{\left\|{{\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}}x}\right\|}{\left\|{y}\right\|}.$

As for $J$, we have

$\displaystyle J$
$\displaystyle=\int^{\infty}_{0}\left|{\left\langle\int^{\infty}_{t}{\mathscr{L}}_{{\text{\got
a}}_{1}}T^{{\text{\got a}}_{1}}_{s}x\,\text{\rm
d}s,{\mathscr{L}}^{\beta+1}_{{\text{\got a}}^{*}_{2}}T^{{\text{\got
a}}^{*}_{2}}_{t}y\right\rangle\,\text{\rm d}s}\right|\,\text{\rm d}t$
$\displaystyle\leqslant\int^{\infty}_{1}\int^{\infty}_{0}t\left|{\left\langle{\mathscr{L}}_{{\text{\got
a}}_{1}}T^{{\text{\got a}}_{1}}_{st}x,{\mathscr{L}}^{\beta+1}_{{\text{\got
a}}^{*}_{2}}T^{{\text{\got a}}^{*}_{2}}_{t}y\right\rangle}\right|\,\text{\rm
d}t\,\text{\rm d}s$
$\displaystyle\leqslant\int^{\infty}_{1}s^{\beta-1}\int^{\infty}_{0}\left|{\left\langle\psi_{1-\beta}(st{\mathscr{L}}_{{\text{\got
a}}_{1}}){\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}}x,{\mathscr{L}}_{{\text{\got
a}}^{*}_{2}}\psi_{\beta}(t{\mathscr{L}}_{{\text{\got
a}}^{*}_{2}})y\right\rangle}\right|\,\text{\rm d}t\,\text{\rm d}s.$

By analyticity of the semigroup $(T^{{\text{\got a}}_{1}}_{t})_{t>0}$ we have

$\psi_{1-\beta}(st{\mathscr{L}}_{{\text{\got
a}}_{1}}){\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}}x\in\operatorname{D}({\mathscr{L}}_{{\text{\got
a}}_{1}})\subseteq\operatorname{D}({\text{\got
a}}_{1})=\operatorname{D}({\text{\got a}}_{2})=\operatorname{D}({\text{\got
a}}^{*}_{2}).$ (29)

It follows that

$\displaystyle\left\langle\psi_{1-\beta}(st{\mathscr{L}}_{{\text{\got
a}}_{1}}){\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}}x,{\mathscr{L}}_{{\text{\got
a}}^{*}_{2}}\psi_{\beta}(t{\mathscr{L}}_{{\text{\got
a}}^{*}_{2}})y\right\rangle$ $\displaystyle=\overline{{\text{\got
a}}^{*}_{2}\left(\psi_{\beta}(t{\mathscr{L}}_{{\text{\got
a}}^{*}_{2}})y,\psi_{1-\beta}(st{\mathscr{L}}_{{\text{\got
a}}_{1}}){\mathscr{L}}^{\beta}_{{\text{\got a}}_{1}}x\right)}$
$\displaystyle={\text{\got
a}}_{2}\left(\psi_{1-\beta}(st{\mathscr{L}}_{{\text{\got
a}}_{1}}){\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}}x,\psi_{\beta}(t{\mathscr{L}}_{{\text{\got a}}^{*}_{2}})y\right).$

Since ${\text{\got a}}_{2}$ is sectorial, there exists $C>0$, that does not
depend on $x,y$ and $\beta$, such that

$\left|{{\text{\got a}}_{2}\left(\psi_{1-\beta}(st{\mathscr{L}}_{{\text{\got
a}}_{1}}){\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}}x,\psi_{\beta}(t{\mathscr{L}}_{{\text{\got
a}}^{*}_{2}})y\right)}\right|\leqslant C\sqrt{\operatorname{Re}{\text{\got
a}}_{2}\left(\psi_{1-\beta}(st{\mathscr{L}}_{{\text{\got
a}}_{1}}){\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}}x\right)}\sqrt{\operatorname{Re}{\text{\got
a}}_{2}\left(\psi_{\beta}(t{\mathscr{L}}_{{\text{\got a}}^{*}_{2}})y\right)}.$

