[Main result]\label{thm:intro-main}
Let \(0<r<R\). Then
\[
\|C_{A(r,R)}\|_{L^2\to L^2}=\frac{2}{\sqrt{\mu_1^{ND}(r,R)}},
\]
where \(\mu_1^{ND}(r,R)\) is the first eigenvalue of
\[
-\Delta U=\mu U \ \text{in }A(r,R),\qquad
\partial_\nu U=0 \ \text{on }|z|=r,\qquad
U=0 \ \text{on }|z|=R.
\]
Equivalently, \(\mu_1^{ND}(r,R)=\kappa_1^2\), where \(\kappa_1\) is the smallest
positive root of a Bessel determinant (see Theorem~\ref{thm:norm-annulus}).

The Cauchy transform on a planar domain $\Omega\subset\mathbb{C}$,
\[
(C_\Omega f)(z)=\frac1\pi\int_\Omega \frac{f(w)}{z-w}\,dA(w),
\]
is a fundamental operator in complex analysis and elliptic PDE. It gives a
canonical solution operator for the $\bar\partial$-equation in the sense that
\[
\bar\partial\,(C_\Omega f)=f\qquad\text{in }\Omega
\]
(in the distributional sense, and pointwise whenever $f$ is sufficiently
regular). Consequently, sharp bounds for the operator norm
$\|C_\Omega\|_{L^2(\Omega)\to L^2(\Omega)}$ are closely connected to
quantitative $\bar\partial$ estimates and to spectral data of associated
elliptic boundary value problems.

On smoothly bounded connected domains one has the spectral lower bound
\[
\|C_\Omega\|_{L^2(\Omega)\to L^2(\Omega)}\ge \frac{2}{\sqrt{\lambda_1(\Omega)}},
\]
where $\lambda_1(\Omega)$ denotes the first Dirichlet eigenvalue of $-\Delta$
on $\Omega$; see \cite{AndersonKhavinsonLomonosov1992}. More precisely, the
estimate can be expressed in the equivalent form
\begin{proposition}[Norm estimate for the Cauchy transform]
Let $\Omega\subset\mathbb{R}^2$ be a smoothly bounded domain. Then
\[
\left\|\frac{C_\Omega^{*}C_\Omega}{4}\right\|
=\frac14\,\|C_\Omega\|^2
\ge \frac{1}{\lambda_1(\Omega)},
\]
where $\lambda_1(\Omega)$ is the smallest positive eigenvalue for the boundary
value problem \emph{(1.2)} in $\Omega$.
\end{proposition}

For the unit disk $D=\{z\in\mathbb{C}:\ |z|<1\}$, Anderson and Hinkkanen computed the
exact $L^2$-operator norm. If $j_{0,1}$ denotes the smallest positive zero of
the Bessel function $J_0$, then
\[
\|C_D\|_{L^2(D)\to L^2(D)}=\frac{2}{j_{0,1}},
\qquad\text{equivalently}\qquad
\frac14\,\|C_D\|^2=\frac{1}{j_{0,1}^2}.
\]
Since $\lambda_1(D)=j_{0,1}^2$, this shows that equality holds in the disk case:
\[
\frac14\,\|C_D\|^2=\frac{1}{\lambda_1(D)}.
\]
We refer to \cite{AndersonHinkkanen1989} for the sharp constant and explicit
extremizers. In contrast, for multiply connected domains additional holomorphic
components may influence sharpness.

In this paper we compute the exact $L^2$-operator norm of $C_\Omega$ when
$\Omega$ is the circular annulus
\[
A(r,R)=\{z\in\mathbb{C}:\ r<|z|<R\}.
\]

Our main result identifies the norm with the first mixed eigenvalue of the
Laplacian (Neumann on \(|z|=r\), Dirichlet on \(|z|=R\)).

\begin{theorem}[Sharp \(L^2\)-norm on a circular annulus]\label{thm:norm-annulus}
Let \(0<r<R\), \(A=A(r,R)=\{z\in\C:\ r<|z|<R\}\), and
\[
(C_A f)(z)=\frac1\pi\int_A \frac{f(w)}{z-w}\,dA(w),\qquad f\in L^2(A).
\]
Then
\[
\|C_A\|_{L^2(A)\to L^2(A)}=\frac{2}{\kappa_1},
\]
where \(\kappa_1>0\) is the smallest positive solution of
\begin{equation}\label{eq:kappa1}
J_1(\kappa r)\,Y_0(\kappa R)-Y_1(\kappa r)\,J_0(\kappa R)=0 .
\end{equation}
\end{theorem}

On smoothly bounded connected domains one has the spectral lower bound
\[
\|C_\Omega\|_{L^2(\Omega)\to L^2(\Omega)}\ge \frac{2}{\sqrt{\lambda_1(\Omega)}},
\]
where $\lambda_1(\Omega)$ denotes the first Dirichlet eigenvalue of $-\Delta$
on $\Omega$; see \cite{AndersonKhavinsonLomonosov1992}. More precisely, the
estimate can be expressed in the equivalent form
\begin{proposition}[Norm estimate for the Cauchy transform]
Let $\Omega\subset\R^2$ be a smoothly bounded domain. Then
\[
\left\|\frac{C_\Omega^{*}C_\Omega}{4}\right\|
=\frac14\,\|C_\Omega\|^2
\ge \frac{1}{\lambda_1(\Omega)},
\]
where $\lambda_1(\Omega)$ is the smallest positive eigenvalue for the boundary
value problem \emph{(1.2)} in $\Omega$.
\end{proposition}

Our main result identifies the norm with the first mixed eigenvalue of the
Laplacian (Neumann on \(|z|=r\), Dirichlet on \(|z|=R\)).

