# Dirichlet spaces over chord-arc domains

Huaying Wei  Department of Mathematics and Statistics, Jiangsu Normal
University Xuzhou 221116, PR China<EMAIL_ADDRESS>and  Michel Zinsmeister
Institut Denis Poisson, Université d’ Orléans, Orléans, 45067, France
<EMAIL_ADDRESS>
###### Abstract.

If $U$ is a $C^{\infty}$ function with compact support in the plane, we let
$u$ be its restriction to the unit circle $\mathbb{S}$, and denote by
$U_{i},\,U_{e}$ the harmonic extensions of $u$ respectively in the interior
and the exterior of $\mathbb{S}$ on the Riemann sphere. About a hundred years
ago, Douglas [9] has shown that

$\displaystyle\iint_{\mathbb{D}}|\nabla U_{i}|^{2}(z)dxdy$
$\displaystyle=\iint_{\bar{\mathbb{C}}\backslash\bar{\mathbb{D}}}|\nabla
U_{e}|^{2}(z)dxdy$
$\displaystyle=\frac{1}{2\pi}\iint_{\mathbb{S}\times\mathbb{S}}\left|\frac{u(z_{1})-u(z_{2})}{z_{1}-z_{2}}\right|^{2}|dz_{1}||dz_{2}|,$

thus giving three ways to express the Dirichlet norm of $u$. On a rectifiable
Jordan curve $\Gamma$ we have obvious analogues of these three expressions,
which will of course not be equal in general. The main goal of this paper is
to show that these $3$ (semi-)norms are equivalent if and only if $\Gamma$ is
a chord-arc curve.

###### Key words and phrases:

Dirichlet space, chord-arc curve, quasicircle, Ahlfors-regular curve, Douglas
formula

###### 2020 Mathematics Subject Classification:

31A05, 31C25, 30C62

Research supported by the National Natural Science Foundation of China (Grant
No. 12271218).

##  1\. Introduction

If $\Omega$ is a domain of the Riemann sphere, the Dirichlet space
$\mathcal{D}(\Omega)$ is the space of harmonic functions
$U:\Omega\mapsto\mathbb{C}$ such that

$\iint_{\Omega}|\nabla U|^{2}(z)dxdy<\infty.$

The Dirichlet space over the unit disk $\mathbb{D}$ is, together with the
Hardy and Bergman spaces, one of three most classical Hilbert spaces in the
unit disk. The key references are survey articles [5, 24] and the book [12].
It plays an important role in domains as distinct as Minimal Surfaces,
Operator Theory and Teichmüller Theory (e.g.[22]). Concerning the latter,
there has been recently a regain of interest with the explosion of results
about Weil-Petersson curves; that is the connected component of the identity
in the universal Teichmüller space viewed as a complex Hilbert manifold; see
[7, 27, 28, 30].

In the case of unit disk, we will see in the next section that functions
$U_{i}$ in $\mathcal{D}(\mathbb{D})$ have well-defined limits $u$ on the unit
circle $\mathbb{S}$ that moreover characterize $U_{i}$. We may thus consider
the Dirichlet space $\mathcal{D}(\mathbb{D})$ as a quotient space of a space
of functions defined on $\mathbb{S}$ modulo constants. Moreover, using the
reflection $z\mapsto 1/\bar{z}$ we have that this latter space coincides with
the space of boundary values of functions $U_{e}$ in
$\mathcal{D}(\mathbb{D}_{e})$ , where $\mathbb{D}_{e}$ is the unbounded
component of $\bar{\mathbb{C}}\backslash\mathbb{S}$. A deep theorem of Douglas
[9] asserts that this space of functions coincides with the space
$H^{1/2}(\mathbb{S})$. Its norm may thus be expressed in three equal ways as
follows:

$\displaystyle\|u\|_{i}^{2}:=\frac{1}{2\pi}\iint_{\mathbb{D}}|\nabla
U_{i}|^{2}(z)dxdy$ $\displaystyle=$
$\displaystyle\|u\|_{e}^{2}:=\frac{1}{2\pi}\iint_{\mathbb{D}_{e}}|\nabla
U_{e}|^{2}(z)dxdy$ $\displaystyle=$
$\displaystyle\|u\|_{H^{1/2}(\mathbb{S})}^{2}:=\iint_{\mathbb{S}\times\mathbb{S}}\Big{|}\frac{u(z_{1})-u(z_{2})}{z_{1}-z_{2}}\Big{|}^{2}\frac{|dz_{1}|}{2\pi}\frac{|dz_{2}|}{2\pi},$

where $\|u\|_{i}^{2}$ is called the interior energy of $u$, while
$\|u\|_{e}^{2}$ is the exterior energy. $\|\cdot\|_{H^{1/2}(\mathbb{S})}$ is
called the Douglas norm, and the equality of the first and third integrals is
the Douglas formula, as introduced by Douglas in his solution of the Plateau
problem ([9],[3]). The formula inspired important developments in the theory
of Dirichlet forms; see [13].

Inspired by Problem 38 of [5] which consists in developing a theory of
Dirichlet spaces in planar domains, our aim in this paper is to start
investigating Dirichlet spaces over planar domains and more specifically to
find the right class of Jordan domains for which these three norms make sense
and are equivalent (we cannot of course expect the equality in general). Since
the analogue of the Douglas norm makes sense only for rectifiable curves we
will , from now on, restrict our study to rectifiable Jordan curves. For such
a curve $\Gamma$ we thus define the space (modulo constants) $H^{1/2}(\Gamma)$
of mesurable functions $u:\Gamma\to\mathbb{C}$ such that

$\|u\|_{H^{1/2}(\Gamma)}^{2}:=\frac{1}{4\pi^{2}}\iint_{\Gamma\times\Gamma}\Big{|}\frac{u(z_{1})-u(z_{2})}{z_{1}-z_{2}}\Big{|}^{2}|dz_{1}||dz_{2}|<\infty.$

About the other two norms $\|\cdot\|_{i},\,\|\cdot\|_{e}$, we will show that
it happens to be true in the good case that the spaces of boundary functions
of $\mathcal{D}(\Omega)$ and $\mathcal{D}(\Omega_{e})$ coincide, where
$\Omega$ and $\Omega_{e}$ denote the bounded and unbounded components of the
complement of $\Gamma$, respectively. In order to make an explanation on this
claim at this stage, we consider instead the space $E(\Gamma)$ of restrictions
to $\Gamma$ of $C^{\infty}$ functions with compact support in the whole plane,
and will see that the norms $\|\cdot\|_{i},\,\|\cdot\|_{e}$ are equivalent on
$E(\Gamma)$ for a rectifiable Jordan curve $\Gamma$ if and only if it is a
quasicircle. We may define, for such a curve, $\mathcal{H}(\Gamma)$ as the
completion of $E(\Gamma)$ with respect to one of these two equivalent norms
(but this is not the order in which we will derive things).

The main result of this paper will be that
$H^{1/2}(\Gamma)=\mathcal{H}(\Gamma)$ if and only if $\Gamma$ is a chord-arc
curve, a rectifiable quasicircle with the property of Ahlfors-regularity.

The paper is structured as follows: In Section 2 we give the precise
definition of the Dirichlet space over any Jordan domain. In Section 3 we
study the case of quasicircles and define precisely the “two-sided” space
$\mathcal{H}(\Gamma)$. Section 4 is devoted to prove that
$\mathcal{H}(\Gamma)=H^{1/2}(\Gamma)$ when $\Gamma$ is a chord-arc curve. In
Section 5 we consider the sharpness of the result in Section 4: we prove more
precisely that if $\Gamma$ is a rectifiable quasicircle such that
$\mathcal{H}(\Gamma)\subset H^{1/2}(\Gamma)$ then $\Gamma$ must be chord-arc
and the two spaces are equal. On the other hand we construct a rectifiable
quasicircle $\Gamma$ that is not chord-arc but such that
$H^{1/2}(\Gamma)\subset\mathcal{H}(\Gamma)$.

##  2\. Dirichlet spaces over a Jordan domain

Recall that if $\Omega$ is a bounded Jordan domain of the complex plane
$\mathbb{C}$, the Dirichlet space $\mathcal{D}(\Omega)$ is the space of
harmonic functions $F$ on $\Omega$ with finite Dirichlet energy $D(F)$, where
the energy of any $C^{1}$ map on $\Omega$ is defined as the
$L^{2}(\Omega)$-norm of the gradient vector $\nabla F(w)=(F_{u},F_{v})$.
Precisely,

$D(F):=\frac{1}{2\pi}\iint_{\Omega}|\nabla F|^{2}(w)dudv<+\infty.$ (1)

The complex notation is much more convenient. Let us first note
$F_{\bar{w}}=(F_{u}+iF_{v})/2$ and $F_{w}=(F_{u}-iF_{v})/2$. This gives
$D(F)=1/\pi\iint_{\Omega}(|F_{w}|^{2}+|F_{\bar{w}}|^{2})dudv$. The space
$\mathcal{D}(\Omega)$ is a Hilbert space modulo constant functions. Let
$\varphi$ map another bounded Jordan domain $\Omega^{\prime}$ conformally onto
$\Omega$. Using

$|G_{z}|^{2}+|G_{\bar{z}}|^{2}=(|F_{w}|^{2}+|F_{\bar{w}}|^{2})\circ\varphi(z)|\varphi^{\prime}(z)|^{2},$

we see that $F\mapsto G:=F\circ\varphi$ is a bijective isometry between
$\mathcal{D}(\Omega)$ and $\mathcal{D}(\Omega^{\prime})$. Similarly, an anti-
conformal mapping $\varphi(\bar{z})$ also induces the invariance of Dirichlet
energies.

In the case of classical Dirichlet space $\mathcal{D}(\mathbb{D})$, one may
make the theory precise by the use of Fourier series. For a real-valued
function $u\in\mathcal{D}(\mathbb{D})$, let $v$ be the unique harmonic
conjugate function of $u$ in $\mathbb{D}$ with the requirement $v(0)=0$ so
that $\Phi=u+iv$ is holomorphic. An easy calculation involving Parseval
formula leads to, writing $\Phi(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$,

$D(\Phi)=\sum_{n=1}^{\infty}n|a_{n}|^{2}.$

By Cauchy-Riemann equations, $D(u)=D(v)$, and then,
$D(u)=\frac{1}{2}\sum_{n=1}^{\infty}n|a_{n}|^{2}$. In particular,
$\sum_{n=1}^{\infty}|a_{n}|^{2}<\infty$, which means that $\Phi\in
H^{2}(\mathbb{D})$, the Hardy space of analytic functions on $\mathbb{D}$, and
that the function $u$ belongs to the Hardy space of harmonic functions
$h^{2}(\mathbb{D})$. As a consequence, $u$ has angular limits almost
everywhere on the unit circle $\mathbb{S}$.

Suppose now that $\Omega$ is a domain bounded by a rectifiable Jordan curve
$\Gamma$. Let $\varphi$ map $\mathbb{D}$ conformally onto $\Omega$. Then
$\varphi$ extends to a homeomorphism of the closures
$\mathbb{D}\cup\mathbb{S}$ and $\Omega\cup\Gamma$. Using F. and M. Riesz
theorem for the curve $\Gamma$ being rectifiable (see [18]), we have that a
subset of $\mathbb{S}$ has measure zero if and only if its image under
$\varphi$ has length zero, and it also makes sense to speak of a tangential
direction almost everywhere on $\Gamma$. Furthermore, the mapping $\varphi$
preserves angles at almost every boundary point on $\Gamma$, see [10, 23] for
details. Consequently, for any $F\in\mathcal{D}(\Omega)$, since
$G:=F\circ\varphi\in\mathcal{D}(\mathbb{D})$ one may say that $F$ has angular
limits $f(w)$ almost everywhere on $\Gamma$ such that $f\circ\varphi\in
L^{2}(\mathbb{S})$. So $F$ can be recovered from its boundary function by the
“Poisson integral” of $f$, in the sense that

$F=P(f\circ\varphi)\circ\varphi^{-1}$ (2)

where $P$ stands for the classical Poisson integral in the unit disk
$\mathbb{D}$. If $f_{1}$ and $f_{2}$ are boundary functions of $F_{1}$ and
$F_{2}$ in $\mathcal{D}(\Omega)$, respectively, and $f_{1}=f_{2}$ almost
everywhere on $\Gamma$, we then have $F_{1}=F_{2}$ by (2). Hence, we may say
that $f_{1}=f_{2}$ if they are equal except possibly on a subset with length
zero. The one to one correspondence $F\leftrightarrow f$ allows us to view the
Dirichlet space on $\Omega$ as a space of functions defined on $\Gamma$. We
denote it by $\mathcal{H}(\Gamma,\Omega)$.