Recall that we are assuming that $\operatorname{Re}{\text{\got
a}}_{2}(h)\sim\operatorname{Re}{\text{\got a}}_{1}(h)$ and note that
$\operatorname{Re}{\text{\got a}}^{*}_{2}(h)=\operatorname{Re}{\text{\got
a}}_{2}(h)$. Hence, by Cauchy-Schwartz inequality,

$J\,\text{\posebni\char
46\relax}\,\int^{\infty}_{1}s^{\beta-1}\left(\int^{\infty}_{0}\operatorname{Re}{\text{\got
a}}_{1}\left(\psi_{1-\beta}(st{\mathscr{L}}_{{\text{\got
a}}_{1}}){\mathscr{L}}^{\beta}_{{\text{\got a}}_{1}}x\right)\,\text{\rm
d}t\right)^{1/2}\left(\int^{\infty}_{0}\operatorname{Re}{\text{\got
a}}^{*}_{2}\left(\psi_{\beta}(t{\mathscr{L}}_{{\text{\got
a}}^{*}_{2}})y\right)\,\text{\rm d}t\right)^{1/2}\,\text{\rm d}s.$

Then a change of variable in the first integral with respect to the variable
$t$ gives

$J\,\text{\posebni\char
46\relax}\,\frac{1}{1/2-\beta}\left(\int^{\infty}_{0}\operatorname{Re}{\text{\got
a}}_{1}\left(\psi_{1-\beta}(t{\mathscr{L}}_{{\text{\got
a}}_{1}}){\mathscr{L}}^{\beta}_{{\text{\got a}}_{1}}x\right)\,\text{\rm
d}t\right)^{1/2}\left(\int^{\infty}_{0}\operatorname{Re}{\text{\got
a}}^{*}_{2}\left(\psi_{\beta}(t{\mathscr{L}}_{{\text{\got
a}}^{*}_{2}})y\right)\,\text{\rm d}t\right)^{1/2}.$

We now observe that, by (29),

$\displaystyle{\text{\got
a}}_{1}\left(\psi_{1-\beta}(t{\mathscr{L}}_{{\text{\got
a}}_{1}}){\mathscr{L}}^{\beta}_{{\text{\got a}}_{1}}x\right)\,\text{\rm
d}t=\left\langle t{\mathscr{L}}_{{\text{\got
a}}_{1}}\psi_{1-\beta}(t{\mathscr{L}}_{{\text{\got
a}}_{1}}){\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}}x,\psi_{1-\beta}(t{\mathscr{L}}_{{\text{\got
a}}_{1}}){\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}}x\right\rangle\frac{\,\text{\rm d}t}{t}$ $\displaystyle{\text{\got
a}}^{*}_{2}\left(\psi_{\beta}(t{\mathscr{L}}_{{\text{\got
a}}^{*}_{2}})y\right)\,\text{\rm d}t=\left\langle t{\mathscr{L}}_{{\text{\got
a}}^{*}_{2}}\psi_{\beta}(t{\mathscr{L}}_{{\text{\got
a}}^{*}_{2}})y,\psi_{\beta}(t{\mathscr{L}}_{{\text{\got
a}}^{*}_{2}})y\right\rangle\frac{\,\text{\rm d}t}{t}.$

We now use Cauchy-Schwartz inequality and Lemma 14. It follows that

$J\,\text{\posebni\char
46\relax}\,\frac{1}{1/2-\beta}{\left\|{{\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}}x}\right\|}{\left\|{y}\right\|}$

and combining the estimate above with the similar estimate for $I$, we
conclude that