The proof exploits rotational symmetry: a Fourier decomposition reduces
\(C_{A(r,R)}\) to a family of one-dimensional weighted Hardy-type operators on
radial profiles. A variational argument then yields a Sturm--Liouville problem
whose first eigenvalue controls the sharp operator norm; the Bessel determinant
arises from solving the radial eigenvalue equation explicitly.
The extremizers are given in terms of the corresponding mixed eigenfunction,
revealing a structural distinction between simply and doubly connected domains.
\subsection{Background and related work}
Norm bounds and spectral properties of the Cauchy transform and related
operators have been studied from several perspectives.
Mapping properties on bounded domains were initiated in
\cite{AndersonHinkkanen1989}, while a spectral approach to potential-theoretic
integral operators was developed in \cite{AndersonKhavinsonLomonosov1992}.
Sharp \(L^2\) estimates and their relation to spectral data appear in
\cite{ArazyKhavinson1992} and subsequent work of Dostani\'c
\cite{Dostanic1996,Dostanic1998PLMS,Dostanic2005IEOT}.
Connections with Poisson-type problems and quantitative PDE estimates were
studied in \cite{Kalaj2012AIM,Kalaj2012IEOT}, with further norm estimates for
Cauchy-type operators in \cite{ZhuKalaj2020JFA} and sharp \(L^p\)-theory on the
disk in \cite{KalajMelentijevicZhu2022JFA}.
Related spectral questions on Bergman-type spaces and in doubly connected
geometry can be found in \cite{Vujadinovic2016JMAA,Vujadinovic2025PA}.

\begin{theorem}[Sharp \(L^2\)-norm on a circular annulus]\label{thm:norm-annulus}
Let \(0<r<R\), \(A=A(r,R)=\{z\in\C:\ r<|z|<R\}\), and
\[
(C_A f)(z)=\frac1\pi\int_A \frac{f(w)}{z-w}\,dA(w),\qquad f\in L^2(A).
\]
Then
\[
\|C_A\|_{L^2(A)\to L^2(A)}=\frac{2}{\kappa_1},
\]
where \(\kappa_1>0\) is the smallest positive solution of
\begin{equation}\label{eq:kappa1}
J_1(\kappa r)\,Y_0(\kappa R)-Y_1(\kappa r)\,J_0(\kappa R)=0 .
\end{equation}
\end{theorem}

\begin{proof}
By Lemmas~\ref{lem:mode-action} and \ref{lem:hardy-m},
\[
\|C_A\|=\sup_{m\in\mathbb Z}\|C_A\|_{H_m\to H_{m-1}},
\qquad
\|C_A\|_{H_m\to H_{m-1}}=
\begin{cases}
\|T_m\|,& m\ge 1,\\
\|T_{1-m}\|,& m\le 0,
\end{cases}
\]
and for \(m\ge 1\),
\[
\|T_m\|=\frac{2}{\sqrt{\mu_m}},
\]
where \(\mu_m\) is the first eigenvalue of \eqref{eq:SL-m}.
Lemma~\ref{lem:monotone} gives \(\mu_m\ge \mu_1\) for all \(m\ge 1\), hence the
supremum is attained at \(m=1\) and
\[
\|C_A\|=\|T_1\|=\frac{2}{\sqrt{\mu_1}}.
\]
Finally, Lemma~\ref{lem:bessel-m} (with \(m=1\)) yields \(\mu_1=\kappa_1^2\), where
\(\kappa_1\) is the smallest positive root of \eqref{eq:kappa1}. Therefore
\(\|C_A\|=2/\kappa_1\).

Moreover, if \(F>0\) is the first eigenfunction of \eqref{eq:SL-m} for \(m=1\) and
\(f(\rho e^{i\theta})=-F'(\rho)e^{i\theta}\), then Lemma~\ref{lem:mode-action} gives
\[
(C_A f)(\rho e^{i\theta})=2F(\rho),
\]
so \(f\) is an extremizer, see remark below for details.
\end{proof}
\begin{remark}[An explicit extremizer and the derivative computation]\label{rem:extremizer-annulus}
Let \(\kappa_1>0\) be the smallest positive solution of \eqref{eq:kappa1}. Define
\[
F(\rho):=c\Big( Y_0(\kappa_1 R)\,J_0(\kappa_1\rho)-J_0(\kappa_1 R)\,Y_0(\kappa_1\rho)\Big),
\qquad r<\rho<R,
\]
where \(c\neq 0\) is chosen so that \(F>0\) on \((r,R)\). Then \(F(R)=0\) holds by
construction. Moreover, using the identities \(J_0'(x)=-J_1(x)\) and
\(Y_0'(x)=-Y_1(x)\) together with the chain rule, we compute
\[
\begin{aligned}
F'(\rho)
&=c\Big( Y_0(\kappa_1 R)\,\frac{d}{d\rho}J_0(\kappa_1\rho)
 -J_0(\kappa_1 R)\,\frac{d}{d\rho}Y_0(\kappa_1\rho)\Big)\\
&=c\,\kappa_1\Big( Y_0(\kappa_1 R)\,J_0'(\kappa_1\rho)
 -J_0(\kappa_1 R)\,Y_0'(\kappa_1\rho)\Big)\\
&=c\,\kappa_1\Big(J_0(\kappa_1 R)\,Y_1(\kappa_1\rho)-Y_0(\kappa_1 R)\,J_1(\kappa_1\rho)\Big).
\end{aligned}
\]
In particular,
\[
F'(r)=0
\quad\Longleftrightarrow\quad
J_1(\kappa_1 r)\,Y_0(\kappa_1 R)-Y_1(\kappa_1 r)\,J_0(\kappa_1 R)=0,
\]
so \(\kappa_1\) is precisely the parameter for which the boundary conditions
\(F'(r)=0\) and \(F(R)=0\) are satisfied. Consequently,
\[
f(\rho e^{i\theta})=-F'(\rho)\,e^{i\theta}
\]
is an extremizer for \(\|C_A\|\) in Theorem~\ref{thm:norm-annulus}.
\end{remark}

