\section{Introduction}

\subsection{Background}
Let $\mathbb{D} \subset \mathbb{C}$ denote the open unit disc and let
\begin{equation*}
  H^2 = \Big\{ f \in \mathcal{O}(\mathbb{D}) : \sup_{0 \le r < 1} \int_{0}^{2 \pi} |f (r e^{ i t})|^2 \, dt < \infty \Big \}
\end{equation*}
be the classical Hardy space.
It is well known that $H^2$ can be identified with the closed subspace of all functions in
$L^2(\partial \mathbb{D})$
whose negative Fourier coefficients vanish.
Correspondingly, subsets of $\partial \mathbb{D}$ of linear Lebesgue
measure zero frequently play the role of small or negligible sets
in the theory of $H^2$ and related spaces.
For instance, a classical theorem of Fatou shows that every function in $H^2$
has radial limits outside of a subset of $\partial \mathbb{D}$ of Lebesgue measure zero;
see for instance \cite[Chapter 3]{Hoffman62}.
For the disc algebra
\begin{equation*}
  A(\mathbb{D}) = \{ f \in C (\overline{\mathbb{D}}): f \big|_{\mathbb{D}} \in \mathcal{O}(\mathbb{D}) \},
\end{equation*}
the Rudin--Carleson theorem shows that
every compact set $E \subset \partial \mathbb{D}$ of Lebesgue measure zero is an interpolation set for $A(\mathbb{D})$,
meaning that for each $g \in C(E)$, there exists $f \in A(\mathbb{D})$ with $f \big|_E = g$.
In fact, one can achieve that $|f(z)| < \|g\|_\infty$ for $z \in \overline{\mathbb{D}} \setminus E$
(provided that $g$ is not identically zero); this is called peak interpolation.
In particular, there exists $f \in A(\mathbb{D})$ with $f \big|_E = 1$ and $|f(z)| < 1$
for $z \in \overline{\mathbb{D}} \setminus E$, meaning that $E$ is peak set
for $A(\mathbb{D})$. Conversely, every peak set and every interpolation set has Lebesgue measure
zero. For background on this material, see \cite[Chapter II]{Gamelin69}.

In the theory of the classical Dirichlet space
\begin{equation*}
  \mathcal{D} = \Big\{ f \in \mathcal{O}(\mathbb{D}): \int_{\mathbb{D}} |f'|^2 \, dA < \infty \Big\},
\end{equation*}
a frequently used notion of smallness of subsets of $\partial \mathbb{D}$ is that of having
logarithmic capacity zero; see \cite[Chapter II]{EKM+14} for an introduction.
This notion is particularly important in the potential theoretic approch to the Dirichlet space.
We will review the definition in Section \ref{sec:prelim}.
A theorem of Beurling shows that every function in $\mathcal{D}$ has radial limits outside
of a subset of $\partial \mathbb{D}$ of (outer) logarithmic capacity zero; see \cite[Section 3.2]{EKM+14}.
In the context of boundary interpolation, Peller and Khrushch\"{e}v \cite{PK82} showed
that a compact set $E \subset \partial \mathbb{D}$ is an interpolation set
for $A(\mathbb{D}) \cap \mathcal{D}$ if and only if $E$ has logarithmic capacity zero.
Many of these considerations have been extended to standard weighted Dirichlet spaces
and their associated capacities, and more generally to Hardy--Sobolev spaces
on the Euclidean unit ball $\mathbb{B}_d$ of $\mathbb{C}^d$ by Cohn \cite{Cohn89}
and by Cohn and Verbitsky \cite{CV95}.

The Hardy space $H^2$, the Dirichlet space $\mathcal{D}$ and more generally
Hardy--Sobolev spaces on the ball belong to a large class
of reproducing kernel Hilbert spaces of holomorphic functions on the ball,
called regular unitarily invariant spaces. We will recall the precise definition
in Section \ref{sec:prelim}.

In studying regular unitarily invariant spaces $\mathcal{H}$ and especially their multipliers,
a functional analytic smallness condition of subsets of $\partial \mathbb{B}_d$
has proved to be very useful in recent years.
This smallness condition has its roots in the study of the ball algebra
\[
A(\mathbb B_d) = \{ f \in C( \overline{\mathbb B_d}) : f \big|_{\mathbb B_d} \in \mathcal O(\mathbb B_d) \}
\]
as explained in Rudin's book \cite[Chapter 10]{Rudin08}.
We let
\begin{equation*}
  \Mult(\mathcal{H}) = \{ \varphi: \mathbb{B}_d \to \mathbb{C}: \varphi \cdot f \in \mathcal{H}
  \text{ whenever } f \in \mathcal{H} \}
\end{equation*}
denote the multiplier algebra of $\mathcal{H}$.
If $\varphi \in \Mult(\mathcal{H})$, its multiplier norm $\|\varphi\|_{\Mult(\mathcal{H})}$
is the norm of the multiplication operator $f \mapsto \varphi \cdot f$ on $\mathcal{H}$.
A complex regular Borel measure $\mu$ on $\partial \mathbb{B}_d$ is said to be \emph{$\Mult(\mathcal{H})$-Henkin}
if whenever $(p_n)$ is a sequence of polynomials satisfying $\|p_n\|_{\Mult(\mathcal{H})} \le 1$ for all $n \in \mathbb{N}$
and $\lim_{n \to \infty} p_n(z) = 0$ for all $z \in \mathbb{B}_d$, we have
\begin{equation*}
  \lim_{n \to \infty} \int_{\partial \mathbb{B}_d} p_n \, d \mu = 0.
\end{equation*}
A Borel subset $E \subset \partial \mathbb{B}_d$ is said to be \emph{$\Mult(\mathcal{H})$-totally null}
if $|\mu|(E) = 0$ for all $\Mult(\mathcal{H})$-Henkin measures $\mu$.
The Henkin condition can be rephrased in terms of a weak-$*$ continuity property;
see Section \ref{sec:prelim} for this reformulation and for more background.

In the case of $H^2$, a measure is Henkin if and only if it is absolutely continuous
with respect to Lebesgue measure on $\partial \mathbb{D}$. Hence the totally
null sets are simply the sets of Lebesgue measure $0$.
Beyond the ball algebra, these notions were first studied
by Clou\^atre and Davidson for the Drury--Arveson space \cite{CD16}, and then for more
general regular unitarily invariant spaces by Bickel, M\textsuperscript{c}Carthy and the second named author \cite{BHM17}.
Just as in the case of the ball algebra, Henkin measures and totally null sets appear naturally
when studying the dual space of algebras of multipliers \cite{CD16,DH20}, ideals of multipliers \cite{CD18,DH20},
functional calculi \cite{BHM17,CD16a}, and peak interpolation problems for multipliers \cite{CD16,DH20}.

\subsection{Main results}
In this article, we will compare the functional
analytic notion of being totally null with the potential theoretic notion of having capacity zero.
As was pointed out in \cite{DH20}, for the Dirichlet space $\mathcal{D}$,
the energy characterization of logarithmic capacity easily implies that every compact subset of $\partial \mathbb{D}$ that is $\Mult(\mathcal{D})$-totally
null necessarily has logarithmic capacity zero.
We will show that for Hardy--Sobolev spaces spaces on the ball, including the the Dirichlet space on the disc, the two notions of smallness in fact agree.

To state the result, let us recall the definition of Hardy--Sobelev spaces
(a.k.a. Besov--Sobelev spaces) on the ball.
Let $\sigma$ denote the normalized surface measure on $\partial \bB_d$
and let
\[
H^2(\mathbb B_d) = \Big\{f \in \cO(\bB_d) : \sup_{0 \le r < 1} \int_{\partial \bB_d} |f( r \zeta)|^2 d \sigma(\zeta) < \infty \Big\}
\]
be the Hardy space on the unit ball. Let $s \in \bR$.
If $f \in \mathcal{O}(\mathbb{B}_d)$ has a homogeneous decomposition $ f= \sum_{n=0}^\infty f_n$, we let
\begin{equation*}
  \|f\|_s^2 = \sum_{n=0}^\infty (n+1)^{2 s} \|f_n\|^2_{H^2(\mathbb B_d)}
\end{equation*}
and define
\begin{equation*}
  \mathcal{H}_s = \{ f \in \mathcal{O}(\mathbb{B}_d): \|f\|_s < \infty \}.
\end{equation*}
Thus, if $R^s f = \sum_{n=1}^\infty n^s f_n$ denotes the fractional radial derivative, then
\begin{equation*}
  \mathcal{H}_s = \{f \in \mathcal{O}(\mathbb{B}_d): R^s f \in H^2(\mathbb B_d) \}.
\end{equation*}
There are also natural $L^p$-versions of these spaces, but we will exclusively work in the Hilbert space setting.
If $s < 0$, then $\mathcal{H}_s$ is a weighted Bergman space on the ball,
and clearly $\mathcal{H}_0 = H^2(\mathbb B_d)$.
If $d=1$, then $\mathcal{H}_{1/2} = \mathcal{D}$, the classical Dirichlet space on the disc, and if $0 < s < \frac{1}{2}$,
then the spaces $\mathcal{H}_s$ are the standard weighted Dirichlet spaces on the disc.
For $d \ge 1$, the space  $\mathcal{H}_{d/2}$ is sometimes called the Dirichlet space on the ball,
and the spaces $\mathcal{H}_{s}$ for $ \frac{d-1}{2} < s < \frac{d}{2}$ are multivariable versions
of the standard weighted Dirichlet spaces on the disc.
If $s > \frac{d}{2}$, then every function in $\mathcal{H}_s$ extends to a continuous function on $\ol{\mathbb{B}_d}$.

In the range $s \le \frac{d}{2}$, there is a different description of $\mathcal{H}_s$,
which is also sometimes used as the definition.
For $a > 0$, let
\begin{equation*}
  K_a(z,w) = \frac{1}{(1 - \langle z,w \rangle)^a }
\end{equation*}
and
\begin{equation*}
  K_0(z,w) = \log \Big( \frac{e}{1 - \langle z,w \rangle} \Big),
\end{equation*}
and let $\mathcal{D}_a$ denote the reproducing kernel Hilbert space on $\mathbb{B}_d$
with kernel $K_a$.
It is well known that if $a = d - 2s$, then $\mathcal{D}_a = \mathcal{H}_s$, with equivalence of norms.
(This follows by expanding $K_a$ in a power series and comparing the coefficients with
$\|z_1^n\|_{\mathcal H_s}^{-2}$ with the help of Stirlings' formula and the known formula
for $\|z_1^n\|_{H^2(\mathbb B_d)}$; see \cite[Proposition 1.4.9]{Rudin08}).
The space $\mathcal{D}_1$ is usually called the \emph{Drury--Arveson space $H^2_d$} and plays
a key role in multivariable operator theory \cite{Arveson98} and in the
theory of complete Pick spaces \cite{AM02}.
For more background on these spaces, we refer the reader to \cite{ZZ08}.

For each of the spaces $\mathcal{H}_s$ for $\frac{d-1}{2} < s \le \frac{d}{2}$,
there is a natural notion of (non-isotropic Bessel) capacity $C_{s,2}(\cdot)$,
introduced by Ahern and Cohn \cite{AC89}.
Equivalently, for the spaces $\mathcal D_a$, there is a notion of capacity
that can be defined in terms of the reproducing kernel of $\mathcal D_a$.
We will review these definitions and show their equivalence in Section \ref{sec:prelim}.

Our main result concerning the Hardy--Sobolev spaces $\mathcal H_s$ is the following.

\begin{thm}
  \label{thm:main_dirichlet}
  Let $d \in \mathbb{N}$ and let $\frac{d-1}{2} < s \le \frac{d}{2}$.
  A compact subset $E \subset \partial \mathbb{B}_d$ is $\Mult(\mathcal{H}_s)$-totally null
  if and only if $C_{s,2}(E) = 0$.
\end{thm}

In particular, taking $d=1$ and $s=\frac{1}{2}$, we see that in the context of the classical
Dirichlet space $\mathcal D$, a compact subset $E \subset \partial \mathbb D$ is
$\Mult(\mathcal D)$-totally null if and only if it has logarithmic capacity zero.

A direct proof of Theorem \ref{thm:main_dirichlet} will be provided in Section \ref{sec:Dirichlet}.

Moreover, we will prove an abstract result about totally null sets,
which, in combination with work on exceptional sets by Ahern and Cohn \cite{AC89} and by Cohn and Verbitsky \cite{CV95},
will yield a second proof of Theorem \ref{thm:main_dirichlet}.
This result applies to some spaces that are not covered by Theorem \ref{thm:main_dirichlet}, such as the Drury--Arveson space.

It is possible to interpret the capacity zero condition as a condition involving the reproducing
kernel Hilbert space $\mathcal{H}$ (cf.\ Proposition \ref{prop:energy_fa} below), whereas the totally null condition is a condition
on the multiplier algebra $\Mult(\mathcal{H})$.
Complete Pick spaces form a class of spaces in which it is frequently possible
to go back and forth between $\mathcal{H}$ and $\Mult(\mathcal{H})$;
see the book \cite{AM02} and Section \ref{sec:prelim} of the present article for more background.
For now, let us simply mention that the spaces $\mathcal{D}_a$ for $0 \le a \le 1$ are complete Pick spaces.
(For $a=0$, one needs to pass to a suitable equivalent norm.)