$\left|{\left\langle x,{\mathscr{L}}^{\beta}_{{\text{\got
a}}^{*}_{2}}y\right\rangle}\right|\,\text{\posebni\char
46\relax}\,_{\beta}{\left\|{{\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}}x}\right\|}{\left\|{y}\right\|},\quad\forall
x\in\operatorname{D}({\mathscr{L}}^{\beta}_{{\text{\got a}}_{1}}),\quad
y\in\operatorname{D}({\mathscr{L}}^{\beta}_{{\text{\got
a}}^{*}_{2}}),\quad\forall\beta\in(0,1/2)$

which is equivalent to

$\operatorname{D}({\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}})\subseteq\operatorname{D}({\mathscr{L}}^{\beta}_{{\text{\got
a}}_{2}})\quad\textrm{and}\quad{\left\|{{\mathscr{L}}^{\beta}_{{\text{\got
a}}_{2}}h}\right\|}\,\text{\posebni\char
46\relax}\,_{\beta}{\left\|{{\mathscr{L}}^{\beta}_{{\text{\got
a}}_{1}}h}\right\|}\quad\forall
h\in\operatorname{D}({\mathscr{L}}^{\beta}_{{\text{\got a}}_{1}}).$

The reverse inclusion and inequality (and thus (28)) follow by reversing the
role of ${\text{\got a}}_{1}$ and ${\text{\got a}}_{2}$ in the previous proof.

### Acknowledgements

A. Carbonaro was partially supported by the “National Group for Mathematical
Analysis, Probability and their Applications” (GNAMPA-INdAM).

O. Dragičević was partially supported by the Slovenian Research Agency, ARRS
(research grant J1-1690 and research program P1-0291).

## References


# Validating transformers for redaction of text from electronic health records
in real-world healthcare

1st Zeljko Kraljevic  Biostatistics and Health Informatics, IoPPN
King’s College London
London, United Kingdom
<EMAIL_ADDRESS>2nd Anthony Shek  Biostatistics and Health
Informatics, IoPPN
King’s College London
London, United Kingdom
<EMAIL_ADDRESS>3rd Joshua Au Yeung  Department of Neurology
King’s College Hospital
London, United Kingdom
<EMAIL_ADDRESS>4th Ewart Jonathan Sheldon  Precision Medicine
King’s College Hospital
London, United Kingdom
<EMAIL_ADDRESS>5th Haris Shuaib  Guy’s and St Thomas’ NHS Foundation
Trust
London, United Kingdom
<EMAIL_ADDRESS>6th Mohammad Al-Agil  Precision Medicine
King’s College Hospital
London, United Kingdom
<EMAIL_ADDRESS>7th Xi Bai  Institute of Health Informatics
University College London
London, United Kingdom
<EMAIL_ADDRESS>8th Kawsar Noor  Institute of Health Informatics
University College London
London, United Kingdom
<EMAIL_ADDRESS>9th Anoop D. Shah  Institute of Health Informatics
University College London
London, United Kingdom
<EMAIL_ADDRESS>10th Richard Dobson  Biostatistics and Health Informatics,
IoPPN
King’s College London
London, United Kingdom
<EMAIL_ADDRESS>11th James Teo  Department of Neurology
King’s College Hospital
London, United Kingdom
<EMAIL_ADDRESS>
###### Abstract

Protecting patient privacy in healthcare records is a top priority, and
redaction is a commonly used method for obscuring directly identifiable
information in text. Rule-based methods have been widely used, but their
precision is often low causing over-redaction of text and frequently not being
adaptable enough for non-standardised or unconventional structures of personal
health information. Deep learning techniques have emerged as a promising
solution, but implementing them in real-world environments poses challenges
due to the differences in patient record structure and language across
different departments, hospitals, and countries.

In this study, we present AnonCAT, a transformer-based model and a blueprint
on how deidentification models can be deployed in real-world healthcare.
AnonCAT was trained through a process involving manually annotated redactions
of real-world documents from three UK hospitals with different electronic
health record systems and 3116 documents. The model achieved high performance
in all three hospitals with a Recall of 0.99, 0.99 and 0.96.

Our findings demonstrate the potential of deep learning techniques for
improving the efficiency and accuracy of redaction in global healthcare data
and highlight the importance of building workflows which not just use these
models but are also able to continually fine-tune and audit the performance of
these algorithms to ensure continuing effectiveness in real-world settings.
This approach provides a blueprint for the real-world use of de-identifying
algorithms through fine-tuning and localisation, the code together with
tutorials is available on GitHub (https://github.com/CogStack/MedCAT).