\begin{remark}[On the graph of \(2/\kappa_{1,1}(r)\) for \(R=1\) and scaling in \(R\)]\label{rem:graph-scaling}
Theorem~\ref{thm:norm-annulus} gives
\[
\|C_{A(r,R)}\|_{2\to 2}=\frac{2}{\kappa_{1,1}(r,R)},
\]
where \(\kappa_{1,1}(r,R)\) is the smallest positive root of the transcendental
Bessel determinant \eqref{eq:kappa1}. In general there is no closed-form
expression for \(\kappa_{1,1}(r,R)\) in terms of elementary functions, so an
\emph{explicit} formula for the function \(r\mapsto 2/\kappa_{1,1}(r,R)\) is not
available. Nevertheless, for each fixed \(r\in(0,1)\) one can compute
\(\kappa_{1,1}(r,1)\) numerically by solving
\[
D_r(\kappa):=J_1(\kappa r)Y_0(\kappa)-Y_1(\kappa r)J_0(\kappa)=0
\]
for its smallest positive root (e.g.\ by bisection, Newton's method, or standard
special-function solvers). This produces the graph of
\[
r\longmapsto \frac{2}{\kappa_{1,1}(r,1)},\qquad 0<r<1 (\text{see Figure~1}).
\]

Let $\lambda_1(A)$ be the first Dirichlet eigenvalue of $-\Delta$ on $A$ and set
\begin{equation}\label{eq:k1def}
k_1:=\sqrt{\lambda_1(A)}.
\end{equation}
In the planar case, $k_1$ is characterized as the smallest positive root of the Bessel cross--product equation
\begin{equation}\label{eq:kEqGeneral}
J_0(kr)\,Y_0(kR)-J_0(kR)\,Y_0(kr)=0 .
\end{equation}

The proof of Theorem~\ref{thm:intro-main} "exploits rotational symmetry": one performs a Fourier decomposition to reduce \(C_{A(r,R)}\) to "a family of one-dimensional weighted Hardy-type operators on radial profiles." Next, "a variational argument" produces a Sturm--Liouville problem "whose first eigenvalue controls the sharp operator norm." Finally, "the Bessel determinant arises from solving the radial eigenvalue equation explicitly." The "extremizers are given in terms of the corresponding mixed eigenfunction," highlighting "a structural distinction between simply and doubly connected domains."

[Main result]\label{thm:intro-main}
Let \(0<r<R\). Then
\[
\|C_{A(r,R)}\|_{L^2\to L^2}=\frac{2}{\sqrt{\mu_1^{ND}(r,R)}},
\]
where \(\mu_1^{ND}(r,R)\) is the first eigenvalue of
\[
-\Delta U=\mu U \ \text{in }A(r,R),\qquad
\partial_\nu U=0 \ \text{on }|z|=r,\qquad
U=0 \ \text{on }|z|=R.
\]
Equivalently, \(\mu_1^{ND}(r,R)=\kappa_1^2\), where \(\kappa_1\) is the smallest
positive root of a Bessel determinant, in establishing the main theorem.

We use the following theorem.
\begin{theorem}[Sharp \(L^2\)-norm on a circular annulus]\label{thm:norm-annulus}
Let \(0<r<R\), \(A=A(r,R)=\{z\in\C:\ r<|z|<R\}\), and
\[
(C_A f)(z)=\frac1\pi\int_A \frac{f(w)}{z-w}\,dA(w),\qquad f\in L^2(A).
\]
Then
\[
\|C_A\|_{L^2(A)\to L^2(A)}=\frac{2}{\kappa_1},
\]
where \(\kappa_1>0\) is the smallest positive solution of
\begin{equation}\label{eq:kappa1}
J_1(\kappa r)\,Y_0(\kappa R)-Y_1(\kappa r)\,J_0(\kappa R)=0 .
\end{equation}
\end{theorem}

Question:
Let \(0<r<R\), let the circular annulus be \(A(r,R)=\{z\in\mathbb C:r<|z|<R\}\), and let the Cauchy transform on \(A(r,R)\) be
\[
(C_{A(r,R)}f)(z)=\frac1\pi\int_{A(r,R)}\frac{f(w)}{z-w}\,dA(w),\qquad f\in L^2(A(r,R)).
\]
Let \(\mu_1^{ND}(r,R)\) denote the first eigenvalue of the mixed Neumann-Dirichlet problem
\[
-\Delta U=\mu U \ \text{in }A(r,R),\qquad \partial_\nu U=0 \ \text{on }|z|=r,\qquad U=0 \ \text{on }|z|=R,
\]
where \(\partial_\nu\) is the outward normal derivative. Which quantitative estimate holds for the \(L^2\)-operator norm of \(C_{A(r,R)}\)?