If $\cH$ is a reproducing kernel Hilbert space on $\mathbb B_d$, let us say that a compact subset $E \subset \partial \mathbb{B}_d$ is an \emph{unboundedness set for $\mathcal{H}$}
if there exists $f \in \mathcal{H}$ so that $\lim_{r \nearrow 1} |f(r \zeta)|$ for all $\zeta \in E$.
The following result covers the spaces in Theorem \ref{thm:main_dirichlet}, but it also
applies, for example, to the Drury--Arveson space, which corresponds to the end point $s = \frac{d-1}{2}$.

\begin{thm}
  \label{thm:main_CNP}
  Let $\mathcal{H}$ be a regular unitarily invariant complete Pick space on $\mathbb{B}_d$.
  A compact set $E \subset \partial \mathbb{B}_d$ is an unboundedness set for $\mathcal{H}$
  if and only if $E$ is $\Mult(\mathcal{H})$-totally null.
\end{thm}

A refinement of this result will be proved in Section \ref{sec:main_CNP}.
The results of Ahern and Cohn \cite{AC89} and of Cohn and Verbitsky \cite{CV95} on exceptional
sets show that in the case of the spaces $\mathcal{H}_s$ for $\frac{d-1}{2} < s \le \frac{d}{2}$,
a compact subset $E \subset \partial \mathbb B_d$ is an unboundedness set for $\mathcal H_s$
if and only if $C_{s,2}(E) = 0$. Indeed,
the ``only if'' part follows from \cite[Theorem B]{AC89},
the ``if'' part is contained in the construction on p.\ 443 of \cite{AC89}; see also \cite[p. 94]{CV95}.
Thus, we obtain another proof of Theorem \ref{thm:main_dirichlet}.

\subsection{Applications}
We close the introduction by mentioning some applications of Theorem \ref{thm:main_dirichlet}.
The first application concerns peak interpolation.
Extending the work of Peller and Khrushch\"{e}v \cite{PK82} on boundary interpolation in the Dirichlet space,
Cohn and Verbitsky \cite[Theorem 3]{CV95} showed that every compact subset $E \subset \partial \mathbb{B}_d$
with $C_{s,2}(E) = 0$ is a strong boundary interpolation set for $\mathcal{H}_s \cap A(\mathbb{B}_d)$.
This means that for every $g \in C(E)$, there exists $f \in \mathcal{H}_s \cap A(\mathbb{B}_d)$ with
$f \big|_E = g$ and $\max( \|f\|_{\mathcal{H}_s}, \|f\|_{A(\mathbb{B}_d)}) \le \|f\|_{C(E)}$.
Combining Theorem \ref{thm:main_dirichlet} with a peak interpolation result for totally
null sets of Davidson and the second named author \cite{DH20}, we can strengthen the result of Cohn and Verbitsky
in two ways. Firstly, we replace $\mathcal{H}_s \cap A(\mathbb{B}_d)$ with the smaller space $A(\mathcal{H}_s)$,
which is defined to be the multiplier norm closure of the polynomials in $\Mult(\mathcal{H}_s)$. Thus,
\begin{equation*}
  A(\mathcal{H}_s) \subset \Mult(\mathcal{H}_s) \cap A(\mathbb{B}_d) \subset \mathcal{H}_s \cap A(\mathbb{B}_d)
\end{equation*}
with contractive inclusions. Secondly, we obtain a strict pointwise inequality off of $E$.

\begin{thm}
  \label{thm:peak_interpolation}
  Let $d \in \mathbb{N}$, let $\frac{d-1}{2} < s \le \frac{d}{2}$ and let $E \subset \partial \mathbb{B}_d$ be compact with $C_{s,2}(E) = 0$. Then for each $g \in C(E) \setminus \{0\}$, there exists
  $f \in A(\mathcal{H}_s)$ so that
  \begin{enumerate}
    \item $f \big|_E = g$,
    \item $|f(z)| < \|g\|_\infty$ for every $z \in \overline{\mathbb{B}_d} \setminus E$, and
    \item $\|f\|_{\Mult(\mathcal{H}_s)} = \|g\|_\infty$.
  \end{enumerate}
\end{thm}

\begin{proof}
  According to \cite[Theorem 1.4]{DH20}, the conclusion holds when $\mathcal{H}_s$ is replaced with any
  regular unitarily invariant space $\mathcal{H}$ and $E$ is $\Mult(\mathcal{H})$-totally null.
  Combined with Theorem \ref{thm:main_dirichlet}, the result follows.
\end{proof}

In fact, in the setting of Theorem \ref{thm:peak_interpolation},
there exists an isometric linear operator $L: C(E) \to A(\mathcal H_s)$ of
peak interpolation; see \cite[Theorem 8.3]{DH20}.
In a similar fashion, one can now apply other results of \cite{DH20}
in the context of the spaces $\mathcal{H}_s$,
replacing the totally null condition with the capacity zero condition.
In particular, this yields a joint Pick and peak interpolation result (cf.\ \cite[Theorem 1.5]{DH20})
and a result about boundary interpolation in the context of interpolation sequences (cf. \cite[Theorem 6.6]{DH20}).

Our second application concerns cyclic functions. Recall that a function $f \in \mathcal{H}_s$ is said to be \emph{cyclic}
if the space of polynomial multiples of $f$ in dense in $\mathcal{H}_s$.
It is a theorem of Brown and Cohn \cite{BC85} that if $E \subset \partial \mathbb{D}$ has logarithmic
capacity zero, then there exists a function $f \in \mathcal{D} \cap A(\mathbb{D})$ that is cyclic
for $\mathcal{D}$ so that $f \big|_E = 0$; see also \cite{EL19} for an extension to other Dirichlet type
spaces on the disc.
The following result extends the theorem of Brown and Cohn to the spaces $\mathcal{H}_s$ on the ball,
and moreover achieves that $f \in A(\mathcal{H}_s)$, so in particular, $f$ is a multiplier.

\begin{cor}
  Let $d \in \mathbb{N}$, let $\frac{d-1}{2} < s \le \frac{d}{2}$ and let $E \subset \partial \mathbb{B}_d$
  be compact with $C_{s,2}(E)= 0$. Then there exists $f \in A(\mathcal{H}_s)$
  that is cyclic for $\mathcal H_s$
  so that $E = \{z \in \overline{\mathbb{B}_d}: f(z) = 0 \}$.
\end{cor}

\begin{proof}
  Applying Theorem \ref{thm:peak_interpolation} to the constant function $g = 1$, we find $h \in A(\mathcal{H}_s)$
  so that $h \big|_E = 1, |h(z)| < 1$ for $z \in \overline{\mathbb{B}_d} \setminus E$ and $\|h\|_{\Mult(\mathcal{H}_s)} = 1$.
  Set $f = 1 - h$. Clearly, $f$ vanishes precisely on $E$. The fact that $\|h\|_{\Mult(\mathcal{H}_s)} \le 1$ easily
  implies that $f$ is cyclic; see for instance \cite[Lemma 2.3]{AHM+17a} and its proof.
\end{proof}

\section{Preliminaries}
\label{sec:prelim}

\subsection{Regular unitarily invariant spaces and totally null sets}

Throughout, let $d \in \mathbb N$.
A \emph{regular unitarily invariant space} is a reproducing kernel Hilbert space $\mathcal{H}$
on $\mathbb{B}_d$ whose reproducing kernel is of the form
\begin{equation}
\label{eqn:unit_inv_kernel}
  K(z,w) = \sum_{n=0}^\infty a_n \langle z,w \rangle^n,
\end{equation}
where $a_0 = 1$, $a_n > 0$ for all $n \in \mathbb{N}$ and $\lim_{n \to \infty} \frac{a_n}{a_{n+1}} = 1$.
We think of the last condition as a regularity condition, as it is natural
to assume that the power series defining $K$ has radius of convergence $1$, since $\mathcal{H}$
is a space of functions on the ball of radius $1$. Under this assumption, the limit, if it exists,
necessarily equals $1$. We recover $H^2$ and $\mathcal{D}$ by choosing $d=1$
and $a_n = 1$, respectively $a_n = \frac{1}{n+1}$, for all $n \in \mathbb{N}$.
Expanding the reproducing kernels of $\cD_a$ into a power series, one easily checks that $\cD_a$ is a regular unitarily
invariant space for all $a \ge 0$.
While the class of regular unitarily invariant spaces is not stable under passing to an equivalent norm,
one can also check that the spaces $\cH_s$ are regular unitarily invariant spaces for all $s \in \bR$.
Indeed, each space $\mathcal H_s$ has a reproducing kernel as in \eqref{eqn:unit_inv_kernel}, where
\begin{equation*}
    a_n = \|z_1^n\|_{\mathcal H_s}^{-2}.
\end{equation*}
More background on these spaces can be found in \cite{BHM17,DH20,GHX04}.

Let $\mathcal{H}$ be a regular unitarily invariant space. We let $\Mult(\mathcal{H})$ denote the multiplier algebra of $\mathcal{H}$. Identifying a multiplier $\varphi$ with the corresponding multiplication operator on $\mathcal{H}$,
we can regard $\Mult(\mathcal{H})$ as a WOT closed subalgebra of $\mathcal{B}(\mathcal{H})$, the algebra
of all bounded linear operators on $\mathcal{H}$. By trace duality, $\Mult(\mathcal{H})$ becomes
a dual space in this way, and hence is equipped with a weak-$*$ topology.
The density of the linear span of kernel functions in $\mathcal{H}$ implies that on bounded
subsets of $\Mult(\mathcal{H})$, the weak-$*$ topology agrees with the topology
of pointwise convergence on $\mathbb{B}_d$.
In a few places, we will use the following basic and well known fact, which we state as a lemma for easier reference.
For a proof, see for instance \cite[Lemma 2.2]{DH20}.

\begin{lem}
  \label{lem:dilations_convergence}
  Let $\mathcal{H}$ be a regular unitarily invariant space and let $\varphi \in \Mult(\mathcal{H})$.
  Let $\varphi_r(z) = \varphi(r z)$ for $0 \le r \le 1$ and $z \in \mathbb{B}_d$.
  Then $\|\varphi_r\|_{\Mult(\mathcal{H})} \le \|\varphi\|_{\Mult(\mathcal{H})}$
  for all $0 \le r \le 1$ and $\lim_{r \nearrow 1} \varphi_r = \varphi$ in the weak-$*$ topology
  of $\Mult(\mathcal{H})$.
\end{lem}

Let $M(\partial \mathbb{B}_d)$ be the space of complex regular Borel measures on $\partial \mathbb{B}_d$.

\begin{defn}
\label{defn:Henkin_TN}
  Let $\mathcal{H}$ be a regular unitarily invariant space.
  \begin{enumerate}[label=\normalfont{(\alph*)}]
      \item A measure $\mu \in M( \partial \mathbb{B}_d)$ is said to be $\Mult(\mathcal{H})$-Henkin if the functional
\begin{equation*}
  \Mult(\mathcal{H}) \supset \mathbb{C}[z_1,\ldots,z_d] \to \mathbb{C},
  \quad p \mapsto \int_{\partial \mathbb{B}_d} p \, d \mu,
\end{equation*}
extends to a weak-$*$ continuous functional on $\Mult(\mathcal{H})$.
\item A Borel subset $E \subset \partial \mathbb{B}_d$ is said to be $\Mult(\mathcal{H})$-totally null
if $|\mu|(E) = 0$ for all $\Mult(\mathcal{H})$-Henkin measures $\mu$.
  \end{enumerate}
\end{defn}

By \cite[Lemma 3.1]{BHM17}, the definition of Henkin measure given here
is equivalent to the one given in the introduction in terms of sequences of polynomials converging pointwise to zero.
The set of $\Mult(\mathcal{H})$-Henkin measures forms a band (see \cite[Lemma 3.3]{BHM17}), meaning in particular
that $\mu$ is $\Mult(\mathcal{H})$-Henkin if and only if $|\mu|$ is Henkin.
This band property implies that a compact set $E$ is $\Mult(\mathcal{H})$-totally
null if and only if $\mu(E) = 0$ for every positive $\Mult(\mathcal{H})$-Henkin measure
$\mu$ that is supported on $E$; see \cite[Lemma 2.5]{DH20}.

Finally, we require the notion of a \emph{complete Pick space}.
Complete Pick spaces are reproducing kernel Hilbert spaces that are
defined in terms of an interpolation condition for multipliers; see the book \cite{AM02}
for more background.
In the context of regular unitarily invariant spaces, there is a concrete
characterization in terms of the reproducing kernel. If the reproducing
kernel of $\mathcal{H}$ is $K(z,w) = \sum_{n=0}^\infty a_n \langle z,w \rangle^n$,
then $\mathcal{H}$ is a complete Pick space if and only if the sequence $(b_n)_{n=1}^\infty$
defined by the power series identity
\begin{equation*}
  \sum_{n=1}^\infty b_n t^n = 1 - \frac{1}{\sum_{n=0}^\infty a_n t^n}
\end{equation*}
tsatisfies $b_n \ge 0$ for all $n \in \mathbb{N}$ (this is a straightforward generalization of \cite[Theorem 7.33]{AM02}). In particular, the spaces $\mathcal{D}_a$ are complete Pick spaces in the range $0 \le a \le 1$ ; cf.\ \cite[Lemma 7.38]{AM02}. (For $a=0$, one needs to pass to the equivalent norm induced by the reproducing
kernel $\frac{1}{\langle z, w \rangle} \log\big( \frac{1}{1 - \langle z,w \rangle} \big)$.)

\subsection{Capacity}

When defining capacity for (compact) sets $E \subset \partial \bB_d$ induced by the  Hardy-Sobolev spaces $\cH_s$, there are at least two possible approaches.
Each one can be viewed as natural depending the perspective,
and in fact we require both approaches in the proof of Theorem \ref{thm:main_dirichlet}, one for each implication.
The two definitions turn out to be equivalent in the sense that the capacities defined are comparable with absolute constants. In particular, capacity zero sets coincide in both senses. We shall briefly discuss the equivalence of the two definitions.