Since the function $F$ in (2) is the solution to Poisson’s equation $\Delta
F=0$ in the domain $\Omega$ with boundary condition $F|_{\Gamma}=f$,
Dirichlet’s principle says that $F$ can be obtained as the minimizer of
Dirichlet energies $D(V)$ amongst all $C^{1}$ extensions $V$ to $\Omega$ of
$f$. We call $\|f\|_{i}^{2}:=D(F)$ the interior energy of $f$. Clearly,
$\mathcal{H}(\Gamma,\Omega)$ is the function space consisting of all $f$ with
a finite semi-norm $\|f\|_{i}$.

Let us make one remark on Dirichlet’s principle: The requirement for $C^{1}$
extensions can be relaxed. Precisely, let $\dot{W}^{1,2}(\Omega)$ be the
homogeneous Sobolev space of locally integrable functions on any domain
$\Omega$ of the Riemann sphere with $L^{2}$-integrable gradient taken in the
sense of distributions, equipped with the natural $L^{2}(\Omega)$-norm of the
gradient vector $\nabla F$ as in (1). It has an important subspace
$\dot{W}^{1,2}_{0}(\Omega)$ defined as the closure in $\dot{W}^{1,2}(\Omega)$
of $C_{0}^{\infty}(\Omega)$, the space of $C^{\infty}$ functions with compact
support included in $\Omega$. By the Meyers-Serrin theorem [21], the space
$C^{\infty}(\Omega)\cap\dot{W}^{1,2}(\Omega)$ is dense in
$\dot{W}^{1,2}(\Omega)$. With this setting, a simple approximation argument
shows that

$D(F)=\min\\{D(V):V\;\text{ranges over all functions
in}\;\dot{W}^{1,2}(\Omega)\;\text{with}\;V-F\in\dot{W}_{0}^{1,2}(\Omega)\\}.$
(3)

By the Jordan curve theorem the complement of the boundary Jordan curve
$\Gamma$ of $\Omega$ has two components: one is $\Omega=\Omega_{i}$, the so-
called interior of $\Gamma$, and the second is $\Omega_{e}$, the exterior of
$\Gamma$. In order to define a Dirichlet space over $\Omega_{e}$, we may first
assume without loss of generality that $0\in\Omega$, and we consider the
reflection $\iota(z)=\frac{1}{\bar{z}}.$ It maps $\Omega_{e}$ onto a Jordan
domain $\tilde{\Omega}=\tilde{\Omega}_{i}$ bounded by a bounded Jordan curve
$\tilde{\Gamma}$. The image of $\Omega$ is $\tilde{\Omega}_{e}$, the exterior
of $\tilde{\Gamma}$. One may see that the point $0=\iota(\infty)$ is an
interior point of $\tilde{\Omega}$. Define $\mathcal{D}(\Omega_{e})$ as the
set $\\{F:=\tilde{F}\circ\iota,\,\tilde{F}\in\mathcal{D}(\tilde{\Omega})\\}$.
Then the Dirichlet energy of $F$ over $\Omega_{e}$ can be written as

$D(F)=\frac{1}{2\pi}\iint_{\Omega_{e}\backslash{\\{\infty\\}}}|\nabla
F|^{2}(w)dudv.$

That is, the point $\infty$ can be removed from the domain of integration
without changing the convergence properties or the value of integral. This is
justified as follows: if $F\in\mathcal{D}(\Omega_{e})$ then for any
$\epsilon>0$ there is an $R>0$ such that

$\iint_{|w|>R}|\nabla F|^{2}(w)dudv=\iint_{|z|\leq
1/R}|\nabla\tilde{F}|^{2}(z)dxdy<\epsilon.$

Let $\tilde{\varphi}$ map $\mathbb{D}=\iota(\mathbb{D}_{e})$ conformally onto
$\tilde{\Omega}=\iota(\Omega_{e})$. The one to one correspondence
$\tilde{F}=P(\tilde{f}\circ\tilde{\varphi})\circ\tilde{\varphi}^{-1}\leftrightarrow\tilde{f}$
between the elements of $\mathcal{D}(\tilde{\Omega})$ and
$\mathcal{H}(\tilde{\Gamma},\tilde{\Omega})$ leads to the one to one
correspondence $F=\tilde{F}\circ\iota\leftrightarrow f=\tilde{f}\circ\iota$
between the elements of $\mathcal{D}(\Omega_{e})$ and
$\mathcal{H}(\Gamma,\Omega_{e})$. Here, $\mathcal{H}(\Gamma,\Omega_{e})$ is a
function space consisting of all boundary functions $f$ of elements $F$ of
$\mathcal{D}(\Omega_{e})$ assigned a semi-norm $\|f\|_{e}$, where
$\|f\|_{e}^{2}:=D(F)$. We call $\|f\|_{e}^{2}$ the exterior energy of $f$.

For the simple case $\Omega=\mathbb{D}$, we notice that
$\tilde{\Omega}=\mathbb{D}$ by $\iota(z)=z$ on $\mathbb{S}$. Then, the
identity operator
$\mathcal{H}(\mathbb{S},\mathbb{D})\to\mathcal{H}(\mathbb{S},\mathbb{D}_{e})$
is an isometric isomorphism with respect to $\|\cdot\|_{i}$ and
$\|\cdot\|_{e}$. In the next section, we investigate the Jordan domain to
which this property may be generalized.

##  3\. Dirichlet spaces over quasidisks

In this section we look for a sufficient and necessary condition on the
rectifiable Jordan curve $\Gamma$ so that the interior and exterior energies
on $\Gamma$ are equivalent. Precisely, we will show the following

######  Theorem 1.

With the above notation, the identity operator
$\mathcal{H}(\Gamma,\Omega)\to\mathcal{H}(\Gamma,\Omega_{e})$ is a bounded
isomorphism with respect to $\|\cdot\|_{i}$ and $\|\cdot\|_{e}$ if and only if
$\Gamma$ is a quasicircle.

Before starting the proof of Theorem 1, we recall some preliminary facts about
quasicircles and quasisymmetric mappings; see [2] for additional background. A
quasicircle is the image of a circle under a quasiconformal mapping of the
complex plane $\mathbb{C}$, and the inner domain of a quasicircle is called a
quasidisk. Here, by quasiconformal mapping $f$ of $\mathbb{C}$ we mean a
homeomorphism $f$ whose gradient in the sense of distribution belongs to
$L^{2}_{loc}(\mathbb{C})$ and satisfies

$f_{\bar{z}}=\mu(z)f_{z}$

for an essentially uniformly bounded function $\mu\in L^{\infty}(\mathbb{C})$
bounded by some constant $k<1$. Here, $f$ may also be called
$k$-quasiconformal to specify the constant $k$.

A sense-preserving homeomorphism $h$ of $\mathbb{S}$ is called a conformal
welding of the Jordan curve $\Gamma$ if $h=\psi^{-1}\circ\varphi$ where
$\varphi$ and $\psi$ are conformal maps from $\mathbb{D}$ onto $\Omega$ and
from $\mathbb{D}_{e}$ onto $\Omega_{e}$, respectively. So there are many
weldings of $\Gamma$ but they differ from each other by compositions with
Möbius transformations of $\mathbb{S}$. In particular, the conformal welding
of a quasicircle is exactly a quasisymmetric homeomorphism of $\mathbb{S}$,
and the conformal maps $\varphi$ and $\psi$ for a quasicircle can be extended
to $\mathbb{C}$ quasiconformally. Saying a homeomorphism $h$ of $\mathbb{S}$
is quasisymmetric means that there exists a constant $C_{1}>0$ such that

$C_{1}^{-1}\leq\frac{|h(e^{i(\theta+\alpha)})-h(e^{i\theta})|}{|h(e^{i\theta})-h(e^{i(\theta-\alpha)})|}\leq
C_{1}$

for all $\theta\in\mathbb{R}$ and $-\pi/2<\alpha\leq\pi/2$. Here, the optimal
constant $C_{1}$ is called the quasisymmetry constant of $h$. A sense-
preserving homeomorphism $h$ of $\mathbb{S}$ is quasisymmetric if and only if
$h$ preserves the modules of quadrilaterals quasi-invariantly, namely, there
exists a constant $C_{2}>0$ such that for any quadrilateral $Q$ it holds that

$C_{2}^{-1}m(Q)\leq m(h(Q))\leq C_{2}m(Q).$

Moreover, the optimal constant $C_{2}$ and the quasisymmetry constant $C_{1}$
of $h$ only depend on each other. Here, by quadrilateral $Q$ we mean the unit
disk $\mathbb{D}$ together with a pair of disjoint closed arcs on the boundary
$\mathbb{S}$. It is a well-known fact that for a quadrilateral $Q$ with two
disjoint closed arcs $\alpha_{1}$, $\beta_{1}$ on $\mathbb{S}$, its module
$m(Q)$ multiplied by $\frac{1}{2\pi}$ is exactly the minimum of Dirichlet
energies $D(P(f))$ of harmonic functions $P(f)$ on $\mathbb{D}$ ranging over
all boundary values $f$ with $f=0$ on $\alpha_{1}$ and $f=1$ on $\beta_{1}$
(see [6]). Then, by the definition of $\|f\|_{i}^{2}$, we have
$\frac{1}{2\pi}m(Q)=\min\|f\|_{i}^{2}$.

The above concept of quasisymmetry of $\mathbb{S}$ onto $\mathbb{S}$ was
introduced by Beurling and Ahlfors [6], and later formulated for general
metric spaces by Tukia and Väisälä [29]. For our purpose, we only need to
consider the quasisymmetric mapping from a curve $\Gamma_{1}$ onto the other
$\Gamma_{2}$. Let $h:\Gamma_{1}\to\Gamma_{2}$ be a sense-preserving mapping.
Let $\eta:[0,+\infty)\to[0,+\infty)$ be an increasing homeomorphism with
$\lim_{t\to+\infty}\eta(t)=+\infty$. We say that $h$ is $\eta$-quasisymmetric
if for each triple $z_{0}$, $z_{1}$, $z_{2}\in\Gamma_{1}$ we have

$\frac{|h(z_{0})-h(z_{1})|}{|h(z_{0})-h(z_{2})|}\leq\eta\bigg{(}\frac{|z_{0}-z_{1}|}{|z_{0}-z_{2}|}\bigg{)}.$

The more general quasisymmetric mapping on an open subset of $\mathbb{C}$ can
be defined in the same way. It is known that if $h:\mathbb{C}\to\mathbb{C}$ is
a $k$-quasiconformal homeomorphism of $\mathbb{C}$ then $h$ is
$\eta$-quasisymmetric where $\eta$ depends only on $k$. Conversely, if
$h:D\to\mathbb{C}$ is an $\eta$-quasisymmetric mapping on a domain $D$ then it
is quasiconformal (see e.g. Chapter 3 of [4]).

Concerning the quasisymmetric homeomorphism of $\mathbb{S}$ onto $\mathbb{S}$,
the following result of Nag-Sullivan [22] is well-known.

######  Proposition 1.

A sense-preserving homeomorphism $h$ of $\mathbb{S}$ is quasisymmetric if and
only if the composition operator $V_{h}:g\mapsto g\circ h$ gives an
isomorphism of $H^{1/2}(\mathbb{S})$, that is, $V_{h}$ and $(V_{h})^{-1}$ (or
$V_{h^{-1}}$) are bounded linear operators. Here, the operator norm
$\|V_{h}\|$ depends only on the quasisymmetry constant of $h$.

We will also give its generalization to the quasisymmetry of $\mathbb{S}$ onto
a curve $\Gamma$ as three corollaries of main theorems in the remainder, which
can be summarized by

######  Proposition 2.

Let $\Gamma$ be a rectifiable quasicircle. Let $h$ be a sense-preserving
homeomorphism of $\mathbb{S}$ onto $\Gamma$. The composition operator
$V_{h}:g\mapsto g\circ h$ is defined on $H^{1/2}(\Gamma)$. Consider the
following four statements:

  * (a)

$\Gamma$ is chord-arc,

  * (b)

$h$ is quasisymmetric,

  * (c1)

$(V_{h})^{-1}$ is a bounded linear operator from $H^{1/2}(\mathbb{S})$ into
$H^{1/2}(\Gamma)$,

  * (c2)

$V_{h}$ is a bounded linear operator from $H^{1/2}(\Gamma)$ into
$H^{1/2}(\mathbb{S})$.