###### Index Terms:

electronic health records, text deidentification, transformers

##  I Introduction

Healthcare systems contain vast amounts of unstructured health data stored in
text form. While the associated structured metadata is typically anonymised
according to various anonymisation guidelines (e.g. UK government guidelines
and USA Health Insurance Portability and Accountability Act (HIPAA) Safe
Harbour guidelines), directly identifiable information embedded in free text
is harder to remove.

One commonly used method for protecting patient privacy is redaction, which
involves removing or obscuring sensitive information from health records.
Currently, rule-based methods such as the Protected Health Information filter
(Philter)[1] and Regular Expression (RegEx) are widely used for redacting
sensitive information from healthcare records. However, these methods have
significant limitations, including being time-consuming to develop and
maintain, and their inability to adapt to new or evolving data formats.
Moreover, although these techniques achieve high recall rates, their precision
is often low, leading to the concurrent redaction of clinically valuable
information (e.g. eponymous conditions like Parkinson’s Disease or Smith
Fracture) [2, 3, 4].

To overcome these challenges, deep learning techniques have emerged as a
promising solution for automating the redaction process. Unlike rule-based
methods, deep learning algorithms can adapt to new or evolving data formats
while requiring minimal manual effort. Furthermore, deep learning techniques
have been shown to achieve high precision rates in redaction, minimising the
risk of redacting clinically valuable information [5, 6, 7, 8].

In a real-world deployment, implementing deep learning techniques for
redacting sensitive information in healthcare records poses several
challenges: the initial challenge is generalisability primarily due to the
differences in the patient record structure and language used across various
departments, hospitals and countries - identifiable information in text blocks
of addresses, names and official identity documents are extremely varied
between geographies. After implementation, data drift which can affect both
rules-based and pre-trained deep learning models will significantly impact any
redacting algorithm performance over time. This is also a bigger feature where
a healthcare economy has a high socio-ethnic diversity of patients who would
have a more varied structure of their personal health information (e.g.
inversion of first name and last name; use of initials; multiple addresses in
different countries; different types of identity documents).

The findings presented in this paper demonstrate the potential of deep
learning techniques in improving the efficiency and accuracy of redaction in
diverse healthcare settings. Additionally, we propose how to continually
monitor and audit the performance of new and existing deep learning algorithms
to ensure they remain effective and accurate and to update or fine-tune these
algorithms as required.

##  II Methods

We present AnonCAT, which builds on the Medical Concept Annotation Toolkit
(MedCAT) [9], a widely used tool for Named Entity Recognition and Linking
(NER+L) of free text from electronic health records (EHRs). This is because
deidentification can be seen as NER with one additional step where we replace
or remove the detected Personal/Protected Health Information (PHI) entities.
It also allows us to create pipelines that can be easily deployed in hospitals
and allow for easy finetuning. AnonCAT supports both the removal of PHI data
and the replacement of PHI data with pseudo identifiers, we do not
differentiate between these two modes of work as essentially they both depend
on the ability of the model to find/detect PHI data (which is what we explore
and validate).

###  II-A Ontology Creation

The UK government guidelines and USA Health Insurance Portability and
Accountability Act (HIPAA) Safe Harbour guidelines were used to specify key
types of data that required redaction [10]. We created an ontology from the
key types (Fig. 1) and populated a MedCAT concept database, where each key
type was represented with a MedCAT concept. The concept database was localised
to the UK setting (e.g. zip code $\rightarrow$ postcode), and an ontology was
chosen as it allows us to easily control the output granularity.

Figure 1: Sunburst hierarchical ontology structure of terms for redaction.
Five broad terms serve as the primary child nodes to which more specific terms
child nodes, e.g. postcode, were connected. The terminal leaf nodes were used
to annotate documents with patient data terms. Non-healthcare identifiers
(green) were not prevalent enough in the text so were not used for the
evaluation.

###  II-B Data Annotation

All annotations were done using MedCATtrainer [11]. Clinical annotators in
each hospital (who already had visibility of the PHI data) were presented with
a random document and asked to annotate all occurrences of PHI. Annotations
were done by two annotators at each of the hospitals (KCH, GSTT, UCLH). To
ensure standardisation between annotators and sites, we created annotation
guidelines. The documents in each of the EHRs were of varying length and PHI
mentions were scattered all throughout the document. As such, for very long
documents, it is possible that annotators could miss a piece of PHI data. This
would degrade the model performance, and also invalidate our metrics. To
improve the quality of the data we used an iterative annotation approach (Fig.
3) to ensure all manual annotations were correct. The process is as follows:

  * •

Take a random sample of a dataset from the EHR.