The first definition, introduced in \cite[p. 489]{AC89}, is motivated by the fact that the spaces $\cH_s$ can be understood as potential spaces and
fits into the framework of the general potential theory of Adams and Hedberg \cite{Adams1996}.
Let $M^+(\partial \mathbb{B}_d)$ denote the set of positive regular Borel measures on $\partial \mathbb{B}_d$.
We let $\sigma$ be the normalized surface measure on $\partial \mathbb B_d$.
If $E \subset \partial \mathbb B_d$ is compact, let $M^+(E)$ be the set of all
measures in $M^+(\partial \mathbb B_d)$ that are supported on $E$.
For $0 \le s < d$, consider the kernel
\[
k_s(z,w) = \frac{1}{|1 - \langle z,w \rangle|^{d-s}} \quad (z,w \in \overline{\bB_d})
\]
and also set
\[
k_d(z,w) = \log \frac{e}{|1 - \langle z,w \rangle|} \quad (z,w \in \overline{\bB_d}).
\]

\begin{defn}
Let $0 \le s \le d$, let $\mu \in M^+(\partial \mathbb B_d)$ and let $E \subset \partial \mathbb B_d$ be compact.
  \begin{enumerate}[label=\normalfont{(\alph*)}]
  \item The \emph{non-isotropic Riesz potential} of $\mu$ is
\[ \cI_s(\mu)(z) =
\int_{\partial \bB_d} k_s(z,w) d \mu(w) \quad (z \in \partial \mathbb B_d).\]
We extend the definition to non-negative measurable functions $f \in L^1(\partial \bB_d, d\sigma ) $ by letting
$\cI_s(f) = \cI_s(f \, d\sigma )$.
\item The \emph{non-isotropic Bessel capacity} of $E$ is defined by
\[ C_{s,2}(E)= \inf \{ \norm f ^2_{L^2(\partial \bB_d , d\sigma )} : \cI_s(f) \geq 1 \,\,\, \text{on} \,\,\, E, f \geq 0 \}. \]
\item The quantity $\|\cI_s(\mu)\|_{L^2(\partial \bB_d, d \sigma)}^2 \in [0,\infty]$ is called the \emph{energy} of $\mu$.
  \end{enumerate}
\end{defn}

By \cite[Theorem 2.5.1]{Adams1996}, we have the following ``dual'' expression for the capacity $C_{s,2}(\cdot)$,
\begin{equation}
\label{eqn:dual}
     C_{s,2}(E)^{1/2} = \sup \{ \mu(E): \mu \in M^+(E), \,\,\, \norm { \cI_s(\mu) }_{L^2(\partial \bB_d, d\sigma)} \leq 1  \}.
\end{equation}
In particular, $C_{s,2}(E) > 0$ if and only if $E$ supports a probability measure of finite energy.

A different approach, which can be justified by regarding $\cH_s$ as a reproducing kernel Hilbert space, is the following; cf. \cite[Chapter 2]{EKM+14}.
Recall that if $a=d-2s$, then $\mathcal H_s = \mathcal D_a$ with equivalent norms. Moreover, we have $k_{2s} = |K_a|$.

\begin{defn}
    Let $\frac{d-1}{2} < s \le \frac{d}{2}$, let $a= d- 2 s$, let $\mu \in M^+(\partial \mathbb{B}_d)$ and let $E \subset \partial \mathbb{B}_d$ be compact.
  \begin{enumerate}[label=\normalfont{(\alph*)}]
    \item The $\mathcal D_a$-\emph{potential} of $\mu$ is
      \begin{equation*}
        \cI_{2s}(\mu)(z) = \int_{\partial \mathbb B_d} |K_a(z,w)| \, d \mu  (w).
      \end{equation*}
    \item The $\mathcal D_a$-\emph{energy}  of $\mu$ is defined by
      \begin{equation*}
        \mathcal{E}(\mu, \mathcal D_a)
        = \int_{\partial \bB_d} \int_{\partial \bB_d} |K_a(z,w)| d \mu(z) d \mu(w).
      \end{equation*}
    \item The $\mathcal D_a$-\emph{capacity} of $E$ is defined by
      \begin{equation*}
        \capac{E}{\mathcal D_a}^{1/2} =  \sup \{ \mu(E): \mu \in M^+(E) , \,\, \mathcal{E}(\mu, \mathcal D_a) \leq 1 \}.
      \end{equation*}
  \end{enumerate}
\end{defn}

As for the Bessel capacity, $\capac{E}{\mathcal{D}_a}> 0$ if and only if $E$ supports a probability measure of finite energy.
The formula $d(z,w) = | 1 - \langle z, w \rangle|^{1/2}$ defines a metric on $\partial \mathbb B_d$,
called the \emph{Koranyi metric}; see \cite[Proposition 5.1.2]{Rudin08}.
Thus, the capacities $\capac{\cdot}{\mathcal D_a}$
fit into the framework of capacities on compact metric spaces developed in \cite[Chapter 2]{EKM+14}.

It appears to be well known to experts that the capacities $\capac{\cdot}{\mathcal D_a}$ and $C_{s,2}(\cdot)$ are equivalent
if $a = d- 2s$,
the point being that the corresponding energies are comparable.
A proof in the case $d=1, s=\frac{1}{2}$ can be found in \cite[Lemma 2.2]{Cascante2012}.
In the case $s \neq \frac{d}{2}$, the crucial estimate is stated in \cite[Remark 2.1]{CO95} without proof.
A proof of the estimate in one direction in this case is contained in \cite[p.442-442]{AC89}.
For the sake of completeness, we provide an argument that applies to all cases under consideration.
We adapt the proof in \cite[Lemma 2.2]{Cascante2012} to the non-isotropic geometry of $\partial \bB_d$.

Throughout, we write $A \lesssim B$ to mean that there exists a constant $C \in (0,\infty)$ so that $A \le C B$,
and $A \approx B$ to mean that $A \lesssim B$ and $B \lesssim A$.

\begin{lem}\label{lem:equiv_riesz_pot}
  Let $\frac{d-1}{2} < s \leq \frac{d}{2}$ and $\mu \in M^+(\partial \bB_d ) $. Then
  \[  \cI_s(\cI_s(\mu)) \approx \cI_{2s}(\mu), \]
  where the implied constants only depend on $s$ and $d$.
\end{lem}

\begin{proof}

We will show that
\[
\int_{\partial \bB_d} k_s(z, \zeta) k_s(\zeta,w) d \sigma(\zeta) \approx k_{2s}(z,w) \quad (z,w \in \partial \bB_d).
\]
The statement then follows by integrating both sides with respect to $\mu$ and using Fubini's theorem.

Let
\[ d(z,w)=|1-\inner{z}{w}|^{\frac{1}{2}}. \quad (z,w \in \partial \mathbb B_d) \]
be the Koranyi metric; see \cite[Proposition 5.1.2]{Rudin08}.
If $\delta>0$ we let $Q_\delta(z) = \{ \zeta \in \partial \bB_d: d(\zeta,z) \le \delta\}$ be the Koranyi ball centered at $z$ with radius $\delta.$
We will repeatedly use the fact that $\sigma(Q_{\delta}(z)) \approx \delta^{2 d}$ for $0 \le \delta \le \sqrt{2}$; see \cite[Proposition 5.1.4]{Rudin08}.

Now let $z,w\in \partial \bB_d$ and set $\delta=\frac{d(z,w)}{2}$. Then, in order to estimate the kernel
\[ \int_{\partial   \bB_d} \frac{d\sigma(\zeta)}{|1-\inner{z}{\zeta}|^{d-s}|1-\inner{\zeta}{w}|^{d-s}}
= \int_{\partial \bB_d} \frac{d \sigma(\zeta)}{d(z,\zeta)^{2(d-s)} d(\zeta,w)^{2(d-s)}},
\]
we split the domain of integration $\partial \bB_d$ as follows
\[ \partial \bB_d = Q_\delta(z) \cup Q_\delta(w) \cup  \{ d(\zeta,z) \leq d(\zeta,w) \} \setminus Q_\delta(z) \cup  \{ d(\zeta,w) \leq d(\zeta,z) \} \setminus Q_\delta(w).  \]
We denote by \MakeUppercase{\romannumeral 1}, \MakeUppercase{\romannumeral 1}$'$, \MakeUppercase{\romannumeral 2}, \MakeUppercase{\romannumeral 2}$'$ the corresponding integrals. By the symmetry of the problem it suffices to estimate \MakeUppercase{\romannumeral 1} and \MakeUppercase{\romannumeral 2}.

For \MakeUppercase{\romannumeral 1}, we note that if $\zeta \in Q_{\delta}(z)$, then $\delta \le d(\zeta,w) \le 3 \delta$
by the triangle inequality for $d$.
Hence, integrating with the help of the the distribution function, we find that
\begin{align*}
    \text{\MakeUppercase{\romannumeral 1}} & \approx \frac{1}{\delta^{2(d-s)}} \int_{Q_\delta(z)}\frac{d\sigma(\zeta)}{d(z,\zeta)^{2(d-s)}} \\
    &= \frac{1}{\delta^{2(d-s)}} \int_{0}^\infty \sigma( \{ \zeta \in Q_{\delta}(z) : d(z,\zeta) \le t^{\frac{-1}{2 (d-s)}} \}) dt\\
           & = \frac{\sigma ( Q_\delta(z) )}{\delta^{4(d-s)}}   +  \frac{1}{\delta^{2(d-s)}} \int_{\delta^{-2(d-s)}}^\infty \sigma ( \{ \zeta \in \partial \mathbb B_d:  d(z,\zeta) \leq t^{\frac{-1}{2(d-s)}} \} ) dt \\
           & \approx \delta^{-2(d-2s)} + \frac{1}{\delta^{2(d-s)}} \int_{\delta^{-2(d-s)}}^{\infty} t^{\frac{-d}{d-s}}dt \\
           & \approx \delta^{-2(d-2s)}.
\end{align*}

Next, using the fact that $d(z,\zeta) \le \sqrt{2}$ for all $z, \zeta \in \partial \bB_d$, we see that
\begin{align*}
    \textup{II} &\le \int_{\partial \bB_d \setminus Q_{\delta}(z)} \frac{d \sigma(\zeta)}{d(z,\zeta)^{4 (d-s)}} \\
     &= \int_{0}^\infty \sigma( \{ \zeta \in \partial \bB_d : \delta < d(z,\zeta) \le t^{\frac{-1}{4(d-s)}} \} ) dt\\
     &\lesssim 1 + \int_{2^{-2(d-s)}}^{\delta^{-4 (d-s)}} t^{\frac{-d}{2(d-s)}} dt \\
     &\lesssim
     \begin{cases}
        \delta^{-2(d-2s)}, & \text{ if } s < \frac{d}{2}, \\
        \log( \delta^{-2}), & \text { if } s = \frac{d}{2}.
     \end{cases}
\end{align*}
Combining the estimates for \textup{I} and \textup{II} and recalling the definition of $\delta$ we see that
\[
\int_{\partial   \bB_d} \frac{d\sigma(\zeta)}{|1-\inner{z}{\zeta}|^{d-s}|1-\inner{\zeta}{w}|^{d-s}}
\lesssim
\begin{cases}
\frac{1}{|1 - \langle z,w \rangle |^{d - 2 s}}, & \text{ if } s < \frac{d}{2} \\
\log\big( \frac{e}{|1 - \langle z,w \rangle|} \big), & \text{ if } s = \frac{d}{2}.
\end{cases}
\]

To establish the lower bound, it suffices to consider $z,w \in \partial \bB_d$ for which $d(z,w)$ is small.
In the case $s < \frac{d}{2}$, the lower bound follows from the treatment of the integral $\textup{I}$ above.
Let $s = \frac{d}{2}$.
Notice that in the region $ \cU_{z,w}  = \{ \zeta \in \partial \bB_d : d(z,w)  \leq d(w,\zeta) \} $,  the triangle inequality yields $d(\zeta,z) \leq 2d(\zeta,w)$.
Hence integrating again with the distribution function and writing $\delta  = (z,w )$, we estimate
\begin{align*}
    \int_{\partial \bB_d} \frac{d\sigma(\zeta)}{d(\zeta,w)^dd(\zeta,z)^d} & \gtrsim \int_{ \cU_{z,w} } \frac{d\sigma(\zeta)}{d(\zeta,w)^{2d}} \\
    & = \int_0^{\delta^{-{2d}}} \sigma( \{ \zeta \in \partial \bB_d : \delta \leq  d(\zeta,w) \leq t^{\frac{-1}{2d}} \} ) dt \\
    & = \int_{0}^{\delta^{-{2d}}} \sigma ( Q_{t^{-\frac{1}{2d}}}(w) ) dt - \delta^{-{2d}} \sigma( Q_\delta(w) ) \\
    &   \geq c_0 \log ( \delta^{-1} ) - c_1,
\end{align*}
where $c_0,c_1 > 0$ are constants depending only on the dimension $d$. This shows the lower bound for small $\delta$, which concludes the proof.
\end{proof}

From this lemma the equivalence of the capacities $C_{s,2}(\cdot)$ and $\capac{\cdot}{\mathcal D_a}$ for $a = d - 2s$ follows easily.
\begin{cor}
\label{cor:energy_comparable}
  Let $\frac{d-1}{2} < s \leq \frac{d}{2}$, let $a = d - 2s$ and $\mu \in M^+(\partial \bB_d ) $. Then
  \[
  \|\cI_s(\mu)\|^2_{L^2(\partial \bB_d, d \sigma)} \approx \cE(\mu,\mathcal D_a).
  \]
  Hence $C_{s,2}(E) \approx \capac{E}{\cD_a}$ for compact subsets $E \subset \partial \bB_d$.
  Here, all implied constants only depend on $d$ and $s$.
\end{cor}

\begin{proof}
For a measure $\mu \in M^+(\partial \bB_d) $, we compute
\begin{align*}
    \norm{\cI_s(\mu)}^2_{L^2(\partial \bB_d, d\sigma)}
    &  = \int_{\partial \bB_d} \Big( \int_{\partial \bB_d} \frac{d\mu(z)}{|1-\inner{z}{\zeta}|^{d-s}} \Big)^2 d\sigma \\
    &  = \int_{\partial \bB_d} \int_{\partial \bB_d}\int_{\partial   \bB_d} \frac{d\sigma(\zeta)}{|1-\inner{z}{\zeta}|^{d-s}|1-\inner{w}{\zeta}|^{d-s}}  d\mu(z)d\mu(w) \\
    &  = \int_{\partial \bB_d}  \cI_s(\cI_s(\mu)) d\mu(w).
\end{align*}
Thus, Lemma \ref{lem:equiv_riesz_pot} yields that
\[
    \norm{\cI_s(\mu)}^2_{L^2(\partial \bB_d, d\sigma)}
     \approx \int_{\partial \bB_d} \cI_{2s}(\mu)(w) d\mu(w)
    = \cE(\mu, \mathcal D_a).
\]
Since the energies involved are comparable, so are the capacities by \eqref{eqn:dual}.
\end{proof}

\section{Direct proof of Theorem \ref{thm:main_dirichlet}}
\label{sec:Dirichlet}

To prove Theorem \ref{thm:main_dirichlet}, we will make use of holomorphic potentials.
Since several of our proofs involve reproducing kenrel arguments, it is slightly more convenient
to work with the spaces $\mathcal D_a$ rather than with $\mathcal H_s$.

\begin{defn}
  Let $0 \le a < 1$ and let $\mu \in M^+(\partial \mathbb{B}_d)$.
  The holomorphic potential of $\mu$ is the function
  \begin{equation*}
    f_\mu: \mathbb{B}_d \to \mathbb{C}, \quad z \mapsto \int_{\partial \mathbb{B}_d} K_a(z,w) \, d\mu(w).
  \end{equation*}
\end{defn}

Let $A(\mathbb{B}_d)$ denote the ball algebra.
If $\mu \in M^+(\partial \mathbb{B}_d)$, let
\begin{equation*}
  \rho_\mu: A(\mathbb{B}_d) \to \mathbb{C}, \quad f \mapsto \int_{\partial \mathbb{B}_d} f \, d \mu
\end{equation*}
denote the associated integration functional.