One has
\[
\|C_{A(r,R)}\|_{L^2\to L^2}=\frac{2}{\sqrt{\mu_1^{ND}(r,R)}}.
\]
Equivalently, if \(\kappa_1>0\) is the smallest positive solution of
\[
J_1(\kappa r)Y_0(\kappa R)-Y_1(\kappa r)J_0(\kappa R)=0,
\]
then \(\mu_1^{ND}(r,R)=\kappa_1^2\) and
\[
\|C_{A(r,R)}\|_{L^2(A(r,R))\to L^2(A(r,R))}=\frac{2}{\kappa_1}.
\]

Choices:
B.
One has
\[
\|C_{A(r,R)}\|_{L^2\to L^2}=\frac{2}{\sqrt{\lambda_1(A(r,R))}},
\]
where \(\lambda_1(A(r,R))\) is the first Dirichlet eigenvalue of \(-\Delta\) on the annulus. Equivalently, if \(k_1>0\) is the smallest positive solution of
\[
J_0(k r)Y_0(kR)-Y_0(k r)J_0(kR)=0,
\]
then \(\lambda_1(A(r,R))=k_1^2\) and
\[
\|C_{A(r,R)}\|_{L^2(A(r,R))\to L^2(A(r,R))}=\frac{2}{k_1}.
\]
C.
One has the lower bound
\[
\|C_{A(r,R)}\|_{L^2\to L^2}\ge \frac{2}{\sqrt{\mu_1^{ND}(r,R)}}.
\]
Equivalently, if \(\kappa_1>0\) is the smallest positive solution of
\[
J_1(\kappa r)Y_0(\kappa R)-Y_1(\kappa r)J_0(\kappa R)=0,
\]
then \(\mu_1^{ND}(r,R)=\kappa_1^2\) and hence
\[
\|C_{A(r,R)}\|_{L^2(A(r,R))\to L^2(A(r,R))}\ge \frac{2}{\kappa_1}.
\]
D.
One has
\[
\|C_{A(r,R)}\|_{L^2\to L^2}=\frac{2}{\sqrt{\mu_1^{DN}(r,R)}},
\]
where \(\mu_1^{DN}(r,R)\) is the first eigenvalue of
\[
-\Delta U=\mu U\ \text{in }A(r,R),\qquad U=0\ \text{on }|z|=r,\qquad \partial_\nu U=0\ \text{on }|z|=R.
\]
Equivalently, if \(\kappa_1>0\) is the smallest positive solution of
\[
J_0(\kappa r)Y_1(\kappa R)-Y_0(\kappa r)J_1(\kappa R)=0,
\]
then \(\mu_1^{DN}(r,R)=\kappa_1^2\) and
\[
\|C_{A(r,R)}\|_{L^2(A(r,R))\to L^2(A(r,R))}=\frac{2}{\kappa_1}.
\]
E.
One has
\[
\|C_{A(r,R)}\|_{L^2\to L^2}=\frac{2}{\sqrt{\mu_1^{ND}(r,R)}},
\]
but now \(\mu_1^{ND}(r,R)=\kappa_1^2\) is characterized by the smallest positive solution of
\[
J_0(\kappa r)Y_0(\kappa R)-Y_0(\kappa r)J_0(\kappa R)=0.
\]
Thus
\[
\|C_{A(r,R)}\|_{L^2(A(r,R))\to L^2(A(r,R))}=\frac{2}{\kappa_1}.
\]

The stem introduces the mixed Neumann-Dirichlet eigenvalue \(\mu_1^{ND}(r,R)\), which is the key quantity appearing in the correct option, so it gives a strong hint. However, it does not explicitly state the exact norm formula, and several alternatives still compete.

The item is close to a theorem-recall question: it asks which estimate holds for the norm and the correct choice essentially states the target result. The presence of nearby alternatives prevents it from being a pure restatement, but it is only a mild reformulation.

Some reasoning is needed to distinguish exact equality from a weaker lower bound and to identify the correct boundary-condition/eigenvalue pairing. Still, the task mainly tests recognition of the known formula rather than deeper derivation or synthesis.

The distractors are mathematically plausible and target realistic confusions: pure Dirichlet vs mixed boundary data, swapping DN/ND conditions, replacing equality by a weaker true statement, and using an incorrect Bessel-characteristic equation.

A reasonably strong theorem-recognition MCQ with good distractors, but it gives noticeable hints via the stem and only moderately tests genuine generative reasoning.