  * •

Clinicians / Annotators manually annotate the sampled dataset.

  * •

Split the annotated dataset into 5 folds and train 5 different models, each
one using a different fold for the test set.

  * •

Find all false positive (FP) and false negative (FN) examples from the 5 test
sets.

  * •

Manually check each FP and FN, if the FP/FN is a mistake from the annotator,
we fix the annotation (this creates a new dataset) and return to step 3. If
the FNs and FPs represent mistakes by the model, we proceed to the next step.

  * •

Once all FPs/FNs are validated and manual annotations are fixed, the process
is completed.

Figure 2: The experimental setup across sites for AnonCAT. Figure 3: AnonCAT
training cycle.

###  II-C Experimental Setup and Datasets

All annotations were first pre-processed and any terms with less than 10
occurrences were removed (e.g. lab number). Next, we merged forename and
surname into name, because of the observed difficulties during annotations
(sometimes it was very hard to distinguish between a forename or surname).

AnonCAT uses RoBERTa-Large as the base model, in our experiments it
outperformed the more clinically specialised models like BioClinicalBERT or
the more general BERT model. The model was extended with a post-processing
step that allows us to bias the predictions towards the positive classes.
BERT-based models output the probability for each class, our modification
decreases the probability of the negative class by the factor
$\lambda\in[0,1]$, Eq. (1). This often means we are increasing recall at the
cost of precision.

$p_{final}(c)=\begin{cases}p(c),&\text{if $c\neq 0$}\\\ p(c)-\lambda
p(c),&\text{if $c=0$}\end{cases}$ (1)

Where $p(c)\text{ is the probability of a class outputted by the model}$ and
$p_{final}(c)\text{is the adjusted probability}$.

The training was done on 8 GPUs with a batch size of 4 per device, learning
rate set to 4.46e-5, warm-up ratio to 0.01, weight decay to 0.14 and 10 epochs
with early stopping (best results achieved around the 4th epoch). The
hyperparameter tuning was done using population-based training [12] only on
the train set with a 90/10 split (train/validation). On all datasets, we’ve
used a train/test split of 80/20.

For the metrics, we calculated precision (P), recall (R) and F1, as well as
Recall_merged (R_m). Recall_merged ignores mistakes in-between concepts and
only accounts for mistakes where something was supposed to be detected as PHI
but was not. For example, if the token John was detected as Address, even
though it is a name, the merged metrics will not consider this a mistake. But,
if the token John was not detected at all, that would be a mistake. This was
done because, for PHI data, it is a much bigger problem if we do not detect
PHI, compared to mislabeling it as another concept. When testing off-the-shelf
tools for deidentification we calculated Precision_merged (P_m) instead of
Precision, because the concepts we have do not exactly match the entity names
of other tools. The merged metrics can also be understood in another way, we
merge all concepts (entity names) into one called PHI, so the task becomes
detecting if a token is PHI or not.

AnonCAT was tested on three different datasets: 1) A random sample of 2648
documents from the EHR at King’s College Hospital NHS Foundation Trust (KCH)
2) A random sample of 328 documents from an existing recruited patient brain
tumour cohort at Guy’s and St Thomas’ NHS Foundation Trust (GSTT); and 3) A
random sample of 140 documents from the EHR at University College London
Hospitals NHS Foundation Trust. To extract the datasets we used the CogStack
Platform deployed at each site [13]. The statistics for the manually annotated
datasets are shown in Table I.