The following functional analytic interpretation of the holomorphic potential
and of capacity will show that every totally null set has capacity zero.
In the case of the Dirichlet space on the disc, it is closely related to the energy formula
for logarithmic capacity in terms of the Fourier coefficients of a measure; see for instance \cite[Theorem 2.4.4]{EKM+14}.

\begin{prop}
  \label{prop:energy_fa}
  Let $\mu \in M^+(\partial \mathbb{B}_d)$ and let $0 \le a < 1$. The following assertions are equivalent:
  \begin{enumerate}[label=\normalfont{(\roman*)}]
    \item $\mathcal{E}(\mu,\mathcal D_a) < \infty$,
    \item the densely defined functional $\rho_\mu$ is bounded on $\mathcal{D}_a$,
    \item $f_\mu \in \mathcal{D}_a$.
  \end{enumerate}
  In this case,
  \begin{equation*}
    \mathcal{E}(\mu,\mathcal D_a) \approx \|\rho_\mu\|^2_{(\mathcal{D}_a)^*} = \| f_\mu\|_{\mathcal{D}_a}^2,
  \end{equation*}
  where the implied constants only depend on $a$ and $d$,
  and
  \begin{equation*}
    \rho_\mu(g) = \langle g, f_\mu \rangle_{\mathcal{D}_a}
  \end{equation*}
  for all $g \in \mathcal{D}_a$.
\end{prop}

\begin{proof}
  For ease of notation, we write $f = f_\mu$, $\rho = \rho_\mu$ and $k_w(z) = K_a(z,w)$.
  For $0 \le r < 1$, define $f_r(z) = f(rz)$ and $\rho_r(f) = \rho(f_r)$.
  Then each $f_r \in \mathcal{D}_a$ and each $\rho_r$ is a bounded functional on $\mathcal{D}_a$.
  First, we connect $f_r$ and $\rho_r$, which will be useful in all parts of the proof.
  By the reproducing property of the kernel, we find that
  \begin{equation*}
    \langle k_z, f_r \rangle = \overline{f(r z)} = \int_{\partial \mathbb{B}_d} k_{rz}(w) \, d \mu(w)
    = \int_{\partial \mathbb{B}_d} k_z(r w) \, d\mu (w)
    = \rho_r(k_z)
  \end{equation*}
  for all $z \in \mathbb{B}_d$. Since finite linear combinations of kernels are dense in $\mathcal{D}_a$,
  it follows that
  \begin{equation}
    \label{eqn:proof_energy_fa}
    \rho_r(g) =  \langle g, f_r \rangle
  \end{equation}
  for all $g \in \mathcal{D}_a$ and hence $\|\rho_r\|_{(\mathcal{D}_a)^*} = \|f_r\|_{\mathcal{D}_a}$.

  Next, we show the equivalence of (ii) and (iii).
  If $f \in \mathcal{D}_a$, then $\lim_{r \nearrow 1} f_r = f$ in $\mathcal{D}_a$ and hence for all $g \in A(\mathbb{B}_d) \cap \mathcal{D}_a$,
  Equation \eqref{eqn:proof_energy_fa} shows that
  \begin{equation*}
    \rho(g) = \lim_{r \nearrow 1} \rho_r(g) = \lim_{r \nearrow 1} \langle g, f_r \rangle =  \langle g,f \rangle,
  \end{equation*}
  so $\rho$ is bounded on $\mathcal{D}_a$. In this case, $\|\rho\|_{(\mathcal{D}_a)^*} = \|f\|_{\mathcal{D}_a}$,
  which establishes the final statement of the proposition holds as well.
  Conversely, if $\rho$ is bounded on $\mathcal{D}_a$, then since $\|\rho_r\|_{(\mathcal{D}_a)^*}
  \le \|\rho\|_{(\mathcal{D}_a)^*}$, it follows that $\sup_{0 \le r < 1} \|f_r\|_{\mathcal{D}_a} \le \|\rho\|_{(\mathcal{D}_a)^*}$,
  hence $f \in \mathcal{D}_a$.

  It remains to show the equivalence of (i) and (iii) and that $\mathcal{E}(\mu,\mathcal D_a) \approx \|f\|_{\mathcal{D}_a}^2$.
  With the help of Equation \eqref{eqn:proof_energy_fa}, we see that
  \begin{align*}
    \|f_r\|_{\mathcal{D}_a}^2 = \langle f_r, f_r \rangle = \rho_r(f_r)
    &= \int f(r^2 z) \, d \mu(z) \\
    &= \iint K_a(rz,rw) \, d\mu(w) d \mu(z),
  \end{align*}
  where all integrals are taken over $\partial \mathbb{B}_d$. Taking real parts and using the fact that $\Re K_a$ and $|K_a|$
  are comparable, we find that
  \begin{equation*}
    \|f_r\|_{\mathcal{D}_a}^2
    \approx \iint |K_a(r z, rw)| d \mu(z) d \mu(w).
  \end{equation*}
  Thus, if $f \in \mathcal{D}_a$, then Fatou's lemma shows that
  \begin{equation*}
    \mathcal{E}(\mu,\mathcal D_a) = \iint |K_a(z,w)| \, d \mu(z) d \mu(w) \lesssim \|f\|_{\mathcal{D}_a}^2.
  \end{equation*}
  Conversely, if $\mathcal{E}(\mu,\mathcal D_a) < \infty$, we use the basic inequality
  \begin{equation*}
    \Big| \frac{1}{1 - r^2 \langle z,w \rangle} \Big| \le 2 \Big| \frac{1}{ 1 - \langle z,w \rangle} \Big| \quad (z,w \in \mathbb{B}_d)
  \end{equation*}
  and the Lebesgue dominated convergence theorem to find that
  \begin{equation*}
    \lim_{r \nearrow 1} \|f_r\|_{\mathcal{D}_a}^2 \lesssim \mathcal{E}(\mu,\mathcal D_a),
  \end{equation*}
  so $f \in \mathcal{D}_a$ and $\|f\|_{\mathcal{D}}^2 \lesssim \mathcal{E}(\mu,\mathcal D_a)$.
\end{proof}

With this proposition in hand, we can prove the ``only if'' part of Theorem \ref{thm:main_dirichlet},
which we restate in equivalent form (see Corollary \ref{cor:energy_comparable} for the equivalence).
The idea is the same as that in the proof of \cite[Proposition 2.6]{DH20}.

\begin{prop}
\label{prop:only_if}
  Let $0 \le a < 1$ and let $E \subset \partial \mathbb{B}_d$ be compact.
  If $E$ is $\Mult(\mathcal{D}_a)$-totally null, then $\capac{E}{\mathcal{D}_a} = 0$.
\end{prop}

\begin{proof}
Suppose that $\capac{E}{\mathcal{D}_a} > 0$. Then $E$ supports a probability measure $\mu$ of finite energy $\mathcal E(\mu,\mathcal D_a)$.
  By Proposition \ref{prop:energy_fa}, we see that the integration functional $\rho_\mu$ is bounded
  on $\mathcal{D}_a$. In particular, it is weak-$*$ continuous on $\Mult(\mathcal{D}_a)$.
  Hence $E$ is not $\Mult(\mathcal{D}_a)$-totally null.
\end{proof}

To prove the converse, we require the following fundamental properties of the holomorphic potential of a capacitary extremal measure
of a compact subset $E \subset \mathbb B_d$, i.e. a measure for which the supremum in \eqref{eqn:dual} is achieved.
If $a> 0$, these properties are contained in the proof of \cite[Theorem 2.10]{AC89}, see also \cite[Lemma 2.3]{Cascante2012}
for a proof in the case $d=1$ and $a=0$.
An argument that directly works with the capacity $\capac{\cdot}{\mathcal D_0}$ in the case $d=1$ and $a=0$
can be found on pp.\ 40--41 of \cite{EKM+14}.
We briefly sketch the argument in general.
\begin{lem}
  \label{lem:potential_basic}
  Let $E \subset \partial \mathbb{B}_d$ be a compact set with $\capac{E}{\mathcal{D}_a} > 0$.
  There exists a positive measure $\mu$ supported on $E$ so that the corresponding holomorphic
  potential $f_\mu$ satisfies
  \begin{enumerate}[label=\normalfont{(\alph*)}]
    \item $f_\mu \in \mathcal D_a$ with $\|f_\mu\|^2_{\cD_a} \lesssim \capac{E}{\cD_a}$.
    \item $\liminf_{r \nearrow 1} \Re f_\mu(r \zeta) \gtrsim 1$ for all $\zeta \in \operatorname{int}(E)$, and
    \item $|f_\mu(z)| \lesssim 1$ for all $z \in \mathbb{B}_d$.
  \end{enumerate}
  Here, the implied constants only depend on $a$ and $d$.
\end{lem}

\begin{proof}
  Let $s = \frac{d-a}{2}$, so that $\mathcal H_s = \mathcal D_a$ with equivalent norms.
  The general theory of Bessel capacities (see \cite[Theorem 2.5.3]{Adams1996}), combined with the maximum principle
  for the capacity $C_{s,2}(\cdot)$ \cite[Lemma 1.15]{AC89} implies that there exists a positive measure $\mu$ supported on $E$ so that
  \begin{enumerate}[]
      \item $\mu(E) = \|\mathcal I_s(\mu)\|^2_{L^2(\partial \mathbb B_d, d \sigma)} = C_{s,2}(E)$;
      \item $\mathcal{I}_s(\mathcal I_s(\mu)) \ge 1$ on $E \setminus F$, where
      $F$ is a countable union of compact sets of $C_{s,2}$-capacity zero.
      \item $\mathcal{I}_s(\mathcal I_s(\mu)) \lesssim 1$ on $\partial \mathbb B_d$.
  \end{enumerate}
  (See also \cite[Corollary 2.4.3]{EKM+14} for an approach using $\capac{\cdot}{\mathcal D_0}$ in the case $d=1$ and $a=0$.)

  Item (1) and Corollary \ref{cor:energy_comparable} show that $\mathcal E(\mu, \mathcal D_a) \approx \|\mathcal I_{s}(\mu) \|^2_{L^2(\partial \bB_d, \sigma)} \approx \capac{E}{\mathcal D_a}$,
  hence Proposition \ref{prop:energy_fa} yields that (a) holds.

  Lemma \ref{lem:equiv_riesz_pot} and Item (3) show that for $z \in \partial \mathbb B_d$, we have
  \[
     \int_{\partial \mathbb B_d} |K_a(z,w)| d \mu(w) = \mathcal I_{2s}(\mu)(z) \approx \mathcal I_s(\mathcal I_s(\mu))(z) \lesssim 1,
  \]
  so in combination with the basic inequality $| \frac{1}{1 - r \langle z, w \rangle}| \le 2 | \frac{1}{1 - \langle z,w \rangle }|$ for $z,w \in \partial \mathbb B_d$
  and $0 \le r <1$, we see that (c) holds.