[\cite{FCW}]\label{Kn}
Let $n$ be a positive number. Then
$$\tau_{o}(K_{n})=\frac{1}{2^{n}}\sum_{k=0}^{n}\binom{n}{k}(2k-n)^{n-2}.$$

Let $G=(V(G), E(G))$ be a graph without self-loops and $\mathcal{T}(G)$ be the collection of spanning trees of $G$.
We usually call $\tau(G)=|\mathcal{T}(G)|$ the number of spanning trees of $G$. Denote by $K_n$, $K_{m,n}$, and $K_{n_1,n_2,\ldots, n_s}$
be the complete graph of $n$ vertices, the complete bipartite graph that one partite has $m$ vertices and the other partite has $n$ vertices,
and the complete $s$-partite graph that each partite has $n_i$ vertices.

Counting spanning trees is a classic problem originating in the 19th century that continues to be a hot topic today.
Some recent developments can be found in \cite{DG,DKM,GJ,LCY,LY,Yan,YT} for example.

In 2025, motivated by Gallai's classical theorem (\cite{Lovasz}, Section 5, Problem 17) that the vertices of
any graph can be partitioned into two sets
where one induces an even-degree subgraph and the other induces an odd-degree subgraph, and also the concept of
homeomorphically irreducible spanning trees (HISTs, spanning trees containing no vertices of degree two),
Zheng and Wu \cite{ZW} introduced the concept of odd spanning trees. An odd spanning tree of a graph $G$
is a spanning tree $T$ of $G$ where each vertex has odd degree in $T$.

The famous Cayley's formula \cite{Cayley} states that $\tau(K_n)=n^{n-2}$.
Let $\tau_o(G)$ be the number of odd spanning trees of $G$.
Recently, Feng, Chen and Wu counted the number of odd spanning trees in complete graphs via Prüfer code and the exponential generating function as follows.

The famous Cayley's formula \cite{Cayley} states that $\tau(K_n)=n^{n-2}$.
Let $\tau_o(G)$ be the number of odd spanning trees of $G$.
Recently, Feng, Chen and Wu counted the number of odd spanning trees in complete graphs via Prüfer code and the exponential generating function as follows.

Note: in \cite{FCW}, the original theorem is of the form
\[
\tau_{o}(K_{n})=
\begin{cases}
0, & \text{if } n \text{ is odd;} \\
\displaystyle\frac{1}{2^{n}}\sum_{k=0}^{n}\binom{n}{k}(2k-n)^{n-2}, & \text{if } n \text{ is even.}
\end{cases}
\]
We slightly revised the theorem since when $n$ is odd, it is easy to check that $\frac{1}{2^{n}}\sum_{k=0}^{n}\binom{n}{k}(2k-n)^{n-2}=0$.

\begin{lemma}\label{bo}
$$
 \sum_{\substack{y \in \{\pm1\}^n}} \left(\sum_{i=1}^n a_i y_i \right)^m=
 2^n\cdot\sum\limits_{\substack{k_1+k_2+\cdots+k_n=m\\k_i\in 2\mathbb{N} }}\frac{m!}{k_1!k_2!\cdots k_n!}a_1^{k_1}a_2^{k_2}\cdots a_n^{k_n}.
$$
\end{lemma}

\begin{theorem}[\cite{Berge,Lovasz}]\label{Kn-degree}
Let $d_1,d_2,\dots,d_n$ be positive integers summing up to $2n-2$. Then the number of spanning trees of $K_n$ in which the vertex $i$ has degree exactly $d_i$ for all $i=1,2,\dots,n$ equals$$
 \dfrac{(n-2)!}{(d_1-1)!(d_2-1)!\cdots(d_n-1)!}.$$
\end{theorem}

\begin{theorem}\label{Kmn-degree}
Let $K_{m,n}$ be the complete bipartite graph with bipartition $A=\{u_1,u_2,\ldots,u_m\}$ and $B=\{v_1,v_2,\ldots,v_n\}$. Let $a_i$ and $b_j$ be positive integers which $\sum_{i=1}^{m}a_i=\sum_{j=1}^{n}b_j=m+n-1$. Then the number of spanning trees of $K_{m,n}$ in which the vertex $u_i$ and $v_j$ have degree exactly $a_i$ and $b_j$ for $i\in\{1,2,\ldots,m\}$ and $j\in\{1,2,\ldots,n\}$ equals
 $$
 \frac{(m-1)!(n-1)!}{\prod_{i=1}^{m}(a_i-1)!\prod_{j=1}^{n}(b_j-1)!},
 $$
\end{theorem}
\begin{proof}
 We use induction on $m+n$. For $m=1$ or $n=1$, $K_{m,n}$ is a star and the statement is trivial.
Since $\sum_{i=1}^{m}a_i=\sum_{j=1}^{n}b_j=m+n-1$, $\sum_{i=1}^{m}a_i+\sum_{j=1}^{n}b_j=2(m+n)-2<2(m+n)$,
there must exist a leaf, and we may assume that $a_m=1$ where $m>1$.
Remove $u_m$, in any tree under consideration, $u_m$ is adjacent to some $v_j$, $1\leq j\leq n$ and the removal of $u_m$ results in another tree on $\{u_1,\dots,u_{m-1},v_1,\dots,v_n\}$ with degrees $a_1,\dots,a_{m-1},b_1,\cdots,b_j-1,\dots,b_n$.
By the induction hypothesis, the number of trees in $K_{m-1,n}$ that $\{u_1,\dots,u_{m-1},v_1,\dots,v_n\}$ has degrees $a_1,\dots,a_{m-1},b_1,\cdots,b_j-1,\dots,b_n$ is
 $$\frac{(m-2)!(n-1)!}{(a_1-1)!\cdots(a_{m-1}-1)!(b_1-1)!\cdots(b_j-2)!\cdots(b_n-1)!}
 =\frac{(m-2)!(n-1)!(b_j-1)}{\prod_{i=1}^{m}(a_i-1)!\prod_{j=1}^{n}(b_j-1)!}.
 $$
 Thus, the number of trees on $\{u_1,\dots,u_{m},v_1,\dots,v_n\}$ with degrees $a_1,\dots,a_{m},b_1,\dots,b_n$ is