TABLE I: The number of annotations per entity and in total, as well as the total number of documents. | KCH | GSTT | UCLH
---|---|---|---
Address Line | 2075 | 240 | 257
Email | 1740 | 23 | 107
Name | 16407 | 1747 | 1743
Postcode | 2065 | 208 | 230
Emergency department Number | 2176 | 0 | 0
DOB | 896 | 145 | 112
Hosp Num | 1882 | 220 | 104
NHS Num | 778 | 49 | 103
Initials | 1399 | 168 | 100
Telephone Num | 1177 | 398 | 320
Total Annotations | 31182 | 3198 | 3076
Total Documents | 2648 | 328 | 140

The KCH dataset (our largest one by a factor of 10) is the primary dataset
where the base RoBERTa-Large model was first trained. Afterwards, the KCH
model was transferred to GSTT and UCLH for further testing and fine-tuning
(Fig. 2)

###  II-D Ethical approval declarations

The project operated under London South East Research Ethics Committee
(reference 18/LO/2048) approval granted to the King’s Electronic Records
Research Interface (KERRI) with project-specific approvals granted for NLP
work on clinical coding, information governance and audit. For redaction
validation, the GSTT corpus for patients in a directly-recruited Glioblastoma
research study was used, this cohort operated under the Health Research
Authority and Health and Care Research Wales (HCRW) Approval (REC reference:
18/LO/1873). For the UCLH validation, the project was reviewed by the Trust
Data Access Process for Research (DAP-R) committee, the Head of Information
Governance and Data Protection Officer and approved being within the remit of
service development for the overall CogStack infrastructure at UCLH.

##  III Results

In Table II, we show the performance of AnonCAT across the three hospitals,
without any recall bias or modifications. The models used are as follows: 1)
KCH - Roberta-Large trained on 80% of KCH data and tested on 20%; 2) UCLH, we
fine-tuned the KCH model on 80% of the UCLH data and tested on 20%; and 3)
GSTT, we fine-tuned the KCH model on 80% of GSTT data and tested on 20%.

Figure 4: Results of biasing the predictions of the model towards positive classes. Left - the KCH model fine-tuned at GSTT, Right - the KCH model without any fine-tuning at GSTT ($P$ = precision, $R$ = recall, $R_{m}$ = recall merged). In the plot on the left, $R$ and $R_{m}$ overlap. TABLE II: Performance of AnonCAT across the three datasets (P=precision, R=Recall, R_m=Recall merged) | KCH |  | GSTT
---
(tuned)
| UCLH
---
(tuned)
| P | R | R_m | P | R | R_m | P | R | R_m
Address Line | 0.99 | 0.99 | 0.99 | 0.95 | 0.99 | 0.99 | 0.80 | 0.96 | 0.97
Email | 0.98 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.74 | 1.00 | 1.00
Name | 0.99 | 0.99 | 1.00 | 1.00 | 0.99 | 0.99 | 0.70 | 0.96 | 0.98
Postcode | 0.99 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.79 | 0.99 | 0.99
Emergency dept Num | 0.99 | 0.97 | 1.00 | NA | NA | NA | NA | NA | NA
DOB | 0.99 | 0.98 | 1.00 | 1.00 | 1.00 | 1.00 | 0.90 | 1.00 | 1.00
Hosp Num | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.78 | 1.00 | 1.00
NHS Num | 1.00 | 0.98 | 1.00 | 1.00 | 1.00 | 1.00 | 0.94 | 0.94 | 0.99
Initials | 0.98 | 0.94 | 0.95 | 1.00 | 0.97 | 0.97 | 0.81 | 0.81 | 0.81
Telephone Num | 0.98 | 1.00 | 1.00 | 0.95 | 0.99 | 0.99 | 0.83 | 0.96 | 0.96
TABLE III: Performance of Philter and the AnonCAT KCH model without any finetuning at GSTT. We are only calculating merged metrics as the entities in Philter do not match exactly the entities in our concept database. |  | AnonCAT
---
(untuned)
Philter
| P_m | R_m | P_m | R_M
Address Line | 0.79 | 0.99 | 0.37 | 0.79
Email | 0.79 | 0.94 | 0.37 | 1
Name | 0.79 | 0.99 | 0.37 | 0.99
Postcode | 0.79 | 1 | 0.37 | 0.51
Emergency dept Num | 0.79 | NA | NA | NA
DOB | 0.79 | 0.74 | 0.37 | 1
Hosp Num | 0.79 | 0.52 | 0.37 | 1
NHS Num | 0.79 | 1 | 0.37 | 1
Initials | 0.79 | 0.67 | 0.37 | 0.83
Telephone Num | 0.79 | 0.70 | 0.37 | 0.50

Next, in Table III we show the performance on the GSTT dataset for two
different models. AnonCAT KCH, we test the KCH model on the full GSTT dataset
without any fine-tuning, and Philter an off-the-shelf tool showing state-of-
the-art performance on de-identification of clinical notes [1]. The entities
in Philter do not match exactly the entities we have in our concept database,
so we only calculate the merged metrics for both Precision and Recall.