If any two of the above three statements $\rm(a)$,$\rm(b)$,and $\rm(c1)$ hold,
then the third one holds true, while $\rm(a)$ and $\rm(b)$ imply $\rm(c2)$,
$\rm(a)$ and $\rm(c2)$ imply $\rm(b)$, but $\rm(b)$ and $\rm(c2)$ does not
necessarily imply $\rm(a)$.

Actually one can view $\rm(a,c1)\Rightarrow\rm(b)$ and
$\rm(a,c2)\Rightarrow\rm(b)$ are due to Proposition 1. The complete version of
$\rm(a,b)\Rightarrow\rm(c1,c2)$ is Corollary 1. $\rm(b,c1)\Rightarrow\rm(a)$
and $\rm(b,c2)\nRightarrow\rm(a)$ are just Corollary 2 and Corollary 3,
respectively.

We are now ready to prove Theorem 1.

###### Proof.

If $\Gamma$ is a quasicircle, then there is a bi-Lipschitz map $\phi$ that
fixes $\Gamma$ pointwise and exchanges the two complementary components
$\Omega$ and $\Omega_{e}$ of $\Gamma$, see [2]. For any
$f\in\mathcal{H}(\Gamma,\Omega)$, recall that
$F=P(f\circ\varphi)\circ\varphi^{-1}$ extends $f$ to $\Omega$, minimizing
Dirichlet energies for the boundary function $f$. Then $F\circ\phi$ extends
$f$ to $\Omega_{e}$ and

$\frac{1}{2\pi}\iint_{\Omega_{e}\backslash\\{\infty\\}}|\nabla(F\circ\phi)|^{2}(w)dudv\approx\frac{1}{2\pi}\iint_{\Omega}|\nabla
F|^{2}(z)dxdy=\|f\|_{i}^{2}$

where the implied constants depend only on the bi-Lipschitz constant of
$\phi$. We conclude by the definition of $\|f\|_{e}^{2}$ and (3) that

$\|f\|_{e}^{2}\leq\frac{1}{2\pi}\iint_{\Omega_{e}\backslash\\{\infty\\}}|\nabla(F\circ\phi)|^{2}(w)dudv$

and thus $\|f\|_{e}\lesssim\|f\|_{i}$. The roles of $\Omega$ and $\Omega_{e}$
can be switched in the above argument, so that we also have
$\|f\|_{i}\lesssim\|f\|_{e}$ for any $f\in\mathcal{H}(\Gamma,\Omega_{e})$.
This gives a proof of sufficiency of Theorem 1.

Conversely, let $\varphi$ map $\mathbb{D}$ conformally onto $\Omega$ with the
normalizations $\varphi(0)=0$ and $\varphi^{\prime}(0)=1$, and
$\tilde{\varphi}$ map $\mathbb{D}=\iota(\mathbb{D}_{e})$ conformally onto
$\tilde{\Omega}=\iota(\Omega_{e})$ with $\tilde{\varphi}(0)=0$,
$\tilde{\varphi}^{\prime}(0)=1$. We denote
$\iota\circ\tilde{\varphi}\circ\iota$ by $\psi$ that maps $\mathbb{D}_{e}$
onto $\Omega_{e}$. Then, these three maps are homeomorphisms of the closures.
For any $f\in\mathcal{H}(\Gamma,\Omega)$, we see that $f\circ\iota$ denoted by
$\tilde{f}$ is defined on $\tilde{\Gamma}$. See the following commutative
diagram for a picturesque description of these maps. According to it we see

$\displaystyle\tilde{f}\circ\tilde{\varphi}$
$\displaystyle=(f\circ\iota)\circ(\iota\circ\psi\circ\iota)$
$\displaystyle=f\circ\psi\circ\iota=f\circ\varphi\circ(\varphi^{-1}\circ\psi)\circ\iota\;$
$\displaystyle=f\circ\varphi\circ(\varphi^{-1}\circ\psi).$

Set $\psi^{-1}\circ\varphi=h$ so that

$f\circ\varphi=\tilde{f}\circ\tilde{\varphi}\circ h.$ (4)

Denote the quadrilateral with two disjoint closed arcs $\alpha_{1}$,
$\beta_{1}$ on $\mathbb{S}$ by $Q$, and denote the image of $Q$ under $h$ by
$Q^{\prime}$, the quadrilateral with two disjoint closed arcs
$\alpha_{2}:=h(\alpha_{1})$ and $\beta_{2}:=h(\beta_{1})$.

Take $f\in\mathcal{H}(\Gamma,\Omega)$ so that $f\circ\varphi$ is given only on
part of $\mathbb{S}$: $f\circ\varphi=0$ on $\alpha_{1}$ and $f\circ\varphi=1$
on $\beta_{1}$. By (4) we see that $\tilde{f}\circ\tilde{\varphi}=0$ on
$\alpha_{2}$ and $\tilde{f}\circ\tilde{\varphi}=1$ on $\beta_{2}$. Then, we
have $\frac{1}{2\pi}m(Q)=\min\|f\circ\varphi\|_{i}^{2}=\min\|f\|_{i}^{2}$
ranging over all desired boundary functions $f$. Moreover, the minimum value
is attained on the function $f_{0}\circ\varphi$ which is $0,1$ on
$\alpha_{1}$, $\beta_{1}$ and whose normal derivative vanishes on the
complementary arcs (see [6]), that is, $\frac{1}{2\pi}m(Q)=\|f_{0}\|_{i}^{2}$.
Similarly, we have

$\frac{1}{2\pi}m(Q^{\prime})=\min\|\tilde{f}\circ\tilde{\varphi}\|_{i}^{2}=\min
D(P(\tilde{f}\circ\tilde{\varphi})\circ\tilde{\varphi}^{-1}\circ\iota)=\min\|f\|_{e}^{2}\leq\|f_{0}\|_{e}^{2}.$

By the isomorphism of the identity operator
$\mathcal{H}(\Gamma,\Omega)\to\mathcal{H}(\Gamma,\Omega_{e})$, we have that
$\|f_{0}\|_{e}^{2}\approx\|f_{0}\|_{i}^{2}$, and thus $m(Q^{\prime})\lesssim
m(Q)$. The above reasoning clearly implies the other inequality $m(Q)\lesssim
m(Q^{\prime})$ by exchanging the roles of $Q$ and $Q^{\prime}$. Then from
$m(Q^{\prime})\approx m(Q)$ it follows that $h$ is quasisymmetric, and thus
$\Gamma$ is a quasicircle. ∎

Theorem 1 implies that $\Gamma$ is a quasicircle if and only if
$\mathcal{H}(\Gamma,\Omega)=\mathcal{H}(\Gamma,\Omega_{e})$. Now we define a
“two-sided” space $\mathcal{H}(\Gamma)$ to be
$\mathcal{H}(\Gamma)=\mathcal{H}(\Gamma,\Omega)=\mathcal{H}(\Gamma,\Omega_{e})$.
This is regarded as a Banach space consisting of all functions $f$ whose
harmonic extensions $F$ to $\Omega$ have finite Dirichlet energies, where $f$
is assigned a norm $\|f\|_{\mathcal{H}(\Gamma)}:=\sqrt{D(F)}$ by ignoring the
difference of complex constant functions. Moreover,
$\|f\circ\varphi\|_{H^{1/2}(\mathbb{S})}^{2}$ is, by the Douglas formula,
equal to $D(F)$, so we also have

$\|f\|_{\mathcal{H}(\Gamma)}=\|f\circ\varphi\|_{H^{1/2}(\mathbb{S})}.$ (5)

We end this section with three comments about its content:

  1. (i)

As we have learned from one of the referees, Theorem 1 has been proved by
Schippers-Staubach [25] (also see [26]) with the use of Proposition 1, even
without the assumption of rectifiability, and the space $\mathcal{H}(\Gamma)$
has already been constructed by them. For the sake of completeness, we give a
new proof of Theorem 1 above taking a more direct approach for the rectifiable
case.

  2. (ii)

Let $\varphi$ be a quasiconformal mapping of $\mathbb{C}$ and
$\Omega=\varphi(\mathbb{D})$. Gol’dshtein et al. ([14], see also [15]) have
noticed that if $F\in\mathcal{D}(\Omega)$ then $F$ extended by

$z\mapsto F\circ\varphi\Bigg{(}\frac{1}{\overline{\varphi^{-1}(z)}}\Bigg{)}$
(6)

belongs to the homogeneous Sobolev space $\dot{W}^{1,2}(\mathbb{C})$. If now
$g\in C_{0}^{\infty}(\mathbb{C})$, the set of infinitely smooth functions in
$\mathbb{C}$ with compact support, we clearly have

$\iint_{\Omega}|\nabla g|^{2}(z)dxdy\leq\iint_{\mathbb{C}}|\nabla
g|^{2}(z)dxdy<+\infty,$

and in particular $g|_{\Gamma}\in\mathcal{H}(\Gamma)$ by Dirichlet’s
principle. Let $f\in\mathcal{H}(\Gamma)$ and $F$ its extension to $\mathbb{C}$
of the form (6) as a function in $\dot{W}^{1,2}(\mathbb{C})$. There is a
sequence of functions $F_{n}$ in $C_{0}^{\infty}(\mathbb{C})$ converging to
$F$ in $\dot{W}^{1,2}(\mathbb{C})$ (see e.g. Chapter 11 of [19]); using
Dirichlet’s principle again, we may conclude that the space of restrictions to
$\Gamma$ of $C_{0}^{\infty}(\mathbb{C})$ is dense in $\mathcal{H}(\Gamma)$.

  3. (iii)

Suppose $\varphi$ maps a quasidisk $\Omega^{\prime}$ onto another one $\Omega$
conformally. It is easy to see that $f\mapsto f\circ\varphi$ is an isometric
isomorphism from $\mathcal{H}(\Gamma)$ onto $\mathcal{H}(\Gamma^{\prime})$. We
now claim that if $\varphi$ is $k$-quasiconformal then $f\mapsto
f\circ\varphi$ is still a bounded isomorphism from $\mathcal{H}(\Gamma)$ onto
$\mathcal{H}(\Gamma^{\prime})$. It is a consequence of quasiconformality and
the change of variable formula. Specifically, if $F$ is the harmonic extension
to $\Omega$ of $f\in\mathcal{H}(\Gamma)$ as in (2) then
$D(F\circ\varphi)\lesssim D(F)$, where the implied constant only depends on
$k$. By (3) and Theoem 1, we conclude that

$\|f\circ\varphi\|_{\mathcal{H}(\Gamma^{\prime})}^{2}\leq
D(F\circ\varphi)\lesssim D(F)=\|f\|_{\mathcal{H}(\Gamma)}^{2}.$

Combined with the quasiconformality of $\varphi^{-1}$, the above argument
actually implies the double inequality
$\|f\circ\varphi\|_{\mathcal{H}(\Gamma^{\prime})}\approx\|f\|_{\mathcal{H}(\Gamma)}$.

##  4\. Chord-arc curves

In this section we establish the equivalence between the Banach spaces
$\mathcal{H}(\Gamma)$ and $H^{1/2}(\Gamma)$ when the curve $\Gamma$ is chord-
arc. Let us start with a geometric description of chord-arc curves.

######  Definition 1.

A rectifiable Jordan curve $\Gamma$ is said to be a chord-arc curve (or
$K$-chord-arc curve) if there exists a (least) positive constant $K$, called
the chord-arc constant, such that

$\mathrm{length}(\Gamma(z_{1},z_{2}))\leq K|z_{1}-z_{2}|,\quad\text{for
all}\;z_{1},z_{2}\in\Gamma$

where $\Gamma(z_{1},z_{2})$ is the shorter arc of $\Gamma$ between $z_{1}$ and
$z_{2}$.

Chord-arc curves are also called “Lavrentiev curves”. The inner domain of a
chord-arc curve is called a chord-arc domain. A chord-arc curve is the image
of a circle under a bi-Lipschitz homeomorphism of $\mathbb{C}$. A sense-
preserving bi-Lipschitz map of $\mathbb{C}$ onto $\mathbb{C}$ is
quasiconformal but not vice versa. Hence, a chord-arc curve must be a
quasicircle. Indeed, it is exactly a quasicircle with “ regularity property”
that will be explored in detail in the next section. In this section, We
assume without loss of generality that $\text{length}(\Gamma)=2\pi$.

Now we can state the main result of this section.

######  Theorem 2.

The identity operator $\mathcal{H}(\Gamma)\to H^{1/2}(\Gamma)$ is a bounded
isomorphism with respect to $\|\cdot\|_{\mathcal{H}(\Gamma)}$ and
$\|\cdot\|_{H^{1/2}(\Gamma)}$ if the curve $\Gamma$ is chord-arc.