  To establish (b), notice that (c) implies that $f_\mu \in H^\infty(\mathbb B_d)$,
  so $f_\mu$ has radial boundary limits $f_\mu^*$ almost everywhere with respect to $\sigma$, and $f_\mu = P[f_\mu^*]$, the Poisson
  integral of $f_\mu^*$.
  Fatou's lemma and the fact that $\Re K_a$ and $|K_a|$ are comparable show that for $\sigma$-almost every $z \in \partial \mathbb B_d$,
  the estimate
  \begin{align*}
    \Re f_{\mu}^*(z) = \lim_{r \nearrow 1} \int_{\partial \mathbb B_d} \Re K_a(r z, w) d \mu(w) &\gtrsim \int_{\partial \mathbb B_d} |K_a(z,w)| \, d \mu (w) \\
    &= \mathcal I_{2s}(\mu)(z).
    \end{align*}
    Now $C_{s,2}(K) = 0$ implies that $\sigma(K) = 0$ for compact sets $K \subset \partial \bB_d$.
    (This is because $\sigma \big|_K$ has finite energy, which for instance follows from Proposition \ref{prop:energy_fa} since $\cD_a$ is continuously contained in $H^2(\mathbb B_d)$.)
    Therefore, Item (2) and Lemma \ref{lem:equiv_riesz_pot} imply that
    $\Re f_{\mu}^*(z) \gtrsim 1$ for $\sigma$-almost every $z \in E$. In combination with $\Re f_\mu = P[ \Re f_{\mu}^*]$,
    this easily implies (b).
\end{proof}

In \cite{Cascante2014}, Cascante, F\`{a}brega and Ortega showed that if $0 < a < 1$ and if the holomorphic potential $f_\mu$ is bounded in $\mathbb{B}_d$, then
it is a multiplier of $\mathcal{D}_a$.
They also proved an $L^p$-analogue of this statement.
We will require an explicit estimate for the multiplier norm of $f_\mu$.
It seems likely that the arguments in \cite{Cascante2014} could be used to obtain such an estimate.
Instead, we will provide a different argument in the Hilbert space setting, based on the following result of Aleman, M\textsuperscript{c}Carthy,
Richter and the second named author \cite{AHM+17c}.
It also applies to the case $a=0$ without changes.
The function $V_f$ below is called the \emph{Sarason function}  of $f$.

\begin{thm}[\cite{AHM+17c}]
  \label{thm:Sarason_function}
  Let $0 \le a < 1$, let $f \in \mathcal{D}_a$ and define
  \begin{equation*}
    V_f(z) = 2 \langle  f, K_a(\cdot,z) f \rangle  - \|f\|^2.
  \end{equation*}
  If $\Re V_f$ is bounded in $\mathbb{B}_d$, then $f \in \Mult(\mathcal{D}_a)$ and
  \begin{equation*}
    \|f\|_{\Mult(\mathcal{D}_a)} \lesssim \| \Re V_f\|_{\infty}^{1/2},
  \end{equation*}
  where the implied constant only depends on $a$ and $d$.
\end{thm}

\begin{proof}
  In \cite[Theorem 4.5]{AHM+17c}, it is shown that if $\mathcal{H}$ is a normalized complete Pick space
  that admits an equivalent norm which is given by an $L^2$-norm of derivatives of order at most $N$,
  then boundedness of $\Re V_f$ implies that $f \in \Mult(\mathcal{H})$, and
  \begin{equation*}
    \|f\|_{\Mult(\mathcal{H})} \lesssim (\|\Re V_f\|_\infty + 3)^{N+\frac{1}{2}}.
  \end{equation*}
  This applies in particular to the spaces $\mathcal{D}_a$.
  The improved bound on the multiplier norm of $f$ follows from the scaling properties of both sides
  of the inequality. Indeed, if $t > 0$, then $V_{t f} = t^2 V_f$, so applying the above inequality
  to the function $t f$, we find that
  \begin{equation*}
    \|f\|_{\Mult(\mathcal{D}_a)}^2 \lesssim \frac{1}{t^2} (t^2 \|\Re V_f\|_\infty +3)^{2 N + 1}
  \end{equation*}
  for all $t > 0$.
  If $\|\Re V_f\|_\infty = 0$, then taking $t \to \infty$ above yields $f=0$.
  If $\|\Re V_f\|_\infty \neq 0$, then choosing $t = \| \Re V_f\|_{\infty}^{-1/2}$,
  we obtain the desired estimate.
  (The choice of $t$ could be optimized to improve the implicit constants, but we will not do so here.)
\end{proof}

With the help of Theorem \ref{thm:Sarason_function}, we can establish the
desired multiplier norm estimate of $f_\mu$.
It can be regarded as a quantitative version of the result of Cascante, F\`abrega and Ortega \cite{Cascante2014}
in the Hilbert space setting.

\begin{prop}
  \label{prop:multiplier_norm}
  Let $0 \le a < 1$ and let $\mu \in M^+(\partial \mathbb{B}_d)$. If $f_\mu$ is bounded in $\mathbb{B}_d$,
  then $f_\mu$ is a multiplier of $\mathcal{D}_a$, and
  \begin{equation*}
    \|f_\mu\|_{\Mult(\mathcal{D}_a)} \approx \|f_\mu\|_\infty,
  \end{equation*}
  where the implied constants only depend on $a$ and $d$.
\end{prop}

\begin{proof}
  Since the multiplier norm dominates the supremum norm, we have to show the inequality ``$\lesssim$''.
  Let $f = f_\mu$ and
  \begin{equation}
    \label{eqn:f_r}
    f_r(z) = f(r z) = \int_{\partial \mathbb{B}_d} K_a(r z,w) \, d \mu(w)
    = \int_{\partial \mathbb{B}_d} K_a(z, rw) \, d \mu (w).
  \end{equation}
  We will show that $\|f_r\|_{\Mult} \lesssim \|f_r\|_{\infty}$ for all $0 < r <1$, where the implied constant
  is independent of $f$ and $r$.
  To simplify notation, write $k_w(z) = K_a(z,w)$.
  We will use Theorem \ref{thm:Sarason_function} and instead show that
  \begin{equation*}
    \sup_{z \in \mathbb{B}_d} \Re \langle f_r, k_z f_r \rangle \lesssim \|f_r\|_\infty^2.
  \end{equation*}

  Since the map
  \begin{equation*}
    \partial \mathbb{B}_d \to \mathcal{D}_a, \quad w \mapsto k_{r w},
  \end{equation*}
  is continuous, the integral on the right-hand side of \eqref{eqn:f_r} converges in $\mathcal{H}$.
  Thus, by the reproducing property of the kernel,
  \begin{align*}
    \Re \langle  f_r, k_z f_r \rangle &= \iint \langle k_{r w}, k_z k_{r \zeta} \rangle d \mu(\zeta) d \mu(w) \\
                                      &= \iint \overline{k_{r \zeta}(r w)} \overline{k_z(r w)} \, d \mu(\zeta) d \mu(w).
  \end{align*}
  Equation \eqref{eqn:f_r} shows that
  $\int {k_{r \zeta}(r w)} \, d \mu(\zeta) = f_r(rw)$, hence
  \begin{equation*}
    \Re \langle  f_r, k_z f_r \rangle = \int \overline{f_r(r w)} \overline{ k_z(r w)} \, d \mu(w)
  \end{equation*}
  and so
  \begin{align*}
    \Re \langle f_r, k_z f_r \rangle \le \|f_r\|_\infty \int |k_z(r w)| \, d \mu (w)
    &\lesssim \|f_r\|_\infty \Re \int k_z(r w) \, d \mu(w) \\
    &= \|f_r\|_\infty \Re f_r(z)
    \le \|f_r\|_\infty^2. \qedhere
  \end{align*}
\end{proof}

We are ready to provide the direct proof of Theorem \ref{thm:main_dirichlet}, which we restate
in equivalent form.

\begin{thm}
  Let $0 \le a < 1$ and let $E \subset \partial \mathbb{B}_d$ be compact.
  Then $E$ is $\Mult(\mathcal{D}_a)$-totally null if and only if $\capac{E}{\mathcal{D}_a} = 0$.
\end{thm}

\begin{proof}
    The ``only if'' part was already established in Proposition \ref{prop:only_if}.
  Conversely, suppose that $\capac{E}{\mathcal{D}_a} = 0$. By upper semi-continuity
  of capacity, there exists a decreasing sequence $(E_n)$ of compact neighborhoods of $E$
so that $ \lim_{n \to \infty} \capac{E_n}{\mathcal{D}_a} = 0$; see \cite[Theorem 2.1.6]{EKM+14}.
Let $\mu_n$ be a positive measure supported on $E_n$ as in Lemma \ref{lem:potential_basic}
and let $g^{(n)} = f_{\mu_n}$ be the corresponding holomorphic potential.
We claim that
\begin{enumerate}
  \item $\liminf_{r \nearrow 1} \Re g^{(n)}(r \zeta) \gtrsim 1$ for all $\zeta \in E$ and all $n \in \mathbb{N}$;
  \item the sequence $(g^{(n)})$ converges to $0$ in the weak-$*$ topology of $\Mult(\mathcal{D}_a)$.
\end{enumerate}

Indeed,
Part (1) is immediate from Lemma \ref{lem:potential_basic}.
To see (2), we first observe that Lemma \ref{lem:potential_basic} (c) and Proposition \ref{prop:multiplier_norm}
imply that the sequence $(g^{(n)})$ is bounded in multiplier norm.
Using Lemma \ref{lem:potential_basic} (a), we see that $\|g^{(n)}\|_{\cD_a}^2 \lesssim \capac{E_n}{\cD_a}$,
so $(g^{(n)})$ converges to zero in the norm of $\cD_a$ and in particular pointwise on $\bB_d$, hence (2) holds.

Let now $\nu$ be a positive $\Mult({\mathcal{D}_a})$-Henkin measure that is supported on $E$.
We will finish the proof by showing that $\nu(E) = 0$; see the discussion following Definition \ref{defn:Henkin_TN}.
Item (1) above and Fatou's lemma show that
\begin{equation*}
  \nu(E) = \int_{E} 1 \, d \nu \lesssim \liminf_{r \nearrow 1} \int_{\partial \mathbb{B}_d} \Re g^{(n)}(r \zeta) \, d \nu(\zeta).
\end{equation*}
Since $\nu$ is $\Mult(\mathcal{D}_a)$-Henkin, the associated
integration functional $\rho_\nu$ extends to a weak-$*$ continuous functional on $\Mult(\mathcal{D}_a)$,
which we continue to denote by $\rho_\nu$.
Since $\lim_{r \nearrow 1} g^{(n)}_r = g^{(n)}$ in the weak-$*$ topology of $\Mult(\mathcal{D}_a)$ by Lemma \ref{lem:dilations_convergence},
we find that for all $n \in \mathbb{N}$,
\begin{equation*}
  \lim_{r \nearrow 1} \int_{\partial \mathbb{B}_d} \Re g_n(r \zeta) d \nu(\zeta)
  = \Re \rho_\nu(g_n).
\end{equation*}
Thus,
\begin{equation*}
  \nu(E) \lesssim \Re \rho_\nu(g_n)
\end{equation*}
for all $n \in \mathbb{N}$. Taking the limit $n \to \infty$ and using Item (2),
we see that $\nu(E) = 0$, as desired.
\end{proof}

\section{Proof of Theorem \ref{thm:main_CNP}}
\label{sec:main_CNP}

In this section, we prove a refined version of Theorem \ref{thm:main_CNP}.
Let $\mathcal{H}$ be a regular unitarily invariant space on $\mathbb{B}_d$.
Recall that a compact set $E \subset \mathbb{B}_d$ is said to be an \emph{unboundedness set for $\mathcal{H}$} if
there exists $f \in \mathcal{H}$ with $\lim_{r \nearrow 1} |f(r \zeta)| = \infty$ for all $\zeta \in E$.
We also say that $E$ is a \emph{weak unboundedness for $\mathcal{H}$} if there exists a separable auxiliary Hilbert space $\mathcal{E}$
and $f \in \mathcal{H} \otimes \mathcal{E}$
so that $\lim_{r \nearrow 1} \|f(r \zeta)\| = \infty$ for all $\zeta \in E$.

\begin{thm}
  Let $\mathcal{H}$ be a regular unitarily invariant complete Pick space on $\mathbb{B}_d$.
  The following assertions are equivalent for a compact set $E \subset \mathbb{B}_d$.
  \begin{enumerate}[label=\normalfont{(\roman*)}]
    \item $E$ is $\Mult(\mathcal{H})$-totally null.
    \item $E$ is an unboundedness set for $\mathcal{H}$.
    \item $E$ is a weak unboundedness set for $\mathcal{H}$.
  \end{enumerate}
\end{thm}

\begin{proof}
  (i) $\Rightarrow$ (ii)
  Suppose that $E$ is totally null. In the first step, we will show that for each $M > 1$,
  there exists $f \in \mathcal{H} \cap A(\mathbb B_d)$ satisfying
  \begin{enumerate}
    \item $f \big|_E = M$,
    \item $\|f\|_\mathcal{H} \le 1$, and
    \item  $\Re f \ge 0$ on $\overline{\mathbb{B}_d}$.
  \end{enumerate}
  Let $\varepsilon = 1/M$.
  Since $E$ is totally null, the simultaneous Pick and peak interpolation result \cite[Theorem 1.5]{DH20}
  shows that there exists $\eta \in A(\mathcal{H}) \subset \mathcal H \cap A(\mathbb B_d)$ satisfying
  $\eta \big|_E = (1 - \varepsilon^2)^{1/2}$, $\eta(0) = 0$ and $\|\eta\|_{\Mult(\mathcal{H})} \le 1$.
  It follows that the column multiplier
  \begin{equation*}
    \begin{bmatrix}
      \varepsilon \\ (1 - \varepsilon^2)^{1/2} \eta
    \end{bmatrix}
  \end{equation*}
  has multiplier norm at most one, so the implication (b) $\Rightarrow$ (a) of part (i) of \cite[Theorem 1.1]{AHM+17c}
  implies that the function $f$ defined by
  \begin{equation*}
    f = \frac{\varepsilon}{1 - (1 - \varepsilon^2)^{1/2} \eta}
  \end{equation*}
  belongs to the closed unit ball of $\mathcal{H}$.
  Moreover, since $\|\eta\|_{\Mult(\mathcal{H})} \le 1$, we find that $|\eta(z)| \le 1$
  for all $\zeta \in \overline{\mathbb{B}_d}$,
  from which it follows that $f \in A(\mathbb B_d)$ and $\Re f \ge 0$.
  Clearly, $f \big|_E = \frac{1}{\varepsilon} = M$.
  This observation finishes the construction of $f$.