Lastly, in Fig. 4 we also test does biasing the model towards predicting the
positive classes work, and what is the tradeoff between precision and recall.
We do the test at GSTT and check the performance for two different models -
AnonCAT trained only on the KCH.

##  IV Discussion

The findings of this study suggest that with appropriate fine-tuning and
maintenance, deep learning techniques offer an adaptable solution for
improving the accuracy and efficiency of the redaction of EHR data. This lays
out a route for a foundational redaction NLP model that can be rapidly
localised to a healthcare environment thereby reducing the rate of under-
redaction (due to the uniqueness of patient data in new hospitals).

We show that as few as 150-300 documents are needed to fine-tune a redaction
model on a new dataset, with a higher performance the more documents available
for fine-tuning. We also note that during our iterative approach where we
check for disagreements between the model and annotators (but also more
generally validate the annotations), most ( 90%) of mistakes are annotators
overlooking a piece of PHI data, and rarely wrongly annotating something. As
such, we argue that our iterative approach is a more efficient way of creating
a well-annotated dataset, than the more standard double annotation approach,
in cases where finding the entities is difficult, but annotating them
correctly is easy. We plan to explore this in more detail in future work.

Table III shows that state-of-the-art off-the-shelf tools, as well as
transformer-based models, require some fine-tuning when applied to a new
dataset if we want to achieve a Recall of 95%+ (as shown in Table II) We argue
that a deep learning approach is easier to maintain and upgrade, as the only
requirement for adapting it to a new dataset is annotating 150-300 documents
and running the fine-tuning. In fact, the very task of annotating for
redaction is already being done by information governance auditors manually
for records release in subject access requests, so a minor adaptation of the
workflow is all that is needed to build resilience in redaction performance.

##  V Conclusion

We demonstrate that a Transformer-based Deep Learning approach to effective
and efficient text redaction can be achieved through localised fine-tuning of
a foundation model. This use of localised efficient fine-tuning would also
mitigate against performance deterioration of a foundation model when deployed
in closed private datasets from different underlying EHRs. Our workflow can be
easily incorporated into existing activity when the hospital’s auditors redact
text, in this way normal hospital information governance activity would
generate validation metrics while continually fine-tuning models. The
application of our method can significantly enhance patient privacy protection
in healthcare organisations and contribute to the advancement of an efficient
safe life cycle of healthcare data and NLP model management [14, 15].

##  VI Acknowledgements

This work was made possible by funding received from NHS AI Lab, National
Institutes of Health Research, Health Data Research UK, Guy’s & St Thomas’ NHS
Foundation Trust, NIHR Applied Research Collaborative South London, Guy’s and
St Thomas’ Foundation, the Radiological Research Trust, and a National
Institute for Health Research Doctoral Research Fellowship (ref. 20647).
Special thanks as well to the patients of the KERRI committee.

R.D. is supported by the following: (1) NIHR Biomedical Research Centre at
South London and Maudsley NHS Foundation Trust and King’s College London,
United Kingdom; (2) Health Data Research UK; (3) the NIHR University College
London Hospitals Biomedical Research Centre; (4) the UK Research and
Innovation London Medical Imaging and Artificial Intelligence Centre for Value
Based Healthcare; and (5) the NIHR Applied Research Collaboration South London
(NIHR ARC South London) at King’s College Hospital NHS Foundation Trust. ADS
is supported by a postdoctoral fellowship from THIS Institute, NIHR
(AI_AWARD01864 and COV-LT-0009), UKRI (Horizon Europe Guarantee for
DataTools4Heart) and British Heart Foundation Accelerator Award
(AA/18/6/24223).

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