Before proceeding the proof of Theorem 2, we point out the following basic
observation.

######  Lemma 1.

Let $\Gamma$ be a rectifiable Jordan curve with $\text{length}(\Gamma)=2\pi$.
Set $z(s)$, $0\leq s<2\pi$, to be an arc-length parametrization of $\Gamma$.
Then it holds that, for any $s,t\in[0,2\pi)$,

$\frac{\pi}{2}|e^{it}-e^{is}|=\pi\left|\sin{\frac{t-s}{2}}\right|\geq|z(t)-z(s)|.$
(7)

Moreover, if $\Gamma$ is $K$-chord-arc, then

$\frac{1}{K}|e^{it}-e^{is}|=\frac{2}{K}\left|\sin{\frac{t-s}{2}}\right|\leq|z(t)-z(s)|.$
(8)

###### Proof.

If $|t-s|\leq\pi$, it is easy to see that

$\Big{|}\sin{\frac{t-s}{2}}\Big{|}\geq\frac{2}{\pi}\Big{|}\frac{t-s}{2}\Big{|}\geq\frac{1}{\pi}|z(t)-z(s)|.$

If $\pi<|t-s|<2\pi$, we may also see that

$\Big{|}\sin{\frac{t-s}{2}}\Big{|}=\sin{\frac{2\pi-|t-s|}{2}}\geq\frac{2}{\pi}\cdot\frac{2\pi-|t-s|}{2}\geq\frac{1}{\pi}|z(t)-z(s)|.$

This proves the inequality (7).

Further, with the use of the chord-arc condition one may see that if
$|t-s|\leq\pi$,

$\Big{|}\sin{\frac{t-s}{2}}\Big{|}\leq\frac{1}{2}|t-s|\leq\frac{K}{2}|z(t)-z(s)|,$

and if $\pi<|t-s|<2\pi$,

$\Big{|}\sin{\frac{t-s}{2}}\Big{|}\leq\frac{1}{2}(2\pi-|t-s|)\leq\frac{K}{2}|z(t)-z(s)|.$

This gives a proof of the inequality (8). ∎

###### Proof of Theorem 2.

Suppose that $\Gamma$ is a $K$-chord-arc curve with
$\text{length}(\Gamma)=2\pi$. We notice by Lemma 1 that its arc-length
parametrization $z(e^{is})$, $0\leq s<2\pi$, satisfies

$\frac{1}{K}|e^{it}-e^{is}|\leq|z(e^{it})-z(e^{is})|\leq\frac{\pi}{2}|e^{it}-e^{is}|$
(9)

for any $s,t\in[0,2\pi)$. Then, we see that $z$ is a bi-Lipschitz embedding of
$\mathbb{S}$ into $\mathbb{C}$. Recall that $\varphi$ is a conformal map of
$\mathbb{D}$ onto the chord-arc domain $\Omega$, and a homeomorphism of
closures $\mathbb{D}\cup\mathbb{S}$ onto $\Omega\cup\Gamma$. It follows that
$\varphi$ restricted to $\mathbb{S}$ is a quasisymmetric mapping of
$\mathbb{S}$ onto $\Gamma$. As a consequence, $z^{-1}\circ\varphi$ is a
quasisymmetry of $\mathbb{S}$. Note that

$\|f\|_{H^{1/2}(\Gamma)}^{2}=\frac{1}{4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}\Big{|}\frac{f(z(e^{it}))-f(z(e^{is}))}{z(e^{it})-z(e^{is})}\Big{|}^{2}dtds$

and

$\|f\circ
z\|_{H^{1/2}(\mathbb{S})}^{2}=\frac{1}{4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}\Big{|}\frac{f(z(e^{it}))-f(z(e^{is}))}{e^{it}-e^{is}}\Big{|}^{2}dtds.$

Using (9) gives

$\frac{4}{\pi^{2}}\|f\circ
z\|_{H^{1/2}(\mathbb{S})}^{2}\leq\|f\|_{H^{1/2}(\Gamma)}^{2}\leq K^{2}\|f\circ
z\|_{H^{1/2}(\mathbb{S})}^{2}.$

By Proposition 1, the quasisymmetry of $z^{-1}\circ\varphi$ implies

$\|f\circ\varphi\|_{H^{1/2}(\mathbb{S})}=\|f\circ
z\circ(z^{-1}\circ\varphi)\|_{H^{1/2}(\mathbb{S})}\approx\|f\circ
z\|_{H^{1/2}(\mathbb{S})}.$

Then, we conclude that

$\|f\circ\varphi\|_{H^{1/2}(\mathbb{S})}\approx\|f\|_{H^{1/2}(\Gamma)}.$ (10)

It follows from (5) that
$\|f\|_{\mathcal{H}(\Gamma)}\approx\|f\|_{H^{1/2}(\Gamma)}.$ This completes
the proof of Theorem 2. ∎

We noticed recently that the corresponding result to Theorem 2 for the
critical Besov space has been obtained in [20] via different reasoning. By
checking this proof we can see that (10) is still valid when replacing the
conformal mapping $\varphi$ by any quasisymmetric mapping $h$ of $\mathbb{S}$
onto $\Gamma$, and thus we have the following

######  Corollary 1.

Let $h$ be a quasisymmetric mapping of $\mathbb{S}$ onto a chord-arc curve
$\Gamma$. Then the composition operator $V_{h}:f\mapsto f\circ h$ gives a
bounded isomorphism of $H^{1/2}(\Gamma)$ onto $H^{1/2}(\mathbb{S})$.

##  5\. About the necessity of the chord-arc condition

In the last section (i.e. Theorem 2) we gave a proof of the fact that if
$\Gamma$ is a chord-arc curve then $\mathcal{H}(\Gamma)=H^{1/2}(\Gamma)$. In
other words we have proven the existence of a constant $C>0$ such that the two
following inequalities hold for chord-arc curves:

  * (a)

$\|\cdot\|_{H^{1/2}(\Gamma)}\leq C\|\cdot\|_{\mathcal{H}(\Gamma)}$.

  * (b)

$\|\cdot\|_{\mathcal{H}(\Gamma)}\leq C\|\cdot\|_{H^{1/2}(\Gamma)}$.

Inequality (a) is equivalent to $\mathcal{H}(\Gamma)\subset H^{1/2}(\Gamma)$
and Inequality (b) to $H^{1/2}(\Gamma)\subset\mathcal{H}(\Gamma)$. The purpose
of this section is to examine the possible converse to Theorem 2; that is the
following theorem.

######  Theorem 3.

Let $\Gamma$ be a rectifiable quasicircle. The following two statements hold.

  * (1)

If (a) holds then $\Gamma$ is a chord-arc curve.

  * (2)

There exists a non-chord-arc rectifiable quasicircle $\Gamma$ such that (b)
holds.

We will discuss notions of Ahlfors-regularity in the next subsection; that
will be the key tool in the proof of part $(1)$ of Theorem 3 and also have
independent interests of their own. We then prove parts $(1)$ and $(2)$ of
Theorem 3 in the final two subsections.

###  5.1. Ahlfors-regular curves

In this section we discuss about Ahlfors-regular curves, sometimes also named
Ahlfors-David regular curves. Ahlfors’ name appears because this author
already considered this condition in [1] and David’s one because this author
proved that these curves are precisely the curves for which the Cauchy
operator is bounded on $L^{2}$ (see [8]). In the sequel we will simply call
regular these curves whose precise definition is

######  Definition 2.

Let $\Gamma$ be a rectifiable curve in the plane. We say that $\Gamma$ is
M-regular if for any $z\in\mathbb{C}$ and $r>0$,

$\mathrm{length}(\Gamma\cap D(z,r))\leq Mr,$

where $D(z,r)$ stands for the open disk centered at $z$ and of radius $r$.

Our aim is to prove the following theorem giving two equivalent properties of
regularity. The first of these properties involves the Riemann sphere $S^{2}$,
which is $\mathbb{C}\cup\\{\infty\\}$ equipped with the metric

$d\rho=\frac{|dz|}{1+|z|^{2}}.$

We will denote by s-length the spherical length of a curve. The group of
conformal automorphisms of $S^{2}$ is the group of Möbius transformations

$z\mapsto\frac{az+b}{cz+d},\quad ad-bc=1,$

which is thus isomorphic to ${\rm PSL}(2,\mathbb{C})$.

######  Lemma 2.

If $\Gamma$ is an $M$-regular curve then $T(\Gamma)$ is, for any Möbius
transformation $T$, $12M$-regular.

###### Proof.

Every Möbius transformation $T$ may be factorized as $T=S_{1}\circ\Theta\circ
S_{2}$ where $S_{i},\,i=1,2$, are similitudes, and $\Theta(z)=1/z$. The
property of regularity, along with the constant $M$, is obviously invariant
for similitudes so it suffices to prove the lemma for $\Theta$.

Let us consider an $M$-regular curve $\Gamma$. Set
$z_{0}=r_{0}e^{it}\in\mathbb{C},\,r>0,\,D=D(z_{0},r)$. Without loss of
generality we may assume that $z_{0}$ is real positive (i.e., $t=0$). Set
$\Delta=\Theta^{-1}(D)$. If $r<r_{0}/2$ then $\Delta$ is the disk centered at
$\frac{r_{0}}{r_{0}^{2}-r^{2}}$ with radius
$\frac{r}{r_{0}^{2}-r^{2}}<\frac{4r}{3r_{0}^{2}}$. Moreover if
$z\in\Delta,\,|z|\geq\frac{1}{r_{0}+r}>\frac{2}{3r_{0}}$. It follows that

$\displaystyle\mathrm{length}(\Theta(\Gamma)\cap D)$
$\displaystyle=\int_{\Gamma\cap\Delta}\frac{|dz|}{|z|^{2}}\leq\int_{\Gamma\cap\Delta}\frac{|dz|}{\left(2/(3r_{0})\right)^{2}}$
$\displaystyle\leq
M\left(\frac{4r}{3r_{0}^{2}}\right)\left(\frac{3r_{0}}{2}\right)^{2}=3Mr.$

If now $r\geq r_{0}/2$, $D$ is included in $D(0,3r)$ and $\Delta$ in
$\\{|z|>1/(3r)\\}$. We may then write

$\displaystyle\mathrm{length}(\Theta(\Gamma)\cap D)$
$\displaystyle=\int_{\Gamma\cap\Delta}\frac{|dz|}{|z|^{2}}\leq\int_{\Gamma\cap\\{|z|>\frac{1}{3r}\\}}\frac{|dz|}{|z|^{2}}$
$\displaystyle=\sum\limits_{n\geq
0}\int_{\Gamma\cap\\{\frac{2^{n}}{3r}<|z|\leq\frac{2^{n+1}}{3r}\\}}\frac{|dz|}{|z|^{2}}\leq\sum\limits_{n\geq
0}M\frac{2^{n+1}}{3r}\frac{9r^{2}}{2^{2n}}=12Mr.$

∎

We are now ready to state the main result of this subsection.

######  Theorem 4.

Let $\Gamma$ be a rectifiable curve in the plane. Then the following
statements are equivalent.

  * (i)

$\Gamma$ is an $M$-regular curve,

  * (ii)

There exists $K>0$ such that for any Möbius transformation $T$,

$\text{\rm s-length}(T(\Gamma))\leq K,$

  * (iii)

There exists $C>0$ such that for every $w\notin\Gamma$,

$\mathrm{length}(M_{w}(\Gamma))\leq\frac{C}{d(w,\Gamma)},$

where $M_{w}(z)=1/(z-w)$ is an inversion and $d(w,\Gamma)$ is the distance
from $w$ to $\Gamma$.

###### Proof.

We first show that (i) $\Rightarrow$ (ii). Suppose $\Gamma$ is $M$-regular. It
follows from Lemma 2 that for any Möbius transformation $T$, $T(\Gamma)$ is
$12M$-regular. Let $\eta:\,[0,1]\to\mathbb{C}$ be an absolutely continuous
parametrization of $T(\Gamma)$. We see the s-length of $T(\Gamma)$ is

$\int_{0}^{1}\frac{|\eta^{\prime}(t)|dt}{1+|\eta(t)|^{2}}=I+\sum\limits_{n=0}^{\infty}I_{n},$

where

$I=\int_{|\eta(t)|<1}\frac{|\eta^{\prime}(t)|dt}{1+|\eta(t)|^{2}}\leq 12M$

while

$I_{n}=\int_{2^{n}\leq|\eta(t)|<2^{n+1}}\frac{|\eta^{\prime}(t)|dt}{1+|\eta(t)|^{2}}\leq\frac{12M2^{n+1}}{1+2^{2n}}\leq
12M2^{1-n},$

and thus we have $\text{\rm s-length}(T(\Gamma))\leq K$ by taking $K=60M$.