  The above construction yields, for each $n \ge 1$, a function $f_n \in \mathcal{H} \cap A(\mathbb B_d)$ satisfying
  $f_n \big|_E = 1$, $\|f_n\|_{\mathcal{H}} \le 2^{-n}$ and $\Re f_n \ge 0$.
  Define $f = \sum_{n=1}^\infty f_n \in \mathcal{H}$.
  Let $\zeta \in E$.
  Then for each $N \in \mathbb{N}$, we have that
  \begin{equation*}
    \liminf_{r \nearrow 1} \Re f(r \zeta) \ge \sum_{n=1}^N \Re f_n(\zeta) = N.
  \end{equation*}
  Thus, $\lim_{r \nearrow 1} |f(r \zeta)| = \infty$ for all $\zeta \in E$, so $E$ is an unboundedness set for $\mathcal{H}$.

  (ii) $\Rightarrow$ (iii) is trivial.

  (iii) $\Rightarrow$ (i)
  Suppose that $E$ is a weak unboundedness set for $\mathcal{H}$ and let $f \in \mathcal{H} \otimes \mathcal{E}$ satisfy $\|f\| \le 1$ and $\lim_{r \nearrow 1} \|f(r \zeta)\| = \infty$ for all $\zeta \in E$.
  By the implication (a) $\Rightarrow$ (b) of part (i) of \cite[Theorem 1.1]{AHM+17c}, we may write $f = \frac{\Phi}{1 - \psi}$,
  where $\Phi \in \Mult(\mathcal{H}, \mathcal{H} \otimes \mathcal{E}),\psi \in \Mult(\mathcal{H})$ have multiplier norm at most $1$
  and $|\psi(z)| < 1$ for all $z \in \mathbb{B}_d$.
  In particular, $\|\Phi(z)\| \le 1$ for all $z \in \mathbb{B}_d$, hence $\lim_{r \nearrow 1} \psi(r \zeta) = 1$ for all $\zeta \in E$.

  Let now $\mu$ be a positive $\Mult(\mathcal{H})$-Henkin measure that is supported on $E$ and let $\rho_\mu$
  denote the associated weak-$*$ continuous integration functional on $\Mult(\mathcal{H})$.
  We have to show that $\mu(E) = 0$; see the discussion following Definition \ref{defn:Henkin_TN}.
  To this end,
  we write $\psi_r(z) = \psi(r z)$ and let $n \in \mathbb{N}$.
  Applying the dominated convergence theorem and the fact that
  $\lim_{r \nearrow 1} \psi_r^n = \psi^n$ in the weak-$*$ topology of $\Mult(\mathcal{H})$ (see Lemma \ref{lem:dilations_convergence}), we find that
  \begin{equation*}
    \mu(E) = \lim_{r \nearrow 1} \int_{E} \psi_r^n \, d \mu  = \lim_{r \nearrow 1} \int_{\partial \mathbb{B}_d} \psi_r^n \, d \mu = \rho_\mu(\psi^n).
  \end{equation*}
  Since $\psi$ is a contractive multiplier satisfying $|\psi(z)| < 1$ for all $z \in \mathbb{B}_d$, it follows that $\psi^n$ tends to zero
  in the weak-$*$ topology of $\Mult(\mathcal{H})$. So taking the limit $n \to \infty$ above, we conclude that $\mu(E) = 0$, as desired.
\end{proof}

Let us briefly compare the direct proof of the implication ``capacity $0$ implies totally null''
given in Section \ref{sec:Dirichlet} with the proof via Theorem \ref{thm:main_CNP}.
If $E \subset \partial \mathbb B_d$ is a compact set with $C_{s,2}(E) = 0$,
then the work of Ahern and Cohn \cite{AC89} and of Cohn and Verbitsky \cite{CV95}
shows that $E$ is unboundedness set for $\mathcal H_s$.
To show this, they use holomorphic potentials and their fundamental properties
(cf.\ Lemma \ref{lem:potential_basic}) to construct a function $f \in \mathcal H_s$ satisfying
$\lim_{r \nearrow 1} |f(r \zeta)| = 1$ for all $\zeta \in E$.
Proceeding via Theorem \ref{thm:main_CNP}, one then applies the factorization
result \cite[Theorem 1.1]{AHM+17c} to $f$ to obtain a multiplier $\psi$ of $\mathcal H$ of norm
at most $1$ satisfying $\lim_{r \nearrow 1} \psi(r \zeta) = 1$ for all $\zeta \in E$,
from which the totally null property of $E$ can be deduced.

The direct proof given in Section \ref{sec:Dirichlet} uses holomorphic potentials as well, this time
to construct a sequence of functions in $\mathcal H$, which, roughly speaking,
have large radial limits on $E$ compared to their norm.
It is then shown that the holomorphic potentials themselves form a bounded sequence of multipliers,
from which the totally null property of $E$ can once again be deduced.

\bibliographystyle{amsplain}


\section{Introduction}

In past decades, with the idea of AdS/CFT or the more generic holographic principle~\cite{Maldacena:1997re,Gubser:1998bc,Witten:1998qj}, physicists are trying to build a bridge connecting gravity and other areas of modern theoretical physics, such as condensed matter physics (CMT)~\cite{Hartnoll:2009sz,Herzog:2009xv,McGreevy:2009xe,Horowitz:2010gk,Cai:2015cya}, QCD~\cite{Mateos:2007ay,Gubser:2009md,CasalderreySolana:2011us}, cosmology~\cite{Banks:2004eb}, quantum information theory (QIT)~\cite{Swingle:2009bg,Swingle:2012wq,Qi:2013caa} and etc. It is hoped that this bridge may help us get insights into both the strongly coupled problems in the quantum field theory (QFT) side as well as the origin of spacetime in the gravity side. After decades' efforts, several precise correspondences between the two sides are proposed. Recently, Susskind and his collaborators conjecture that complexity of the boundary QFT may be related to the interior geometry of black hole in the gravity side~\cite{Susskind:2014rva}. In QFT (or QIT), complexity of a target state is an important concept defined as the minimum number of unitary operators (or gates) needed to prepare the state starting from some reference state (for example the vacuum). So far, this conjecture has been refined into two concrete proposals, namely the CV (complexity=volume) and CA (complexity=action) conjectures. In the CV conjecture, complexity of a state living on a time slice $\Sigma$ of the boundary equals to the extremal volume of a codimension-one hypersurface ${\cal B}$ in the bulk ending on $\Sigma$ at the boundary~\cite{Stanford:2014jda}, that is
\begin{eqnarray}\label{CV}
C_V (\Sigma) =\max_{\partial {\cal B} = \Sigma} \left(\frac{{\rm Vol} ({\cal B})}{G_N R}\right),
\end{eqnarray}
where $G_N$ is the gravitational constant and $R$ is some typical length scale of the bulk geometry, for example the AdS radius or the horizon radius. While the CA conjecture states that complexity of a state equals to the on-shell gravitational action evaluated on the so-called Wheeler-DeWitt (WDW) patch of the bulk spacetime~\cite{Brown:2015bva,Brown:2015lvg}. Each conjecture has its own merits and demerits respectively~\cite{Hashimoto:2018bmb}. Inspired by these ideas, an amount of work are raised to study the holographic complexity for various gravity models to check these proposals~\cite{Momeni:2016ekm,Cai:2016xho,Brown:2016wib,Couch:2016exn,Yang:2016awy,Chapman:2016hwi,
	Carmi:2016wjl,Pan:2016ecg,Brown:2017jil,Kim:2017lrw,Cai:2017sjv,Alishahiha:2017hwg,Bakhshaei:2017qud,
	Tao:2017fsy,Guo:2017rul,Zangeneh:2017tub,Alishahiha:2017cuk,Abad:2017cgl,Reynolds:2017lwq,Hashimoto:2017fga,Nagasaki:2017kqe,Miao:2017quj,Ge:2017rak,
	Ghodrati:2017roz,Qaemmaqami:2017lzs,Carmi:2017jqz,Kim:2017qrq,Cottrell:2017ayj,Sebastiani:2017rxr,
	Moosa:2017yvt,HosseiniMansoori:2017tsm,Zhang:2017nth,Reynolds:2017jfs,Chapman:2018dem,Chapman:2018lsv,Khan:2018rzm,Caputa:2018kdj,Feng:2018sqm,Liu:2019smx,Jiang:2019qea,Jiang:2019pgc,Jiang:2018sqj,Jiang:2018gft}.

The above two conjectures are for the whole boundary system which both are then extended to be defined on subsystem respectively in Refs.~\cite{Alishahiha:2015rta} and~\cite{Carmi:2016wjl} later, and they are now called holographic subregion complexity (HSC). The subregion CV proposal is a natural extension of the well-known Hubney-Ryu-Takayanagi (HRT) holographic entanglement entropy (HEE) conjecture~\cite{Ryu:2006bv,Hubeny:2007xt}. Namely, complexity of a subregion ${\cal A}$ of the boundary system equals to the volume of the extremal codimension-one hypersurface $\Gamma_{\cal A}$ enclosed by ${\cal A}$ and the corresponding Hubney-Ryu-Takayanagi (HRT) surface $\gamma_{\cal A}$~\cite{Ryu:2006bv,Hubeny:2007xt}, that is
\begin{eqnarray}\label{subCV}
	{\cal C} (\gamma_{\cal A}) = \frac{{\rm Vol (\Gamma_{{\cal A}})}}{8 \pi G_{d+1} L},
\end{eqnarray}
where $L$ is the AdS radius. Later studies suggest that it should be dual to the fidelity susceptibility in QIT~\cite{Alishahiha:2015rta,MIyaji:2015mia}. While in the subregion CA proposal, complexity of subregion ${\cal A}$ is given by the on-shell gravitational action evaluated on the intersection region between WDW patch and the so-called entanglement wedge~\cite{Czech:2012bh,Headrick:2014cta}. Also, lots of work and effort have been devoted to understand the holographic subregion complexity~\cite{Caputa2017,Caputa2017b,Czech1706,subBenAmi2016,subRoy2017,
	subBanerjee2017,subBakhshaei2017,subSarkar2017,subZangeneh2017,subMomeni2017,subRoy2017b,subCarmi2017,Chen:2018mcc,Ageev:2018nye,Ghodrati:2018hss,Zhang:2018qnt,Alishahiha:2018lfv,Alishahiha:2018tep}.

On the other hand, in the so-called "holographic thermalization" topic, the AdS/CFT duality has been applied successfully to study the physics in non-equilibrium processes, especially the thermalization process of hot QCD matter which is strongly coupled and produced in heavy ion collisions at the Relativistic Heavy
Ion Collider~\cite{Gelis:2011xw,Iancu:2012xa,Muller:2011ra,CasalderreySolana:2011us}.
According to the AdS/CFT dictionary, the thermalization process in the
boundary QFT system is dual to
a black hole formation process in the bulk which can be modelled simply by a Vaidya-like metric. There are already lots of work on this topic and many interesting results are obtained. For more details on this topic, please refer to the review~\cite{Balasubramanian:2011ur} and references therein.

Complexity in the holographic thermalization process is also studied to investigate its time evolution behaviours under thermal quench. In Refs.~\cite{Chapman:2018dem,Chapman:2018lsv}, by applying the CV and CA conjectures, it is found that the late time growth of holographic complexity in the Vaidya spacetime is the same as that found for an eternal black hole. In Ref.~\cite{Chen:2018mcc}, the time evolution of subregion complexity is studied in the process with the subregion CV conjecture. And the results show that the subregion complexity is not always a monotonically increasing function of time. Actually, it increases at early time, but after reaching a maximum it decreases quickly and gets to saturation finally. For other related work, please see Refs.~\cite{Ageev:2018nye,Ageev:2019fxn,Ling:2018xpc,Fan:2018xwf,Jiang:2018tlu}

However, it should be noted that the boundary QFTs considered in the above mentioned work are usually living on the flat Minkowski spacetimes. It would be interesting to generalize the discussions to more realistic situations where QFTs lives on curved spacetimes, which may hep us to understand the extremely hot and condensed physics such as in the very early universe. Several holographic models of the quantum field theory in curved spacetimes (QFTCS) have already been proposed in de
Sitter (dS) spacetime and other cosmological
backgrounds (please refer to the review \cite{Marolf:2013ioa} for details). Here we would like to mention the work done in Ref.~\cite{Marolf:2010tg}, where an interesting
holographic model was built to
relate the QFTs living on the dS
boundary to the bulk Einstein gravity. Employing
this model, the
thermalization process of QFTs in dS
spacetime is studied holographically in Ref.~\cite{Fischler:2013fba}. By applying the holographic entanglement
entropy as a probe, the whole
thermalization is found to be similar to
the flat boundary
case~\cite{Liu:2013iza,Liu:2013qca} and can be divided into a sequence of
processes. Moreover, the saturation time is found to depend almost only on the
entanglement sphere radius. When the radius is
small, the saturation time is almost a linear increasing function of the radius, as expected to coincide with the
result of the flat boundary case at this
time~\cite{Balasubramanian:2011ur}.
However, when the radius becomes larger and larger to approach the cosmological horizon, the
saturation time blows up logarithmically. Later, the study is extended to include the effect of higher-derivative terms, such as the Gauss-Bonnet
correction~\cite{Zhang:2014cga}. And it is found that increasing the Gauss-Bonnet coupling will shorten the saturation time. Please also refer to Refs. \cite{Fischler:2014ama,Fischler:2014tka,Nguyen:2017ggc} for other related work on AdS/CFT with dS boundary.

As the deep connection between holographic entanglement entropy (HEE) and holographic subregion CV (HSCV), it would be interesting to study the time evolution of subregion complexity in the thermalization process of the QFTs living on dS spacetime within the above model. It is natural to ask the following questions: How the existence of the cosmological horizon affects the behaviour of HSCV? Whether the time evolution behaviours of HSCV can be used to describe the the whole thermalization process? Is there any difference between behaviours of HSCV and HEE? The main goal of this work is trying to address these questions.