\noindent
{\bf Proof of Theorem \ref{Kn}.}
By Theorem \ref{Kn-degree}, the number of odd spanning trees of $K_n$ in which
the vertex $i$ has degree exactly $d_i=2d_i'+1$ for all $i=1,2,\dots,n$ equals$$
 \dfrac{(n-2)!}{(2d_1')!(2d_2')!\cdots(2d_n')!},$$
 where $d_i\geq0$, and $\sum_{i=1}^{n}(2d_i')=n-2$.
Therefore,
$$\tau_o(K_n)=\sum_{\substack{k_1+k_2+\cdots+k_n=n-2\\k_i\in 2\mathbb{N} }}\frac{(n-2)!}{k_1!k_2!\cdots k_n!}.$$
It follows from Lemma \ref{bo} that
$$
 \sum_{\substack{\varepsilon_i \in \{\pm1\}}}(\varepsilon_1x_1+\varepsilon_2x_2+\cdots\varepsilon_nx_n)^{n-2}= 2^n\cdot\sum_{\substack{k_1+k_2+\cdots+k_n=n-2\\k_i\in 2\mathbb{N} }}\frac{(n-2)!}{k_1!k_2!\cdots k_n!}x_1^{k_1}x_2^{k_2}\cdots x_n^{k_n}.
 $$
Let $x_1=x_2=\cdots=x_n=1$, we obtain that
 $$
 \tau_o(K_n)=\frac{1}{2^n} \sum_{\substack{\varepsilon_i \in \{\pm1\}}}(\varepsilon_1+\varepsilon_2+\cdots\varepsilon_n)^{n-2}\\
 =\frac{1}{2^n}\sum_{k=0}^{n}\binom{n}{k}(2k-n)^{n-2}.
 $$
{\hfill$\Box$}

\begin{theorem}\label{Kmn}
Let $m$ and $n$ be positive numbers. Then
$$\tau_o(K_{m,n})=\frac{1}{2^{m+n}}\left[\sum_{i=0}^{m}\binom{m}{i}(2i-m)^{n-1}\right]\left[\sum_{j=0}^{n}\binom{n}{j}(2j-n)^{m-1}\right].$$
\end{theorem}

\begin{proof}
 The notation and symbols used below are consistent with those in Theorem \ref{Kmn-degree}.
By Theorem \ref{Kmn-degree}, the number of odd spanning trees of $K_{m,n}$
in which the vertex $u_i\in A$ and $v_j\in B$ have degree exactly $d(u_i)=2d'(u_i)+1$ and $d(v_j)=2d'(v_j)+1$
for all $i\in\{1,2,\dots,m\}$ and $j\in\{1,2,\dots,n\}$ equals
$$
 \frac{(m-1)!(n-1)!}{\prod_{i=1}^{m}2d'(u_i)!\prod_{j=1}^{n}2d'(v_j)!},
 $$
 where all $d'\geq0$, $\sum_{i=1}^{m}2d'(u_i)=n-1$ and $\sum_{j=1}^{n}2d'(v_j)=m-1$.
Therefore,
$$\tau_o(K_{m,n})=\sum_{\substack{k_1+k_2+\cdots+k_m=n-1\\k_i\in 2\mathbb{N} }}
 \sum_{\substack{l_1+l_2+\cdots+l_n=m-1\\l_i\in 2\mathbb{N} }}\frac{(m-1)!(n-1)!}{\prod_{i=1}^{m}k_i!\prod_{j=1}^{n}l_j!}.$$

Note: in \cite{FCW}, the original theorem is of the form
\[
\tau_{o}(K_{n})=
\begin{cases}
0, & \text{if } n \text{ is odd;} \\
\displaystyle\frac{1}{2^{n}}\sum_{k=0}^{n}\binom{n}{k}(2k-n)^{n-2}, & \text{if } n \text{ is even.}
\end{cases}
\]
We slightly revised the theorem since when $n$ is odd, it is easy to check that $\frac{1}{2^{n}}\sum_{k=0}^{n}\binom{n}{k}(2k-n)^{n-2}=0$.

In this note, we we give a simple proof via a classical spanning tree enumeration formula and the Boolean function.
We also generalize it to complete bipartite graphs.

To prove Theorem~\ref{Kn}, the note says it will “give a simple proof via a classical spanning tree enumeration formula and the Boolean function.” It also notes that the statement is revised from \cite{FCW} because “when $n$ is odd, it is easy to check that $\frac{1}{2^{n}}\sum_{k=0}^{n}\binom{n}{k}(2k-n)^{n-2}=0$.”