Now we prove (ii) $\Rightarrow$ (iii). Let $\gamma:\,[0,1]\to\mathbb{C}$ be an
absolutely continuous parametrization of $\Gamma$. Set $w\notin\Gamma$ so that

$\mathrm{length}(M_{w}(\Gamma))=\int_{0}^{1}\frac{|\gamma^{\prime}(t)|dt}{|\gamma(t)-w|^{2}}.$

We may write, by definition of $d:=d(w,\Gamma),$

$|\gamma(t)-w|^{2}\geq\frac{1}{2}(|\gamma(t)-w|^{2}+d^{2})$

for all $t\in[0,1]$, so that, if we define $\eta(t)=\frac{\gamma(t)-w}{d},$

$\int_{0}^{1}\frac{|\gamma^{\prime}(t)|dt}{|\gamma(t)-w|^{2}}\leq
2\int_{0}^{1}\frac{|\gamma^{\prime}(t)|dt}{d^{2}+|\gamma(t)-w|^{2}}=\frac{2}{d}\int_{0}^{1}\frac{|\eta^{\prime}(t)|dt}{1+|\eta(t)|^{2}}\leq\frac{2K}{d}$

by (ii).

Finally, we prove (iii) $\Rightarrow$ (i). This is the hardest part of the
proof. We first notice that in order to test the “regularity property” of a
curve we may replace disks by squares of diameter not greater than the
diameter of $\Gamma$, moreover centered on $\Gamma$. Let $\mathcal{C}$ be such
a square. By the invariance of regularity under similitudes, one may assume
that $\mathcal{C}$ has side-length $1$ and center $0$. We define

$L=\mathrm{length}(\mathcal{C}\cap\Gamma).$

The goal is to estimate $L$ from above by a constant depending only on $C$,
the constant in (iii). If $L\leq 100C$, we are done. In the other case let $n$
be the integer part of $\frac{L}{2C}$. It implies that $L<3Cn$. It remains to
prove that $n$ is bounded from above by a constant depending only on $C$. We
will do this in two steps.

We first cut $\mathcal{C}$ into $n^{2}$ sub-squares of side-length
$\frac{1}{n}$, that we call $\mathcal{C}_{j}$. We denote by $k\mathcal{C}_{j}$
( $k>0$) the intersection of $\mathcal{C}$ and the square with the same center
as $\mathcal{C}_{j}$ whose side-length is $k$ times the side-length of
$\mathcal{C}_{j}$. If $z_{0}\in\mathcal{C}\\!\setminus\\!\\!\Gamma$ then we
have on one side

$\int_{\Gamma\cap\mathcal{C}}|z-z_{0}|^{-2}|dz|\geq 2L,$

while, on the other side, by (iii),

$\int_{\Gamma\cap\mathcal{C}}|z-z_{0}|^{-2}|dz|\leq\frac{C}{d(z_{0},\Gamma)}.$

Consequently,

$d(z_{0},\Gamma)\leq\frac{C}{2L}\leq\frac{1}{4n},$ (11)

from which it follows that every $\mathcal{C}_{j}$ meets $\Gamma$. Since
diam$(\mathcal{C}_{j})\leq$diam$(\Gamma)/10$, $\Gamma$ also meets the
complement of $3\mathcal{C}_{j}$ and, as a consequence,

$\mathrm{length}(\Gamma\cap 3\mathcal{C}_{j})\geq\frac{1}{n}.$

Next, note that

$\int_{\Gamma}|z-z_{0}|^{-2}|dz|\geq\frac{1}{9}\sum\limits_{j\in
J}\int_{3\mathcal{C}_{j}\cap\Gamma}|z-z_{0}|^{-2}|dz|,$

where $J=\\{j:z_{0}\notin 27\mathcal{C}_{j}\\}$. If $z,z^{\prime}\in
3\mathcal{C}_{j},\,j\in J,$ then $|z^{\prime}-z_{0}|\leq 2|z-z_{0}|$, from
which it follows easily that

$\displaystyle\iint_{3\mathcal{C}_{j}}|z-z_{0}|^{-2}dxdy$ $\displaystyle\leq
n\times\mathrm{length}(\Gamma\cap
3\mathcal{C}_{j})\times\Big{(}\frac{3}{n}\Big{)}^{2}\times\max_{z\in
3\mathcal{C}_{j}}|z-z_{0}|^{-2}$
$\displaystyle\leq\frac{36}{n}\int_{\Gamma\cap
3\mathcal{C}_{j}}|z-z_{0}|^{-2}|dz|.$

By comparison with an integral we have

$\displaystyle\sum\limits_{j\in J}\iint_{3\mathcal{C}_{j}}|z-z_{0}|^{-2}dxdy$
$\displaystyle\geq\int_{0}^{\pi/4}\int_{12/n}^{1/2}\frac{1}{r}drd\theta$
$\displaystyle\geq\frac{\pi}{4}\log\frac{n}{24}.$

Combining these estimates we deduce that

$\int_{\Gamma}|z-z_{0}|^{-2}|dz|\geq\frac{n}{500}\log\frac{n}{24},$

and this, combined with (iii), implies

$d(z_{0},\Gamma)\leq\frac{500C}{n\log\frac{n}{24}}.$ (12)

Let $n^{\prime}$ be the integer part of $\frac{n\log\frac{n}{24}}{2000C}$. We
can suppose that $n^{\prime}$ is bigger than $n$. If not, $n$ must be bounded
from above by a constant depending only on $C$, and then we are done. By (12)
we have

$d(z_{0},\Gamma)\leq\frac{1}{4n^{\prime}}$ (13)

which is an improvement of (11). Then we repeat the above discussion. We cut
$\mathcal{C}$ into $n^{\prime 2}$ sub-squares of side-length
$\frac{1}{n^{\prime}}$, and we realize that each of these sub-squares meets
$\Gamma$. On the other hand, $\Gamma$ meets the complement of
$3\mathcal{C}_{j}$. Then, we see

$\mathrm{length}(\Gamma\cap 3\mathcal{C}_{j})\geq\frac{1}{n^{\prime}}.$

As a consequence,

$L>\frac{1}{9}\times\sum_{j=1}^{n^{\prime 2}}\mathrm{length}(\Gamma\cap
3\mathcal{C}_{j})>\frac{n^{\prime}}{9}>\frac{1}{10}\times\frac{n\log\frac{n}{24}}{2000C}.$
(14)

Combining this with $L<3Cn$ we conclude that $n<24e^{60000C^{2}}$. Finally, we
deduce that $L<72Ce^{60000C^{2}},$ where $C$ is the constant occurring in
(iii). The proof of the theorem is complete. ∎

###  5.2. Necessity of the chord-arc property: part $(1)$ of Theorem 3

In this subsection we prove part $(1)$ of Theorem 3. To this end we first
consider $w\in\Omega_{e}$ and apply the Inequality (a) to the analytic
function $f(z)=\frac{1}{z-w}$ defined on $\Omega\cup\Gamma$. The square of the
integral on the right hand side is then

$\displaystyle\|f\|_{\mathcal{H}(\Gamma)}^{2}$
$\displaystyle=\frac{1}{\pi}\iint_{\Omega}|f^{\prime}(z)|^{2}=\frac{1}{\pi}\iint_{\Omega}|z-w|^{-4}dxdy$
$\displaystyle\leq\frac{1}{\pi}\iint_{|z-w|>d(w,\Gamma)}|z-w|^{-4}dxdy$
$\displaystyle=d(w,\Gamma)^{-2}$

as we see using polar coordinates. The square of the integral on the left hand
side is

$\displaystyle\|f\|_{H^{1/2}(\Gamma)}^{2}$
$\displaystyle=\frac{1}{4\pi^{2}}\iint_{\Gamma\times\Gamma}\left|\frac{f(z)-f(\zeta)}{z-\zeta}\right|^{2}|dz||d\zeta|$
$\displaystyle=\frac{1}{4\pi^{2}}\iint_{\Gamma\times\Gamma}\left|\frac{\frac{1}{z-w}-\frac{1}{\zeta-w}}{z-\zeta}\right|^{2}|dz||d\zeta|$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{\Gamma}\int_{\Gamma}\frac{1}{|z-w|^{2}|\zeta-w|^{2}}|dz||d\zeta|$
$\displaystyle=\left(\frac{1}{2\pi}\mathrm{length}\left(M_{w}(\Gamma)\right)\right)^{2}$

by Fubini theorem and with the notations of last subsection.

We have thus proven that if (a) is valid then there exists a constant $C>0$
such that

$\mathrm{length}(M_{w}(\Gamma))\leq\frac{C}{d(w,\Gamma)}$ (15)

for any $w\in\Omega_{e}$. Next we consider $w\in\Omega$ and apply $(a)$ to the
analytic function $f(z)=\frac{1}{z-w}$ defined on $\Omega_{e}\cup\Gamma$.
Using the fact that $\mathcal{H}(\Gamma)$ is a “two-sided” space for a
quasicircle $\Gamma$, we may similarly see that the inequality (15) remains
true for any $w\in\Omega$. We can then invoke Theorem 4, and see that $\Gamma$
has to be regular. Consequently, $\Gamma$ is chord-arc since we have assumed
it is a quasicircle. This completes the proof of part $(1)$ of Theorem 3.

As noted above, it follows from Proposition 1 that when $h$ is a
quasisymmetric mapping from $\mathbb{S}$ onto a rectifiable quasicircle
$\Gamma$ it holds that
$\|f\|_{\mathcal{H}(\Gamma)}=\|f\circ\varphi\|_{H^{1/2}(\mathbb{S})}\approx\|f\circ
h\|_{H^{1/2}(\mathbb{S})}$, and thus part $(1)$ of Theorem 3 does immediately
imply the following fact.

######  Corollary 2.

Let $h$ be a quasisymmetric mapping of $\mathbb{S}$ onto a rectifiable
quasicircle $\Gamma$. Let the composition operator $(V_{h})^{-1}:f\mapsto
f\circ h^{-1}$ be a bounded linear operator from $H^{1/2}(\mathbb{S})$ into
$H^{1/2}(\Gamma)$. Then $\Gamma$ is a chord-arc curve.

###  5.3. Non-Smirnov domains: part $(2)$ of Theorem 3

In this subsection we present a counter-example to show part $(2)$ of Theorem
3. When $\Gamma$ is a Jordan curve with $\Omega$ as interior domain and
$\varphi:\mathbb{D}\to\Omega$ a Riemann mapping, we know (F. and M. Riesz
theorem [23]) that $\Gamma$ is rectifiable if and only if $\varphi^{\prime}$
belongs to the Hardy space $H^{1}(\mathbb{D})$.

######  Definition 3.

Let $\Gamma$ be a rectifiable curve in the plane. We say that $\Omega$ is a
Smirnov domain if $\varphi^{\prime}$ is an outer function of
$H^{1}(\mathbb{D})$; that is,

$\log|\varphi^{\prime}(z)|=\int_{\mathbb{S}}p(z,\zeta)\log|\varphi^{\prime}(\zeta)||d\zeta|\quad\text{\rm
for}\;z\in\mathbb{D},$ (16)

where $p$ is the Poisson kernel.

It has been shown by Lavrentiev [18] that chord-arc domains are Smirnov
domains and later by the second author [31] that Jordan domains with Ahlfors-
regular boundary also have Smirnov property. On the other hand there exists a
rectifiable quasidisk $\Omega$ whose Riemann map $\varphi$ satisfies that
$\varphi^{\prime}$ is an inner function, see [11],[17]; that is,

$|\varphi^{\prime}(\zeta)|=1\quad\text{\rm for almost\;all
}\,\zeta\in\mathbb{S},\quad|\varphi^{\prime}(z)|<1\quad\text{\rm
for}\,z\in\mathbb{D}.$ (17)

In particular, harmonic measure on $\Gamma$ with respect to $\varphi(0)$ is
equal to arc-length measure on $\Gamma$ despite the fact that $\Omega$ is not
a disk. But the Smirnov condition (16) is not satisfied because of (17).