The work is organized as follows. In the next section, we will give a brief review of the holographic model of QFTs in dS spacetime proposed in Ref.~\cite{Marolf:2010tg}, including the Vaidya-like solution. Then in Sec. III,
we study in detail the time evolution of HSCV in the thermalization process. Due to the complication of the equations needed to solve, we rely mainly on numerical calculations. The final section is devoted to discussions and summary.

\section{Gravity solutions with dS boundary}

In this section, following Refs.~\cite{Fischler:2013fba,Zhang:2014cga}, we will briefly review the bulk solutions in Einstein gravity with a foliation such that the boundary metric corresponds to a de Sitter spacetime. Three relevant solutions will be presented, including a vacuum AdS, a static AdS black hole and its Vaidya-like cousin.

\subsection{Action}

We consider $(d+1)$-dimensional Einstein-Hilbert action as follows
\begin{eqnarray}\label{Action}
S = \frac{1}{16 \pi G_N} \int d^{d+1} x \sqrt{-g} \left(R - 2\Lambda\right),
\end{eqnarray}
where $G_N$ is the Newton constant and $\Lambda$ negative cosmological constant. The action gives the following equations of motion
\begin{eqnarray}\label{EoMs}
R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R + \Lambda g_{\mu\nu}=0.
\end{eqnarray}

For asymptotically AdS spacetime, the metric can be written in the Fefferman-Graham form~\cite{Fefferman:1985}
\begin{eqnarray}\label{FGForm}
ds^2 = \frac{L^2}{z^2} \left(g_{\mu\nu} (z,x)dx^\mu dx^\nu + dz^2\right),
\end{eqnarray}
where $L$ is the AdS raduis related to the cosmological constant as $\Lambda = -\frac{d(d-1)}{2 L^2}$. The dual quantum field theory lives at the conformal boundary  $z=0$ with a metric $ds^2_\Sigma = g_{\mu\nu} (0,x)dx^\mu dx^\nu$. In this paper, we are interested in cases where the boundary metric $ds^2_\Sigma$ corresponds to a dS spacetime in certain coordinates.

\subsection{AdS vacuum solution}

The equations of motion (\ref{EoMs}) admit an AdS vacuum solution as
\begin{eqnarray}\label{Vacuum}
&&ds^2 = \frac{L^2}{z^2} \left(-f(r) g(z) dt^2 + f^{-1}(r) g(z) dr^2 + r^2 g(z) d\Omega_{d-2}^2 + dz^2\right),\nonumber\\
&&f(r) =1- H^2 r^2 ,\qquad g(z) = \left(1-
\frac{H^2 z^2}{4}\right)^2.
\end{eqnarray}
The conformal boundary locates at $z=0$ with conformally reduced metric
\begin{eqnarray}\label{dS}
ds^2_\Sigma = -f(r) dt^2 + f^{-1} (r) dr^2 + r^2 d\Omega_{d-2}^2,
\end{eqnarray}
which is just the dS spacetime in the static patch with a cosmological horizon at $r=1/H$, where $H$ denotes the Hubble constant.

The AdS vacuum solution is dual to the vacuum state of the dual QFT  with the latter can be taken as the well-known Bunch-Davis or Euclidean vacuum. For a geodesic observer sitting at $r=0$, the Bunch-Davis vacuum appears to have temperature $T_{dS} = H/2\pi$ natural for the existence of the cosmological horizon.

\subsection{AdS black hole solution}

The equations of motion (\ref{EoMs}) also admit an AdS black hole solution with the dS boundary (\ref{dS})
\begin{eqnarray}\label{BH}
&&ds^2 = \frac{L^2}{z^2} \left(-h(z) dt^2 + \frac{dz^2}{h(z)}+ \frac{H^2 L^2}{f(r)^2}dr^2 + \frac{H^2 L^2}{f(r)} r^2 d\Omega_{d-2}^2\right),\nonumber\\
&&h(z) = 1 - \frac{z^2}{L^2} - \frac{m z^d}{L^{2(d-1)}}.
\end{eqnarray}
The event horizon $z_+$ is given by the largest positive root of $h(z)$. The mass parameter $m$ can be written in terms of the horizon as
\begin{eqnarray}
m = \frac{L^{2(d-1)}}{z_+^d} \left(1-\frac{z_+^2}{L^2}\right).
\end{eqnarray}
The Hawking temperature of the black hole is
\begin{eqnarray}\label{Temperature}
T_H = \frac{L^2 d -(d-2) z_+^2}{4\pi L^2 z_+}.
\end{eqnarray}
It should be noted that the zero temperature limit of the black hole solution (\ref{BH}) is the not the solution with $m=0$ which is isometric to the AdS vacuum solution (\ref{Vacuum}). Actually, the zero temperature limit of the solution has the smallest horizon radius and most "negative" mass as
\begin{eqnarray}
z_+^{\textrm ext} = L \sqrt{\frac{d}{d-2}},\qquad m^{\textrm ext} = - \frac{2 L^{2(d-1)}}{(d-2) (z_+^{\textrm ext})^d}
\end{eqnarray}
This means that when the mass is negative in the range $0> m > m^{\textrm ext}$, the black hole still has a regular horizon and reasonable thermodynamics. This is a typical behavior of topological black holes.

Holographically, the black hole solution is dual to the QFT on the static patch of dS spacetime at the temperature given by Eq.~(\ref{Temperature}). Note that this temperature does not have to be the same as the dS temperature $T_{dS}$. For more discussions on this point, please refer to Ref.~\cite{Fischler:2013fba}.

\subsection{Vaidya-like solution}

Our aim is to study the holographic thermalization process of the dual QFT under quench. This process can be simply described holographically by a Vaidya-like geometry in the bulk.

Going to the Eddington-Finskelstein coordinates and introducing a time-dependent mass parameter, from the black hole solution one can obtain its Vaidya-like cousin as
\begin{eqnarray}\label{Vaidya}
&&ds^2 =\frac{L^2}{z^2} \left(-h(v,z) dv^2 -2 dv dz+ \frac{H^2 L^2}{f(r)^2}dr^2 + \frac{H^2 L^2}{f(r)} r^2 d\Omega_{d-2}^2\right),\nonumber\\
&&h(v,z) = 1 - \frac{z^2}{L^2} - \frac{m(v) z^d}{L^{2(d-1)}}.
\end{eqnarray}
External source should be introduced to maintain the equations of motion
\begin{eqnarray}
R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R + \Lambda g_{\mu\nu} = 8\pi G_N T_{\mu\nu}^{\textrm ext},\nonumber\\
8\pi G_N T_{\mu\nu}^{\textrm ext} = \frac{(d-1)z^{d-1}}{2 L^{2(d-1)}} \frac{d m(v)}{dv} \delta_{\mu v} \delta_{\nu v},
\end{eqnarray}
which implies that the infalling shell is made of null dust. We take the form of the mass function as
\begin{eqnarray}
m(v) = \frac{M}{2} \left[1 + \tanh\left(\frac{v}{v_0}\right)\right],
\end{eqnarray}
where $M>0$ is the total mass of the dust shell and $v_0$ of its thickness. Then the solution describes the collapsing of the null dust shell from the boundary into the bulk to form a black hole. At the QFT side, it corresponds to a sudden global injection of energy into the system and then let it evolve from the Bunch-Davis vacuum  to a thermal state with $T>T_{dS}$.

\section{Holographic entanglement entropy and subregion complexity}

In this section, by applying the holographic subregion CV (HSCV) (\ref{subCV}), we will study the time evolution of holographic subregion complexity in the thermalization process which is described by the Vaidya-like geometry holographically.

On the boundary at time $\tilde{t}$, taking into account the symmetry of the Vaidya-like metric (\ref{Vaidya}), it is convenient to choose the subregion ${\cal A}$ to be a $(d-1)$-dimensional sphere centred at $\tilde{r} =0$ ($\tilde{r} \equiv H r$) with raduis $\tilde{R}$. According to the conjecture (\ref{subCV}), the holographic subregion complexity of ${\cal A}$ is given by the extreme volume of the codimension-one hypersurface $\Gamma_{\cal A}$ enclosed by ${\cal A}$ and its corresponding HRT surface $\gamma_{\cal A}$. So, to study the holographic subregion complexity, we should first find the HRT surface $\gamma_{\cal A}$ whose area gives the holographic entanglement entropy.

\subsection{Holographic entanglement entropy}

Considering the symmetry, the HRT surface $\gamma_{\cal A}$ in the bulk can be parameterized by functions $z(\tilde{r})$ and $v(\tilde{r})$, with the boundary conditions
\begin{eqnarray}
z(\tilde{R}) = \epsilon, v(\tilde{R}) = \tilde{t},
\end{eqnarray}
where $\epsilon$ is an UV cutoff constant. At the tip of the HRT surface, taking into account the symmetry, we have
\begin{eqnarray}
z'(0) = v'(0)=0,z(0) = z_\ast, v(0)=v_\ast,
\end{eqnarray}
where $(z_\ast, v_\ast)$ are two parameters labelling the location of the tip and the prime denotes derivative with respect to $\tilde{r}$. The induced metric on $\gamma_{\cal A}$ is
\begin{eqnarray}
ds^2 = \frac{L^2}{z^2} \left(\frac{L^2}{(1-\tilde{r}^2)^2} - h(v,z) v'^2 - 2 v' z'\right) d\tilde{r}^2 + \frac{L^4}{z^2} \frac{\tilde{r}^2}{1-\tilde{r}^2} d\Omega_{d-2}^2.
\end{eqnarray}
The holographic entanglement entropy functional is given by the area of the HRT surface
\begin{eqnarray}\label{HEE}
&&{\cal S} = \frac{L^{2d-3}}{4 G_N} \Omega_{d-2} \int_0^{\tilde{R}} \frac{d\tilde{r}}{z^{d-1}} Q P^{d-2},\\
&&Q \equiv \sqrt{\frac{L^2}{(1-\tilde{r}^2)^2} - h(v, z) v'^2 - 2 v' z'},\qquad P \equiv \frac{\tilde{r}}{\sqrt{1-\tilde{r}^2}}.\nonumber
\end{eqnarray}
To find the extreme value of this functional, we need to solve the two equations of motion, which can be obtained by varying the functional and are rather complicated
\begin{eqnarray}
&&h^2 \left(\tilde{r}^2-1\right)^2 v'^3 \left((d-1) \left(\tilde{r}^2-1\right) \tilde{r} z'+(d-2) z\right)\nonumber\\
&&+v' \left(\left(\tilde{r}^2-1\right) z' \left(\left(\tilde{r}^2-1\right) \left(2 z' \left((d-1) \left(\tilde{r}^2-1\right) \tilde{r} z'+(d-2) z\right)-\tilde{r} \left(\tilde{r}^2-1\right) z z''\right)-(d-1) h L^2 \tilde{r}\right)\right.\nonumber\\
&&\qquad\left.+h L^2 z \left(-d+2 \tilde{r}^2+2\right)\right)\nonumber\\
&&+3 h \left(\tilde{r}^2-1\right)^2 v'^2 z' \left((d-1) \left(\tilde{r}^2-1\right) \tilde{r} z'+(d-2) z\right)-(d-1) L^2 \tilde{r} \left(\tilde{r}^2-1\right) z'^2\nonumber\\
&&+L^2 z \left(-d+2 \tilde{r}^2+2\right) z'+v'' \left(h L^2 \tilde{r} \left(\tilde{r}^2-1\right) z+\tilde{r} \left(\tilde{r}^2-1\right)^3 z z'^2\right)+L^2 \left(\tilde{r}^2-1\right) \tilde{r} z z''=0,\nonumber\\
\end{eqnarray}
\begin{eqnarray}
&&(d-1) h^2 \tilde{r} \left(\tilde{r}^2-1\right)^4 v'^4-2 (d-1) h L^2 \tilde{r} \left(\tilde{r}^2-1\right)^2 v'^2\nonumber\\
&&+z' \left(3 (d-1) h \tilde{r} \left(\tilde{r}^2-1\right)^4 \left(v'\right)^3-3 (d-1) L^2 \tilde{r} \left(\tilde{r}^2-1\right)^2 v'\right.\nonumber\\
&&\qquad\left.+z \left(\tilde{r} \left(\tilde{r}^2-1\right)^4 v' v''-2 (d-2) \left(\tilde{r}^2-1\right)^3 v'^2\right)\right)\nonumber\\
&&+z \left((2-d) h \left(\tilde{r}^2-1\right)^3 v'^3+L^2 \left(\tilde{r}^2-1\right) \left(d-2 \tilde{r}^2-2\right) v'-L^2 \tilde{r} \left(\tilde{r}^2-1\right)^2 v''\right)\nonumber\\
&&+(d-1) L^4 \tilde{r}+2 (d-1) \tilde{r} \left(\tilde{r}^2-1\right)^4 v'^2 z'^2-\tilde{r} \left(\tilde{r}^2-1\right)^4 z v'^2 z''=0.\nonumber\\
\end{eqnarray}
To avoid symbol confusion, we denote the solution of the above two equations as $(v_0(\tilde{r}), z_0(\tilde{r}))$ which parameterize the HRT surface. The relation between $v$ and $z$ on the HRT surface, denoted as $v_0(z)$, can be obtained by eliminating the parameter $\tilde{r}$ from the two functions.