[F.C.W. \cite{FCW}]\label{Kn}
Let $n$ be a positive number. Then
$$\tau_{o}(K_{n})=\frac{1}{2^{n}}\sum_{k=0}^{n}\binom{n}{k}(2k-n)^{n-2}.$$,

Question: Let \(K_n\) denote the complete graph on \(n\) vertices, and let \(\tau_o(G)\) denote the number of odd spanning trees of a graph \(G\), where an odd spanning tree is a spanning tree in which every vertex has odd degree. For every positive integer \(n\), which statement gives \(\tau_o(K_n)\)?

\[\tau_o(K_n)=\frac{1}{2^n}\sum_{k=0}^{n}\binom{n}{k}(2k-n)^{n-2}.\]

Choices:
B. \[\tau_o(K_n)=\frac{1}{2^n}\sum_{k=0}^{n}\binom{n}{k}(2k-n)^{n-1}.\]
C. \[\tau_o(K_n)=0\quad\text{for every odd }n.\]
D. \[\tau_o(K_n)=\frac{1}{2^{n-1}}\sum_{k=0}^{n}\binom{n}{k}(2k-n)^{n-2}.\]
E. \[\tau_o(K_n)=\frac{1}{2^n}\sum_{k=0}^{n}\binom{n}{k}\lvert 2k-n\rvert^{\,n-2}.\]

The stem does not reveal the formula or give wording that points uniquely to choice A; it only defines the quantity and asks for its expression.

This is essentially a direct 'state the formula' question. If the underlying result is the closed form for odd spanning trees of K_n, the MCQ is mostly a restatement of that theorem rather than a new inference task.

There is some reasoning pressure because the distractors are close variants of the true formula and one option is a weaker true statement, but the item still primarily tests recall/recognition of the exact formula rather than deeper generation of an argument.

The distractors are strong: B changes the exponent, D changes the normalization, E removes signed cancellation via absolute values, and C is a plausible weaker truth based on parity. These reflect realistic mathematical failure modes.

A solid formula-recognition MCQ with good distractors and no answer leakage, but it is close to theorem restatement and only moderately tests genuine generative reasoning.


Question: Let \(K_n\) denote the complete graph on \(n\) vertices, and let \(\tau_o(G)\) denote the number of odd spanning trees of a graph \(G\), where an odd spanning tree is a spanning tree in which every vertex has odd degree. For every positive integer \(n\), which statement gives \(\tau_o(K_n)\)?

Choices:
B. \[\tau_o(K_n)=\frac{1}{2^n}\sum_{k=0}^{n}\binom{n}{k}(2k-n)^{n-1}.\]
C. \[\tau_o(K_n)=0\quad\text{for every odd }n.\]
D. \[\tau_o(K_n)=\frac{1}{2^{n-1}}\sum_{k=0}^{n}\binom{n}{k}(2k-n)^{n-2}.\]
E. \[\tau_o(K_n)=\frac{1}{2^n}\sum_{k=0}^{n}\binom{n}{k}\lvert 2k-n\rvert^{\,n-2}.\]

The stem defines the notation and asks for the value of the quantity, but it does not reveal the formula or give direct hints pointing uniquely to choice A.

The item is essentially a theorem-identification question asking for the exact closed form of u03c4_o(K_n). It is not a verbatim restatement of the answer, but it is close to recall of a known formula rather than a substantially transformed application.

There is some reasoning pressure because several options are structurally similar and one distractor is a weaker true statement, so the student must distinguish the strongest correct conclusion. Still, success depends largely on recalling or recognizing the exact formula rather than generating a new argument.

The distractors are plausible and mathematically targeted: wrong exponent, wrong normalization factor, loss of sign cancellation via absolute value, and a weaker true parity consequence. These reflect realistic failure modes.

A solid MCQ with no answer leakage and strong distractors, but it mainly tests recognition of a known formula rather than deep generative reasoning.


\label{thm1x}
Let $\Phi$, $\Psi$ be linear flows on $X$. Then each of the following
four statements implies the other three:
\begin{enumerate}
\item $\Phi \stackrel{1}{\thicksim} \Psi$, i.e., $\Phi$, $\Psi$ are
  Lipschitz equivalent;
  \item there exists $\alpha
    \in \mathbb{R} \setminus \{0\}$ so that $\Phi \stackrel{1}{\cong}
    \Psi_{*\alpha}$, i.e., $\Phi$, $\Psi_{*\alpha}$ are Lipschitz conjugate;
\item there exists $\beta
    \in \mathbb{R} \setminus \{0\}$ so that $A^{\Phi}$, $\beta A^{\Psi}$ are
    Lyapunov similar while $A^{\Phi_{\sf  AD}}$, $\beta A^{\Psi_{\sf
        AD}}$ as well as $A^{\Phi_{\sf  C}}$, $\beta A^{\Psi_{\sf
        C}}$ are similar;
\item there exists $\gamma
    \in \mathbb{R} \setminus \{0\}$ so that $A^{\Phi}$, $\gamma A^{\Psi}$ are
    Lipschitz similar while $A^{\Phi_{\sf  C}}$, $\gamma A^{\Psi_{\sf
        C}}$ are similar.
  \end{enumerate}
  Moreover, $\Phi \stackrel{1}{\cong}\Psi$ if and
only if $A^{\Phi}$, $A^{\Psi}$ are Lipschitz similar while $A^{\Phi_{\sf  C}}$, $ A^{\Psi_{\sf
        C}}$ are similar.