We are going to exploit this fact in order to show that this domain satisfies
Inequality (b) even if it is not chord-arc. This follows immediately from the
following observation

$\displaystyle\|f\|^{2}_{H^{1/2}(\Gamma)}$
$\displaystyle=\int_{0}^{2\pi}\int_{0}^{2\pi}\left|\frac{f(\varphi(e^{it}))-f(\varphi(e^{is}))}{\varphi(e^{it})-\varphi(e^{is})}\right|^{2}\frac{dt}{2\pi}\frac{ds}{2\pi}$
$\displaystyle\geq\frac{4}{\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}\Big{|}\frac{f(\varphi(e^{it}))-f(\varphi(e^{is}))}{e^{it}-e^{is}}\Big{|}^{2}\frac{dt}{2\pi}\frac{ds}{2\pi}$
$\displaystyle=\frac{4}{\pi^{2}}\|f\|^{2}_{\mathcal{H}(\Gamma)},\quad\text{for}\,f\in
H^{1/2}(\Gamma).$

Here, the inequality “$\geq$” is due to the inequality (7). This completes the
proof of part $(2)$ of Theorem 3.

Immediately, by taking the quasisymmetric mapping $h$ to be $\varphi$ on
$\mathbb{S}$ we have the following

######  Corollary 3.

There exists a quasisymmetric mapping $h$ of $\mathbb{S}$ onto a rectifiable
quasicircle $\Gamma$ such that the composition operator $V_{h}:f\mapsto f\circ
h$ is a bounded linear operator from $H^{1/2}(\Gamma)$ into
$H^{1/2}(\mathbb{S})$, but $\Gamma$ is non-chord-arc.

As a side remark, in [16], it is proven that if $\Omega$ is a quasidisk such
that $\varphi^{\prime}$, the derivative of its Riemann map, is a singular
inner function, then $\Omega_{e}$ has to be a Smirnov domain. It follows that
there are counterexamples which are Smirnov domains.

Acknowledgement We would like to thank Yves Meyer for having shared his idea
leading to (iii) $\Rightarrow$ (i) of Theorem 4. We also thank the referees
for many constructive comments, which helped us to greatly improve the quality
of this paper.

## Statements and Declarations

Conflict of interest. On behalf of all authors, the corresponding author
states that there is no conflict of interest regarding the publication of this
paper.

Data Availability. This paper has no associated data.

## References

  * [1] Ahlfors, L.V.: Zur Theorie der Uberlagerungsflachen. Acta. Math. 65 (1935), 157-194.
  * [2] Ahlfors, L.V.: Lectures on quasiconformal mappings. Van Nostrand Math. Studies, 10 (1966).
  * [3] Ahlfors, L.V.: Conformal Invariants, Topics in Geometric Function Theory. McGraw-Hill, 1973.
  * [4] Astala, K., Iwaniec, T. and Martin, G.J.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton University Press, Princeton, 2009.
  * [5] Arcozzi, N., Rochberg, R., Sawyer, T. and Wick, B.: The Dirichlet Space: A Survey. New York J. Math. 17a (2011), 45-86.
  * [6] Ahlfors, L.V. and Beurling A.: The boundary correspondence under quasiconformal mappings. Acta Math. 96 (1956), 125-142.
  * [7] Bishop, C.: Function-Theoretic characterizations of Weil-Petersson Curves. Rev. Math. Iberoam 38 (2022), no. 7, 2355-2384.
  * [8] David, G.: Opérateurs intégraux singuliers sur certaines courbes du plan complexe. Ann. Sci. École Norm. Sup. 17 (1984), no. 4, 157-189.
  * [9] Douglas, J.: Solution of the problem of Plateau. Trans. Amer. Math. Soc. 33 (1931), no. 1, 263-321.
  * [10] Duren, P.L.: Theory of $H^{p}$ spaces. Academic Press, New York, 1970.
  * [11] Duren, P.L., Shapiro, H.S. and Shields, A.L.: Singular measures and domains not of Smirnov type. Duke Math. J. 33 (1966), 247-254.
  * [12] EI-Fallah, O., Kellay, K., Mashreghi, J. and Ransford, T.: A Primer on the Dirichlet space. Cambridge University Press, Cambridge, 2014.
  * [13] Fukushima, M., Oshima, Y. and Takeda, M.: Dirichlet Forms and symmetric Markov Processes. Vol. 19. Walter de Gruyter, 2010.
  * [14] Go’ldshtein, V., Lattuflin, T. and Vodop’yanov, S.: Criteria for extensions of functions of the class $L_{2}^{1}$ from unbounded plane domains. Siberian Math. J. (english translation) 20:2 (1979), 298-301.
  * [15] Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147 (1981), 71-88.
  * [16] Jones, P. and Smirnov, S.: On V. I. Smirnov domains. Ann. Acad. Sci. Fenn. Math. 24 (1999), 105-108.
  * [17] Kahane, J.-P.: Trois notes sur les ensembles parfaits linéaires. Enseign. Math. 15 (1969),185-192.
  * [18] Lavrentiev, M.: Boundary problems in the theory of univalent functions. Amer. Math. Soc. Transl. Ser. 2, 32 (1963), 1-35.
  * [19] Leoni, G.: A First Course in Sobolev Spaces. Volume 18 of Graduate Studies in Mathematics, second edition. American Mathematical Society, Providence, RI, 2017.
  * [20] Liu, T. and Shen, Y.: The jump problem for the critical Besov space. Math. Z. 59, 306 (2024).
  * [21] Meyers, N.G. and Serrin, J.: $H=W$. Proc. Nat. Acad. Sci. U.S.A. 51 (1964), no. 6, 1055-1056.
  * [22] Nag, S. and Sullivan, D.: Teichmüller Theory and the universal periodic mapping via quantum calculus and the $H^{1/2}$ space on the circle. Osaka J. Math. 32 (1995), no. 1, 1-34.
  * [23] Pommerenke, C.: Boundary behavior of conformal maps. Grundlehren Math. Wiss., vol. 299, Springer-Verlag, Berlin, 1992.
  * [24] Ross, W.T.: The classical Dirichlet space. In Recent advances in operator-related function theory, pp. 171-197. Contemp. Math. 393, Amer. Math. Soc., Providence, RI, 2006.
  * [25] Schippers, E. and Staubach, W.: Harmonic reflection in quasicircles and well-posedness of a Riemann-Hilbert problem on quasidisks. J. Math. Anal. Appl. 448 (2017), 864-884.
  * [26] Schippers, E. and Staubach, W.: Analysis on quasidisks: A unified approach through transmission and jump problems. EMS Surv. Math. Sci. 9 (2022), 31-97.
  * [27] Shen, Y.: Weil-Petersson Teichmüller Space. Amer. J. Math. 140 (2018), 1041-1074.
  * [28] Takhtajan, L.A. and Teo, L.-P.: Weil-Petersson metric on the Universal Teichmüller Space. Mem. Amer. Math. Soc. 183 (2006), no. 861.
  * [29] Tukia, P. and Väisälä, J.: Quasisymmetric embeddings of metric spaces. Ann. Acad. Sci. Fenn. Ser. A. I. Math. 5 (1980), 97-114.
  * [30] Wang, Y.: Equivalent descriptions of the Loewner energy. Invent. Math. 218 (2019), no. 2, 573-621.
  * [31] Zinsmeister, M.: Domaines de Lavrentiev. Publ. Math. Orsay, 1985.


# On extropy of past lifetime distribution

Osman Kamari
University of Human Development
Sulaymaniyah, Iraq
Francesco Buono
Università di Napoli Federico II
Italy

###### Abstract

Recently Qiu et al. (2017) have introduced residual extropy as measure of
uncertainty in residual lifetime distributions analogues to residual entropy
(1996). Also, they obtained some properties and applications of that. In this
paper, we study the extropy to measure the uncertainty in a past lifetime
distribution. This measure of uncertainty is called past extropy. Also it is
showed a characterization result about the past extropy of largest order
statistics.

Keywords: Reversed residual lifetime, Past extropy, Characterization, Order
statistics.

AMS Subject Classification: 94A17, 62B10, 62G30

##  1 Introduction

The concept of Shannon entropy as a seminal measure of uncertainty for a
random variable was proposed by Shannon (1948). Shannon entropy $H(f)$ for a
non-negative and absolutely continuous random variable $X$ is defined as
follows:

$H\left(f\right)=-\mathbb{E}[\log f(x)]=-\int_{0}^{+\infty}f\left(x\right)\log
f\left(x\right)\ \mathrm{d}x,$ (1)

where F and f are cumulative distribution function (CDF) and probability
density function (pdf), respectively. There are huge literatures devoted to
the applications, generalizations and properties of Shannon entropy (see, e.g.
Cover and Thomas, 2006).

Recently, a new measure of uncertainty was proposed by Lad et al. (2015)
called extyopy as a complement dual of Shannon entropy (1948). For a non-
negative random variable $X$ the extropy is defined as below:

$J\left(X\right)=-\frac{1}{2}\int_{0}^{+\infty}f^{2}(x)\mathrm{d}x.$ (2)

It’s obvious that $J\left(X\right)\leq 0.$

One of the statistical applications of extropy is to score the forecasting
distributions using the total log scoring rule.

Study of duration is an interest subject in many fields of science such as
reliability, survival analysis, and forensic science. In these areas, the
additional life time given that the component or system or a living organism
has survived up to time $t$ is termed the residual life function of the
component. If $X$ is the life of a component, then
$X_{t}=\left(X-t|X>t\right)$ is called the residual life function. If a
component is known to have survived to age $t$ then extropy is no longer
useful to measure the uncertainty of remaining lifetime of the component.

Therefore, Ebrahimi (1996) defined the entropy for resudial lifetime
$X_{t}=\left(X-t|X>t\right)$ as a dynamic form of uncertainty called the
residual entropy at time $t$ and defined as

$H(X;t)=-\int_{t}^{+\infty}\frac{f(x)}{\overline{F}(t)}\log\frac{f(x)}{\overline{F}(t)}\
\mathrm{d}x,$

where $\overline{F}(t)=\mathbb{P}(X>t)=1-F(t)$ is the survival (reliability)
funltion of $X$.

Analogous to residual entropy, Qiu et al. (2017) defined the extropy for
residual lifetime $X_{t}$ called the residual extropy at time $t$ and defined
as

$J\left(X_{t}\right)=-\frac{1}{2}\int_{0}^{+\infty}f_{X_{t}}^{2}(x)\mathrm{d}x=-\frac{1}{2\overline{F}^{2}(t)}\int_{t}^{+\infty}f^{2}(x)\mathrm{d}x.$
(3)

In many situations, uncertainty can relate to the past. Suppose the random
variable $X$ is the lifetime of a component, system or a living organism,
having an absolutely continuous distribution function $F_{X}(t)$ and the
density function $f_{X}(t)$. For $t>0$, let the random variable
${}_{t}X=(t-X|X<t)$ be the time elapsed after failure till time $t$, given
that the component has already failed at time $t$. We denote the random
variable ${}_{t}X$, the reversed residual life (past lifetime). For instance,
at time $t$, one has under gone a medical test to check for a certain disease.
Suppose that the test result is positive. If $X$ is the age when the patient
was infected, then it is known that $X<t$. Now the question is, how much time
has elapsed since the patient has been infected by this disease? Based on this
idea, Di Crescenzo and Longobardi (2002) introduced the entropy of the
reversed residual lifetime ${}_{t}X$ as a dynamic measure of uncertainty
called past entropy as follows:

$H\left(X;[t]\right)=-\int_{0}^{t}\frac{f(x)}{F(t)}\log\frac{f(x)}{F(t)}\
\mathrm{d}x.$

This measure is dual of residual entropy introduced by Ebrahimi (1996).

In this paper, we study the extropy for ${}_{t}X$ as dual of residual extropy
that is called past extropy and it is defined as below (see also Krishnan et
al. (2020)):

$J\left({}_{t}X\right)=-\frac{1}{2}\int_{0}^{+\infty}f_{{}_{t}X}^{2}(x)\mathrm{d}x=-\frac{1}{2F^{2}(t)}\int_{0}^{t}f^{2}(x)\mathrm{d}x,$
(4)

where $f_{{}_{t}X}(x)=\frac{f(t-x)}{F(t)}$, for $x\in(0,t)$. It can be seen
that for $t\geq 0$, $J\left({}_{t}X\right)$ possesses all the properties of
$J(X)$.

######  Remark 1.

It’s clear that $J\left({}_{+\infty}X\right)=J(X)$.

Past extropy has applications in the context of information theory,
reliability and survival analysis, insurance, forensic science and other
related fields beceuse in that a lifetime distribution truncated above is of
utmost importance.

The paper is organized as follows: in section 2, an approach to measure of
uncertainty in the past lifetime distribution is proposed. Then it is studied
a characterization result with the reversed failure rate. Following a
characterization result is given based on past extropy of the largest order
statistics in section 3.