Generally, the HEE (\ref{HEE}) is ultra-divergent. To remove the divergence and for convenience, we define a normalised HEE as
\begin{eqnarray}\label{RHEE}
\hat{\cal S}  \equiv \frac{4 G_N ({\cal S}_{Vaidya} - {\cal S}_{AdS})}{V_{\cal A}},
\end{eqnarray}
where ${\cal S}_{AdS}$ is the HEE for the same subregion ${\cal A}$ in pure AdS geometry. And $V_{\cal A} \equiv L^{d-1} \Omega_{d-2} \int_0^{\tilde{R}} \frac{\tilde{r}^{d-2}}{(1-\tilde{r}^2)^{d/2}} d\tilde{r} = L^{d-1} \Omega_{d-2} \frac{\tilde{R}^{d-1}}{d-1} \ _2 F_1\left(\frac{d-1}{2},\frac{d}{2},\frac{d+1}{2}, \tilde{R}^2\right)$ is the volume of the subregion ${\cal A}$ \footnote{Actually, it should be noted that $V_{\cal A}$ is divergent as $\tilde{R}$ approaches the cosmological horizon to cover the whole boundary space.}. So, $\hat{\cal S}$ can be seen as a normalised entanglement entropy density.

\subsection{Holographic subregion complexity}

Due to the spherical symmetry, the co-dimension one extreme hypersurface $\Gamma_{\cal A}$, enclosed by ${\cal A}$ and the HRT surface $\gamma_{\cal A}$, can be parameterized by function $v = v(z, \tilde{r})$. The induced metric on $\Gamma_{\cal A}$ is
\begin{eqnarray}
ds^2 = &&\frac{L^2}{z^2} \left[-\left(h \frac{\partial v}{\partial z} + 2\right) \frac{\partial v}{\partial z} d z^2 - 2 \left(h \frac{\partial v}{\partial z} + 1\right) \frac{\partial v}{\partial \tilde{r}} dz d\tilde{r} \right. \nonumber\\
&&+ \left.\left(\frac{L^2}{(1-\tilde{r}^2)^2} - h \left(\frac{\partial v}{\partial \tilde{r}}\right)^2 \right) d\tilde{r}^2 + \frac{L^2 \tilde{r}^2}{1-\tilde{r}^2} d\Omega_{d-2}^2\right].
\end{eqnarray}
According to the HSCV proposal (\ref{subCV}), the holographic subregion complexity functional of $\Gamma_{\cal A}$ is
\begin{eqnarray} \label{HSC}
&&{\cal C} = \frac{L^{2d-2} \Omega_{d-2}}{8\pi G_N L} \int_0^{z_\ast} dz \int_0^{\tilde{r}(z)} d\tilde{r} N P^{d-2}{z^{-d}} ,\\
&&N \equiv \sqrt{-\left[\frac{L^2}{(1-\tilde{r}^2)^2} - h v_{\tilde{r}}^2\right](h v_z + 2) v_z - (h v_z + 1)^2 v_{\tilde{r}}^2},\qquad P \equiv \frac{\tilde{r}}{\sqrt{1-\tilde{r}^2}},\nonumber
\end{eqnarray}
where $v_z \equiv \frac{\partial v}{\partial z}$ and $v_{\tilde{r}} \equiv \frac{\partial v}{\partial \tilde{r}}$. To extremizing the HSCV functional, we need to solve the equation of motion which can be obtained by varying the functional with respect to $v(z, \tilde{r})$
\begin{eqnarray}
&&L^2 \tilde{r} v_z \left(L^2 \left(v_z \left(2 d h \left(h v_z+3\right)-z v_z \left(h_v+h h_z\right)-3 z h_z\right)+4 d\right)-2 \left(\tilde{r}^2-1\right)^2 z v_{rr} \left(h v_z+2\right)\right)\nonumber\\
&&+2 L^2 \left(\tilde{r}^2-1\right) z v_r \left(\left(d-2 \left(\tilde{r}^2+1\right)\right) v_z \left(h v_z+2\right)+2 \left(\tilde{r}^2-1\right) \tilde{r} v_{zr} \left(h v_z+1\right)\right)\nonumber\\
&&+2 L^2 \left(\tilde{r}^2-1\right)^2 \tilde{r} v_r^2 \left(v_z \left(d h-z h_z\right)+d-h z v_{zz}\right)+2 (d-2) \left(\tilde{r}^2-1\right)^3 z v_r^3+2 L^4 \tilde{r} z v_{zz}=0.\nonumber\\
\end{eqnarray}
At first glance, it seems difficult to solve the above equation. However, it is interesting to note that $v(z,\tilde{r})=v_0(z)$ is just the solution (Here we would like to emphasize again that $v_0(z)$ is just the function giving the relation between $v$ and $z$ on the HRT surface), which can be checked directly by plugging $v_0(z)$ into the equation. This simply means that $\Gamma_{\cal A}$ is just formed by dragging the HRT surface $\gamma_{\cal A}$ along the $\tilde{r}$ direction. Similar feature has already been observed in flat boundary case with strip subregion in Ref.~\cite{Chen:2018mcc}.

As HEE, HSC is also ultra-divergent, so we can also define a normalised HSC density as
\begin{eqnarray}
\hat{\cal C} \equiv \frac{8\pi G L ({\cal C}_{Vaidya} - {\cal C}_{AdS}) }{ V_{\cal A}},
\end{eqnarray}
where ${\cal C}_{AdS}$ is the HSCV for ${\cal A}$ in pure AdS geometry.

\subsection{Numerical results}

Having set up the general frame work of HEE and HSCV, now we are ready to study the time evolution of HSC in holographic thermalization. Due to the complication of the equations needed to solve, we rely on numerical method. And for convenience, we set the AdS radius $L=1$.

\subsubsection{General behaviours}

In Fig.~1, we plot the time evolution of normalised HSC density $\hat{\cal C}$ for various $\tilde{R}$ with fixed spacetime dimension. From the figure, one can see that the time evolution of $\hat{\cal C}$ is not a monotonically increasing function of the time. Rather, it can be divided into four stages: After quench, firstly it grows quickly and almost linearly, then the growth slows down; After reaching a maximal value $\hat{\cal C}_{max}$ it starts to drop down fast,and shortly after the drop down stops and it saturates to a constant value $\hat{\cal C}_{sat}$ finally. Moreover, it is interesting to note that the final saturation constant may be negative, which means that the final value of the complexity may be smaller than its initial value. These behaviours are very different from that in CV or CA conjectures, where the complexity is always a monotonically increasing function of time \cite{Chapman:2018dem,Chapman:2018lsv}. Similar behaviours have been observed in flat boundary cases with strip subregion \cite{Chen:2018mcc,Ling:2018xpc}, indicating universality of the behaviours.

\begin{figure}[!htbp]
	\centering
	\includegraphics[width=0.32\textwidth]{CAdS3.pdf}
	\includegraphics[width=0.33\textwidth]{CAdS4.pdf}
	\includegraphics[width=0.32\textwidth]{CAdS5.pdf}
\caption{(colour online) Time evolution of normalised HSCV density $\hat{\cal C}$ for various $\tilde{R}$ with fixed $d$. The mass is fixed as $M=1$.}
\end{figure}

\begin{figure}[!htbp]
	\centering
	\includegraphics[width=0.45\textwidth]{CAdS3R05M.pdf}
	\includegraphics[width=0.45\textwidth]{CAdS5R05M.pdf}
	\caption{(colour online) Time evolution of normalised HSCV density $\hat{\cal C}$ for various $M$ with fixed $d$. The subregion size is fixed as $\tilde{R}=0.5$.}
\end{figure}

\begin{figure}[!htbp]
	\centering
	\includegraphics[width=0.45\textwidth]{CR05.pdf}
	\includegraphics[width=0.45\textwidth]{CR09.pdf}
\caption{(colour online) Time evolution of normalised HSCV density $\hat{\cal C}$ for various $d$ with fixed $\tilde{R}$. The mass is fixed as $M=1$. }
\end{figure}

From Fig.~1-3, one can see that the maximal value $\hat{\cal C}_{max}$ depends on the subregion size $\tilde{R}$, the spacetime dimension $(d+1)$ and the mass parameter $M$. Increasing $\tilde{R}$ or $M$ will yield a bigger $\hat{\cal C}_{max}$, while increasing the dimension will, on the contrary, lower the maximal value.

Moreover, one can also see that the final saturation constant $\hat{\cal C}_{sat}$ also depends on $(\tilde{R}, d, M)$ but in a more complicated way.

\subsubsection{Linear growth stage}

Let us focus on discussing the first stage when $\hat{\cal C}$ grows almost linearly in time, i.e.,
\begin{eqnarray}
\frac{d\hat{\cal C}}{d \tilde{t}} \sim A,
\end{eqnarray}
where $A$ is the proportional constant which may depend on $(\tilde{R}, d, M)$. From Fig.~1-3, one can see that $A$ is nearly independent of $\tilde{R}$ and $d$; While it strongly depends on $M$. By fitting the numerical data, it is found that $A \approx 0.4 M$.

From Fig.~1, one can also see that larger the subregion size $\tilde{R}$ is, longer time the linear growth stage lasts. It is expected that as $\tilde{R}$ approaches the cosmological horizon $\tilde{r} =1$ to cover the entire boundary space, the linear growth stage will last forever which agrees well with the CV conjecture. We can see this point more clearly in Fig.~4 where we take $d=2$ case as an example. We will give more evidences on this point later.

\begin{figure}[!htbp]
	\centering
	\includegraphics[width=0.5\textwidth]{CAdS3R.pdf}
	\caption{(colour online) Time evolution of normalised HSCV density $\hat{\cal C}$ for various $\tilde{R}$ with fixed $d$. The mass is fixed as $M=1$.}
\end{figure}

\subsubsection{Saturation time}

In Refs.~\cite{Fischler:2013fba,Zangeneh:2017tub}, one defines the saturation time as the time HEE approaches a constant. Similarly, for the complexity, we can also define a saturation time $\tilde{t}_{sat}$ as the time $\hat{\cal C}$ reaches its saturation constant $\hat{\cal C}_{sat}$.

In Fig.~5, we plot the time evolution of the two observables, $\hat{\cal S}$ and $\hat{\cal C}$ to make a comparison. From the figure, we can see the well-known fact that $\hat{\cal S}$ is always a monotonically increasing function of time. Moreover, from Fig.~5 and Fig.~2, one can see that $\hat{\cal S}$ and $\hat{\cal C}$ reache their saturation values at almost the same time. And their saturation time $\tilde{t}_{sat}$ is nearly independent of $d$ and $M$.

\begin{figure}[!htbp]
	\centering
	\includegraphics[width=0.45\textwidth]{EECR05.pdf}
	\includegraphics[width=0.45\textwidth]{EECR09.pdf}
	\caption{(colour online) Time evolution of HEE and $\hat{\cal C}$ for fixed $d$. The mass parameter is fixed as $M=1$.}
\end{figure}

In Fig.~6, the saturation time $\tilde{t}_{sat}$ for $\hat{\cal C}$  as a function of the subregion size $\tilde{R}$ is plotted. The numerical results can be well fitted by the function $\tilde{t}_{sat} = \tanh^{-1} (\tilde{R})$, as for the HEE \cite{Fischler:2013fba}. It is interesting to note that the $\tilde{t}_{sat}$ is just the time light takes travelling from the origin $\tilde{r}=0$ to the boundary of the subregion $\tilde{r} = \tilde{R}$.\footnote{We thank J.F. Pedraza for pointing out this.} From the figure and the fitting, one can easily see that $\tilde{t}_{sat}$ is linear in $\tilde{R}$ when $\tilde{R}$ is small; However, as $\tilde{R}$ approaches the cosmological horizon $\tilde{r} =1$, the saturation time diverges logarithmically and thus the linear growth stage will also last forever, as we already mentioned above.

\begin{figure}[!htbp]
	\centering
	\includegraphics[width=0.5\textwidth]{Ctsat.pdf}
	\caption{(colour online) Saturation time $\tilde{t}_{sat}$ as a function of the subregion size $\tilde{R}$. Red dots correspond to numerical results while the solid blue curve represents the fitting function $\tilde{t}_{sat} = \tanh^{-1} (\tilde{R})$. Space dimension and mass parameter are fixed as $d=2$ and $M=1$, respectively.}
\end{figure}

\section{Summary and Discussions}

In this work, we consider the holographic model of thermalization process for QFTs in dS spacetime. By applying the holographic subregion CV conjecture, we study the time evolution of subregion complexity under quench. The subregion ${\cal A}$ is chosen to be a sphere on the boundary time slice. The dual extremal codimension-one hypersurface $\Gamma_{{\cal A}}$ in the bulk, whose volume gives the complexity of ${\cal A}$, is found to be simply swept out by the HRT surface along the $\tilde{r}$-direction. The whole time evolution of subregion complexity can be divided into four stages: It first increases almost linearly; Then its growth slows down and after reaching a maximum it starts to drop down quickly, and shortly after the drop down stops and it gets to saturation finally. This picture is similar to that in flat boundary cases but with a strip subregion~\cite{Chapman:2018lsv,Ling:2018xpc}. This implies that the time evolution behaviours of subregion complexity are very general, and is independent of the subregion shape and the cosmological horizon.

The linear growth rate in the first stage is found to almost only depend on the mass parameter. As the subregion size approaches the cosmological horizon, this stage is expected to last forever, and as the HEE the saturation time is logarithmically divergent. The saturation time is found to depend almost only on the subregion size $\tilde{R}$, and their relation can be well fitted by the function $\tilde{t}_{sat} = \tanh^{-1} (\tilde{R})$. It is interesting to note that the $\tilde{t}_{sat}$ is just the time light takes travelling from the origin $\tilde{r}=0$ to the boundary of the subregion $\tilde{r} = \tilde{R}$. The underlying physical meaning of this fact needs further investigation.

In this work, we only consider the HSCV conjecture. It is interesting to check whether general behaviours of subregion complexity still holds for other conjectures, for example the holographic subregion CA. In Ref.~\cite{Zhang:2014cga}, using HEE as a probe we show that including the Gauss-Bonnet correction will shorten the saturation time. It is also interesting to see how the higher-derivative terms affect the time evolution of subregion complexity. We leave these questions for further investigations.

\section*{Acknowledgement}

This work was supported by National Natural Science Foundation of China (Nos. 11605155 and 11675144). We thank J.F. Pedraza for helpful comments on this manuscript.