##  2 Past extropy and some characterizations

Analogous to residual extropy (Qiu et al. (2017)), the extropy for ${}_{t}X$
is called past extropy and for a non-negative random variable $X$ is as below:

$J\left({}_{t}X\right)=-\frac{1}{2}\int_{0}^{+\infty}f_{{}_{t}X}^{2}(x)\mathrm{d}x=-\frac{1}{2F^{2}(t)}\int_{0}^{t}f^{2}(x)\mathrm{d}x,$
(5)

where $f_{{}_{t}X}(x)=\frac{f(t-x)}{F(t)}$, for $x\in(0,t)$ is the density
function of ${}_{t}X$. It’s clear that $J({{}_{t}}X)\leq 0$ while the residual
entropy of a continuous distribution may take any value on
$[-\infty,+\infty]$. Also, $J\left({}_{+\infty}X\right)=J(X)$.

######  Example 1.

  * a)

If $X\sim Exp(\lambda)$, then
$J\left({}_{t}X\right)=-\frac{\lambda}{4}\frac{1+\mathrm{e}^{-\lambda
t}}{1-\mathrm{e}^{-\lambda t}}$ for $t>0$. This shows that the past extropy of
exponential distribution is an increasing function of t.

  * b)

If $X\sim U(0,b)$, then $J\left({}_{t}X\right)=-\frac{1}{2t}$.

  * c)

If $X$ has power distribution with parameter $\alpha>0$, i.e. $f(x)=\alpha
x^{(\alpha-1)}$, $0<x<1$, then
$J\left({}_{t}X\right)=\frac{-\alpha^{2}}{2(2\alpha-1)t}$.

  * d)

If $X$ has Pareto distribution with parameters $\theta>0,x_{0}>0$, i.e.
$f(x)=\frac{\theta}{x_{0}}\frac{x_{0}^{\theta+1}}{x^{\theta+1}}$, $x>x_{0}$,
then
$J\left({}_{t}X\right)=\frac{\theta^{2}}{2(2\theta+1)(t^{\theta}-x_{0}^{\theta})^{2}}\left[\frac{x_{0}^{2\theta}}{t}-\frac{t^{2\theta}}{x_{0}}\right]$.

There is a functional relation between past extropy and residual extropy as
follows:

$J(X)=F^{2}(t)J\left({}_{t}X\right)+\overline{F}^{2}(t)J\left(X_{t}\right),\forall
t>0.$

In fact

$\displaystyle
F^{2}(t)J\left({}_{t}X\right)+\overline{F}^{2}(t)J\left(X_{t}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{t}^{+\infty}f^{2}(x)\mathrm{d}x-\frac{1}{2}\int_{0}^{t}f^{2}(x)\mathrm{d}x$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{0}^{+\infty}f^{2}(x)\mathrm{d}x=J(X).$

From (4) we can rewrite the following expression for the past extropy:

$J\left({}_{t}X\right)=\frac{-\tau^{2}(t)}{2f^{2}(t)}\int_{0}^{t}f^{2}(x)\mathrm{d}x,$

where $\tau(t)=\frac{f(t)}{F(t)}$ is the reversed failure rate.

######  Definition 1.

A random variable is said to be increasing (decreasing) in past extropy if
$J\left({}_{t}X\right)$ is an increasing (decreasing) function of $t$.

######  Theorem 2.1.

$J\left({}_{t}X\right)$ is increasing (decreasing) if and only if
$J\left({}_{t}X\right)\leq(\geq)\frac{-1}{4}\tau(t)$.

###### Proof.

From (5) we get

$\frac{\mathrm{d}}{\mathrm{d}t}J\left({}_{t}X\right)=-2\tau(t)J\left({}_{t}X\right)-\frac{1}{2}\tau^{2}(t).$

Then $J\left({}_{t}X\right)$ is increasing if and only if

$2\tau(t)J\left({}_{t}X\right)+\frac{1}{2}\tau^{2}(t)\leq 0,$

but $\tau(t)\geq 0$ so this is equivalent to

$J\left({}_{t}X\right)\leq-\frac{1}{4}\tau(t).$

∎

######  Theorem 2.2.

The past extropy $J\left({}_{t}X\right)$ of $X$ is uniquely determined by
$\tau(t)$.

###### Proof.

From (5) we get

$\frac{\mathrm{d}}{\mathrm{d}t}J\left({}_{t}X\right)=-2\tau(t)J\left({}_{t}X\right)-\frac{1}{2}\tau^{2}(t).$

So we have a linear differential equation of order one and it can be solved in
the following way

$J\left({}_{t}X\right)=\mathrm{e}^{-2\int_{t_{0}}^{t}\tau(s)\mathrm{d}s}\left[J\left({}_{t_{0}}X\right)-\int_{t_{0}}^{t}\frac{1}{2}\tau^{2}(s)\mathrm{e}^{2\int_{t_{0}}^{s}\tau(y)\mathrm{d}y}\mathrm{d}s\right],$

where we can use the boundary condition $J\left({}_{+\infty}X\right)=J(X)$, so
we get

$J\left({}_{t}X\right)=\mathrm{e}^{2\int_{t}^{+\infty}\tau(s)\mathrm{d}s}\left[J\left(X\right)+\int_{t}^{+\infty}\frac{1}{2}\tau^{2}(s)\mathrm{e}^{-2\int_{s}^{+\infty}\tau(y)\mathrm{d}y}\mathrm{d}s\right].$
(6)

∎

######  Example 2.

Let $X\sim Exp(\lambda)$, with reversed failure rate
$\tau(t)=\frac{\lambda\mathrm{e}^{-\lambda t}}{1-\mathrm{e}^{-\lambda t}}$. It
follows from (6) that

$\displaystyle J\left({}_{t}X\right)$ $\displaystyle=$
$\displaystyle\mathrm{e}^{2\int_{t}^{+\infty}\frac{\lambda\mathrm{e}^{-\lambda
s}}{1-\mathrm{e}^{-\lambda
s}}\mathrm{d}s}\left[J\left(X\right)+\int_{t}^{+\infty}\frac{1}{2}\frac{\lambda^{2}\mathrm{e}^{-2\lambda
s}}{\left(1-\mathrm{e}^{-\lambda
s}\right)^{2}}\mathrm{e}^{-2\int_{s}^{+\infty}\frac{\lambda\mathrm{e}^{-\lambda
y}}{1-\mathrm{e}^{-\lambda y}}\mathrm{d}y}\mathrm{d}s\right]$ $\displaystyle=$
$\displaystyle\left(1-\mathrm{e}^{-\lambda
t}\right)^{-2}\left[J(X)+\frac{1}{2}\int_{t}^{+\infty}\lambda^{2}\mathrm{e}^{-2\lambda
s}\mathrm{d}s\right]$ $\displaystyle=$
$\displaystyle\frac{\lambda}{4}\frac{\mathrm{e}^{-2\lambda
t}-1}{\left(1-\mathrm{e}^{-\lambda
t}\right)^{2}}=-\frac{\lambda}{4}\frac{1+\mathrm{e}^{-\lambda
t}}{1-\mathrm{e}^{-\lambda t}},$

so we find again the same result of example 1.

Using the following definition (see Shaked and Shanthikumar, 2007), we show
that $J\left({}_{t}X\right)$ is increasing in $t$.

######  Definition 2.

Let $X$ and $Y$ be two non-negative variables with reliability functions
$\overline{F},\overline{G}$ and pdfs $f,g$ respectively. $X$ is smaller than
$Y$

  * a)

in the likelihood ratio order, denoted by $X\leq_{lr}Y$, if
$\frac{f(x)}{g(x)}$ is decreasing in $x\geq 0$;

  * b)

in the usual stochastic order, denoted by $X\leq_{st}Y$ if
$\overline{F}(x)\leq\overline{G}(x)$ for $x\geq 0$.

######  Remark 2.

It is well known that if $X\leq_{lr}Y$ then $X\leq_{st}Y$ and $X\leq_{st}Y$ if
and only if $\mathbb{E}(\varphi(Y))\leq(\geq)\mathbb{E}(\varphi(X))$ for any
decreasing (increasing) function $\varphi$.

######  Theorem 2.3.

Let $X$ be a random variable with CDF $F$ and pdf $f$. If
$f\left(F^{-1}(x)\right)$ is decreasing in $x\geq 0$, then
$J\left({}_{t}X\right)$ is increasing in $t\geq 0$.

###### Proof.

Let $U_{t}$ be a random variable with uniform distribution on $(0,F(t))$ with
pdf $g_{t}(x)=\frac{1}{F(t)}$ for $x\in(0,F(t))$, then based on (4) we have

$\displaystyle J\left({}_{t}X\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{2F^{2}(t)}\int_{0}^{F(t)}f\left(F^{-1}(u)\right)\mathrm{d}u=-\frac{1}{2F(t)}\int_{0}^{F(t)}g_{t}(u)f\left(F^{-1}(u)\right)\mathrm{d}u$
$\displaystyle=$
$\displaystyle-\frac{1}{2F(t)}\mathbb{E}\left[f\left(F^{-1}(U_{t})\right)\right].$

Let $0\leq t_{1}\leq t_{2}$. If $0<x\leq F(t_{1})$, then
$\frac{g_{t_{1}}(x)}{g_{t_{2}}(x)}=\frac{F(t_{2})}{F(t_{1})}$ is a non-
negative constant. If $F(t_{1})<x\leq F(t_{2})$, then
$\frac{g_{t_{1}}(x)}{g_{t_{2}}(x)}=0$. Therefore
$\frac{g_{t_{1}}(x)}{g_{t_{2}}(x)}$ is decreasing in $x\in(0,F(t_{2}))$, which
implies $U_{t_{1}}\leq_{lr}U_{t_{2}}$. Hence $U_{t_{1}}\leq_{st}U_{t_{2}}$ and
so

$0\leq\mathbb{E}\left[f\left(F^{-1}(U_{t_{2}})\right)\right]\leq\mathbb{E}\left[f\left(F^{-1}(U_{t_{1}}\right)\right]$

using the assumption that $f\left(F^{-1}(U_{t})\right)$ is a decreasing
function. Since $0\leq\frac{1}{F(t_{2})}\leq\frac{1}{F(t_{1})}$ then

$J\left({}_{t_{1}}X\right)=-\frac{1}{2F(t_{1})}\mathbb{E}\left[f\left(F^{-1}(U_{t_{1}})\right)\right]\leq-\frac{1}{2F(t_{2})}\mathbb{E}\left[f\left(F^{-1}(U_{t_{2}})\right)\right]=J\left({}_{t_{2}}X\right).$

∎

######  Remark 3.

Let $X$ be a random variable with CDF $F(x)=x^{2}$, for $x\in(0,1)$. Then
$f\left(F^{-1}(x)\right)=2\sqrt{x}$ is increasing in $x\in(0,1)$. However
$J\left({}_{t}X\right)=-\frac{2}{3t}$ is increasing in $t\in(0,1)$. So the
condition in theorem 2.3 that $f\left(F^{-1}(x)\right)$ is decreasing in $x$
is sufficient but not necessary.

##  3 Past extropy of order statistics

Let $X_{1},X_{2},\dots,X_{n}$ be a random sample with distribution function
$F$, the order statistics of the sample are defined by the arrangement
$X_{1},X_{2},\dots,X_{n}$ from the minimum to the maximum by
$X_{(1)},X_{(2)},\dots,X_{(n)}$. Qiu and Jia (2017) defined the residual
extropy of the $i-th$ order statistics and showed that the residual extropy of
order statistics can determine the underlying distribution uniquely. Let
$X_{1},X_{2},\dots,X_{n}$ be continuous and i.i.d. random variables with CDF
$F$ indicate the lifetimes of $n$ components of a parallel system. Also
$X_{1:n},X_{2:n},\dots,X_{n:n}$ be the ordered lifetimes of the components.
Then $X_{n:n}$ represents the lifetime of parallel system with CDF
$F_{X_{n:n}}(x)=(F(x))^{n}$, $x>0$. The CDF of
$\left[t-X_{n:n}|X_{n:n}<t\right]$ is $1-\left(\frac{F(t-x)}{F(t)}\right)^{n}$
where $\left[t-X_{n:n}|X_{n:n}<t\right]$ is called reversed residual lifetime
of the system. Now past extropy for reversed residual lifetime of parallel
system with distribution function $F_{X_{n:n}}(x)$ is as follows:

$J\left({}_{t}X_{n:n}\right)=-\frac{n^{2}}{2(F(t))^{2n}}\int_{0}^{t}f^{2}(x)[F(x)]^{2n-2}\mathrm{d}x.$

######  Theorem 3.1.

If $X$ has an increasing pdf $f$ on $[0,T]$, with $T>t$, then
$J\left({}_{t}X_{n:n}\right)$ is decreasing in $n\geq 1$.

###### Proof.

The pdf of $(X_{n:n}|X_{n:n}\leq t)$ can be expressed as

$g_{n:n}^{t}(x)=\frac{nf(x)F^{n-1}(x)}{F^{n}(t)},\mbox{ }x\leq t.$

We note that

$\frac{g_{2n-1:2n-1}^{t}(x)}{g_{2n+1:2n+1}^{t}(x)}=\frac{2n-1}{2n+1}\frac{F^{2}(t)}{F^{2}(x)}$

is decreasing in $x\in[0,t]$ and so $(X_{2n-1:2n-1}|X_{2n-1:2n-1}\leq
t)\leq_{lr}(X_{2n+1:2n+1}|X_{2n+1:2n+1}\leq t)$ which implies
$(X_{2n-1:2n-1}|X_{2n-1:2n-1}\leq t)\leq_{st}(X_{2n+1:2n+1}|X_{2n+1:2n+1}\leq
t)$. If $f$ is increasing on $[0,T]$ we have

$\mathbb{E}\left[f\left(X_{2n-1:2n-1}\right)|X_{2n-1:2n-1}\leq
t\right]\leq\mathbb{E}\left[f\left(X_{2n+1:2n+1}\right)|X_{2n+1:2n+1}\leq
t\right].$

From the definition of the past extropy it follows that

$\displaystyle J\left({}_{t}X_{n:n}\right)$ $\displaystyle=$
$\displaystyle-\frac{n^{2}}{2F^{2n}(t)}\int_{0}^{t}f^{2}(x)F^{2n-2}(x)\mathrm{d}x$
$\displaystyle=$
$\displaystyle\frac{-n^{2}}{2(2n-1)F(t)}\int_{0}^{t}\frac{(2n-1)F^{2n-2}(x)f(x)}{F^{2n-1}(t)}f(x)\mathrm{d}x$
$\displaystyle=$
$\displaystyle\frac{-n^{2}}{2(2n-1)F(t)}\mathbb{E}\left[f\left(X_{2n-1:2n-1}\right)|X_{2n-1:2n-1}\leq
t\right].$

Then it follows that

$\displaystyle\frac{J\left({}_{t}X_{n:n}\right)}{J\left({}_{t}X_{n+1:n+1}\right)}$
$\displaystyle=$
$\displaystyle\frac{n^{2}}{(n+1)^{2}}\frac{2n-1}{2n+1}\frac{\mathbb{E}\left[f\left(X_{2n-1:2n-1}\right)|X_{2n-1:2n-1}\leq
t\right]}{\mathbb{E}\left[f\left(X_{2n+1:2n+1}\right)|X_{2n+1:2n+1}\leq
t\right]}$ $\displaystyle\leq$
$\displaystyle\frac{\mathbb{E}\left[f\left(X_{2n-1:2n-1}\right)|X_{2n-1:2n-1}\leq
t\right]}{\mathbb{E}\left[f\left(X_{2n+1:2n+1}\right)|X_{2n+1:2n+1}\leq
t\right]}\leq 1.$

Since the past extropy of a random variable is non-negative we have
$J\left({}_{t}X_{n:n}\right)\geq J\left({}_{t}X_{n+1:n+1}\right)$ and the
proof is completed. ∎

######  Example 3.

Let $X$ be a random variable distribuited as a Weibull with two parameters,
$X\sim W2(\alpha,\lambda)$, i.e. $f(x)=\lambda\alpha
x^{\alpha-1}\exp\left(-\lambda x^{\alpha}\right)$. It can be showed that for
$\alpha>1$ this pdf has a maximum point
$T=\left(\frac{\alpha-1}{\lambda\alpha}\right)^{\frac{1}{\alpha}}$. Let us
consider the case in which $X$ has a Weibull distribution with parameters
$\alpha=2$ and $\lambda=1$, $X\sim W2(2,1)$ and so $T=\frac{\sqrt{2}}{2}$. The
hypothesis of the theorem 3.1 are satisfied for $t=0.5<T=\frac{\sqrt{2}}{2}$.
Figure 1 shows that $J\left({}_{0.5}X_{n:n}\right)$ is decreasing in
$n\in\\{1,2,\dots,10\\}$. Moreover the result of the theorem 3.1 does not hold
for the smallest order statistic as shown in figure 2.

Figure 1: $J\left({}_{0.5}X_{n:n}\right)$ of a $W2(2,1)$ for $n=1,2,\dots,10$
Figure 2: $J\left({}_{0.5}X_{1:n}\right)$ of a $W2(2,1)$ for $n=1,2,\dots,10$

In the case in which $X$ has an increasing pdf on $[0,T]$ with $T>t$ we give a
lower bound for $J\left({}_{t}X\right)$.

######  Theorem 3.2.

If $X$ has an increasing pdf $f$ on $[0,T]$, with $T>t$, then
$J\left({}_{t}X\right)\geq-\frac{\tau(t)}{2}$.

###### Proof.

From the definition we get

$\displaystyle J\left({}_{t}X\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{2F^{2}(t)}\int_{0}^{t}f^{2}(x)\mathrm{d}x$
$\displaystyle=$
$\displaystyle\frac{-f(t)}{2F(t)}+\frac{1}{2F^{2}(t)}\int_{0}^{t}F(x)f^{\prime}(x)\mathrm{d}x$
$\displaystyle\geq$ $\displaystyle-\frac{\tau(t)}{2}.$

∎

######  Example 4.

Let $X\sim W2(2,1)$, as in example 3, so we know that its pdf is increasing in
$[0,T]$ with $T=\frac{\sqrt{2}}{2}$. The hypothesis of the theorem 3.2 are
satisfied for $t<T=\frac{\sqrt{2}}{2}$. Figure 3 shows that the function
$-\frac{\tau(t)}{2}$ (in red) is a lower bound for the past extropy (in
black). We remark that the theorem gives information only for $t\in[0,T]$, in
fact for larger values of $t$ the function $-\frac{\tau(t)}{2}$ could not be a
lower bound anymore, as showed in figure 3.

Figure 3: $J\left({}_{t}X\right)$ (in black) and $-\frac{\tau(t)}{2}$ (in red)
of a $W2(2,1)$

Qiu (2016), Qiu and Jia (2017) showed that extropy of the $i-th$ order
statistics and residual extropy of the $i-th$ order statistics can
characterize the underlying distribution uniquely. In the following theorem,
whose proof requires next lemma, we show that the past extropy of the largest
order statistic can uniquely characterize the underlying distribution.

######  Lemma 3.1.

Let $X$ and $Y$ be non-negative random variables such that
$J\left(X_{n:n}\right)=J\left(Y_{n:n}\right)$, $\forall n\geq 1$. Then
$X\overset{d}{=}Y$.

###### Proof.

From the definition of the extropy,
$J\left(X_{n:n}\right)=J\left(Y_{n:n}\right)$ holds if and only if

$\int_{0}^{+\infty}F_{X}^{2n-2}(x)f_{X}^{2}(x)\mathrm{d}x=\int_{0}^{+\infty}F_{Y}^{2n-2}(x)f_{Y}^{2}(x)\mathrm{d}x$

i.e. if and only if

$\int_{0}^{+\infty}F_{X}^{2n-2}(x)\tau_{X}(x)\mathrm{d}F^{2}_{X}(x)=\int_{0}^{+\infty}F_{Y}^{2n-2}(x)\tau_{Y}(x)\mathrm{d}F^{2}_{Y}(x).$

Putting $u=F^{2}_{X}(x)$ in the left side of the above equation and
$u=F^{2}_{Y}(x)$ in the right side we have

$\int_{0}^{1}u^{n-1}\tau_{X}\left(F_{X}^{-1}(\sqrt{u})\right)\mathrm{d}u=\int_{0}^{1}u^{n-1}\tau_{Y}\left(F_{Y}^{-1}(\sqrt{u})\right)\mathrm{d}u.$

that is equivalent to

$\int_{0}^{1}u^{n-1}\left[\tau_{X}\left(F_{X}^{-1}(\sqrt{u})\right)-\tau_{Y}\left(F_{Y}^{-1}(\sqrt{u})\right)\right]\mathrm{d}u=0\mbox{
}\forall n\geq 1.$

Then from Lemma 3.1 of Qui (2017) we get
$\tau_{X}\left(F_{X}^{-1}(\sqrt{u})\right)=\tau_{Y}\left(F_{Y}^{-1}(\sqrt{u})\right)$
for all $u\in(0,1)$. By taking $\sqrt{u}=v$ we have
$\tau_{X}\left(F_{X}^{-1}(v)\right)=\tau_{Y}\left(F_{Y}^{-1}(v)\right)$ and so
$f_{X}\left(F_{X}^{-1}(v)\right)=f_{Y}\left(F_{Y}^{-1}(v)\right)$ for all
$v\in(0,1)$. This is equivalent to
$(F_{X}^{-1})^{\prime}(v)=(F_{Y}^{-1})^{\prime}(v)$ i.e.
$F_{X}^{-1}(v)=F_{Y}^{-1}(v)+C$, for all $v\in(0,1)$ with $C$ constant. But
for $v=0$ we have $F_{X}^{-1}(0)=F_{Y}^{-1}(0)=0$ and so $C=0$. ∎

######  Theorem 3.3.

Let $X$ and $Y$ be two non-negative random variables with cumulative
distribution functions $F(x)$ and $G(x)$, respectively. Then $F$ and $G$
belong to the same family of distributions if and only if for $t\geq 0$,
$n\geq 1$,

$J\left({}_{t}X_{n:n}\right)=J\left({}_{t}Y_{n:n}\right).$

###### Proof.

It sufficies to prove the sufficiency. $J\left({}_{t}X_{n:n}\right)$ is the
past extropy for $X_{n:n}$ but it is also the extropy for the variable
${}_{t}X_{n:n}$. So through lemma 3.1 we get ${}_{t}X\overset{d}{=}\mbox{
}_{t}Y$. Then $\frac{F(t-x)}{F(t)}=\frac{G(t-x)}{G(t)}$, for $x\in(0,t)$. If
exists $t^{\prime}$ such that $F(t^{\prime})\neq G(t^{\prime})$ then in
$(0,t^{\prime})$ $F(x)=\alpha G(x)$ with $\alpha\neq 1$. But for all
$t>t^{\prime}$, exists $x\in(0,t)$ such that $t-x=t^{\prime}$ and so $F(t)\neq
G(t)$ and as in the precedent step we have $F(x)=\alpha G(x)$ for $x\in(0,t)$.
Letting $t$ to $+\infty$ we have a contradiction because $F$ and $G$ are both
distribution function and their limit is 1. ∎

##  4 Conclusion

In this paper we studied a measure of uncertainty, the past extropy. It is the
extropy of the inactivity time. It is important in the moment in which with an
observation we find our system down and we want to investigate about how much
time has elapsed after its fail. Moreover we studied some connections with the
largest order statistic.

##  5 Acknowledgement

Francesco Buono is partially supported by the GNAMPA research group of INdAM
(Istituto Nazionale di Alta Matematica) and MIUR-PRIN 2017, Project
”Stochastic Models for Complex Systems” (No. 2017 JFFHSH).

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

## References

  * [1] Cover, T. M. and Thomas, J. A., 2006. Elements of Information Theory, (2nd edn). Wiley, New York.
  * [2] Di Crescenzo, A., Longobardi, M., 2002. Entropy-based measure of uncertainty in past lifetime distributions, Journal of Applied Probability, 39, 434–440.
  * [3] Ebrahimi, N., 1996. How to measure uncertainty in the residual life time distribution. Sankhya: The Indian Journal of Statistics, Series A, 58, 48–56.
  * [4] Krishnan, A. S., Sunoj S. M., Nair N. U., 2020. Some reliability properties of extropy for residual and past lifetime random variables. Journal of the Korean Statistical Society. https://doi.org/10.1007/s42952-019-00023-x.
  * [5] Lad, F., Sanfilippo, G., Agrò, G., 2015. Extropy: complementary dual of entropy. Statistical Science 30, 40–58.
  * [6] Qiu, G., 2017. The extropy of order statistics and record values. Statistics & Probability Letters, 120, 52–60.
  * [7] Qiu, G., Jia, K., 2018. The residual extropy of order statistics. Statistics & Probability Letters, 133, 15–22.
  * [8] Shaked, M., Shanthikumar, J. G., 2007. Stochastic orders. Springer Science & Business Media.
  * [9] Shannon, C. E., 1948. A mathematical theory of communication. Bell System Technical Journal, 27, 379–423.