\label{sec1}

Let $X\ne \{0\}$ be a finite-dimensional normed space over $\mathbb{R}$ and $\varphi$
a flow on $X$, i.e., $\varphi : \mathbb{R} \times X \to X$ is
continuous with $\varphi(t+s , x) = \varphi \bigl( t, \varphi (s,x)
\bigr)$ and $\varphi(0,x) = x$ for all $t,s\in \mathbb{R}$, $x\in X$. A
fundamental question throughout dynamics is that of classification:
When, precisely, are two flows $\varphi$, $\psi$ on $X$ {\em the
 same\/}? Taking a geometrically motivated approach to this question,
say that $\varphi$, $\psi$ are {\bf
 equivalent}, in symbols $\varphi \stackrel{0}{\thicksim} \psi$, if there exists a
homeomorphism $h:X\to X$ with $h(0)=0$ that maps each $\varphi$-orbit
$\varphi_{\mathbb{R}}(x):= \{\varphi (t, x):t\in \mathbb{R}\}$ onto
a $\psi$-orbit, i.e.,
\begin{equation}\label{eq1_1}
 h \bigl( \varphi_{\mathbb{R}} (x) \bigr) = \psi_{\mathbb{R}} \bigl( h(x)\bigr) \qquad
 \forall x \in X \, .
\end{equation}
If $h$, $h^{-1}$ both are H\"{o}lder continuous (or Lipschitz
continuous, differentiable, linear) then $\varphi$, $\psi$ are
said to be {\bf H\"{o}lder} (or {\bf Lipschitz}, {\bf differentiably},
{\bf linearly}) {\bf equivalent}, in symbols $\varphi
\stackrel{1^-}{\thicksim} \psi$ (or $\varphi
\stackrel{1}{\thicksim} \psi$, $\varphi
\stackrel{{\sf diff}}{\thicksim} \psi$, $\varphi
\stackrel{{\sf lin}}{\thicksim} \psi$). Plainly $\stackrel{\bigstar}{\thicksim}$
yields an equivalence relation
for each $\bigstar\in \{0,1^-,1, {\sf diff}, {\sf lin}\}$, thereby establishing
five natural classifications of all flows on $X$. In addition,
consider the following, much more
restrictive form of equivalence: Say that $\varphi$, $\psi$ are {\bf conjugate}, in symbols
$\varphi \stackrel{0}{\cong} \psi$, if
\begin{equation}\label{eq1_2}
 h \bigl( \varphi (t,x) \bigr) = \psi \bigl( t,h(x)\bigr) \qquad
 \forall t\in \mathbb{R} , x \in X \, .
\end{equation}
Notice that (\ref{eq1_2}) implies (\ref{eq1_1}) but most definitely
not vice versa, and analogously defined
$\stackrel{\bigstar}{\cong}$ again yields an equivalence relation for
each $\bigstar \in \{0, 1^-, 1, {\sf diff}, {\sf lin}\}$. Simple examples
show that no two of these equivalences, or the
classification established by them, coincide, not even when $\mbox{\rm
 dim}\, X=1$.

Building on the classical literature briefly reviewed below, the
present article concludes the authors' earlier work \cite{BW, BW2}
by carrying out a comprehensive analysis of Lipschitz equivalence and conjugacy
for {\em linear\/} flows. Recall that a flow $\varphi$ on $X$ is {\bf linear} if the time-$t$
map $\varphi_t = \varphi (t,\cdot):X\to X$ is linear, or equivalently
if $\varphi_t = e^{t A^{\varphi}}$,
for every $t\in \mathbb{R}$, with a (unique) linear operator $A^{\varphi}$ on
$X$ called the {\bf generator} of $\varphi$. Henceforth, upper case Greek letters
$\Phi$, $\Psi$ are used exclusively to denote linear flows. The challenge, then, is to
characterize $\Phi \stackrel{1}{\thicksim} \Psi$ and $\Phi
\stackrel{1}{\cong} \Psi$ in terms of basic linear algebra properties
of $A^{\Phi}$, $A^{\Psi}$. Correspondingly the main result of this
article, Theorem \ref{thm1x} below, can be
viewed as a {\bf Lipschitz classification theorem}. To preview the result,
note that every linear flow $\Phi$ on $X$ determines unique
decompositions $\Phi \stackrel{{\sf lin}}{\cong} \Phi_{\sf D} \times \Phi_{\sf
 AD}
\stackrel{{\sf lin}}{\cong} \Phi_{\sf S} \times \Phi_{\sf C} \times
\Phi_{\sf U}$ into a {\em diagonal\/}
and {\em anti-diagonal\/} part, as well as into a
{\em stable}, {\em central}, and {\em unstable\/} part; see Section \ref{sec2a} below
for formal details. For convenience denote by
$\Phi_{*\alpha}$ the linear flow generated by $\alpha A^{\Phi}$, for
any $\alpha \in \mathbb{R} \setminus \{0\}$. Thus, $\Phi_{*\alpha}$ simply is
$\Phi$ with its first variable (``time'') rescaled by $\alpha$. Finally, the statement involves
the concepts of {\em Lyapunov\/} and {\em Lipschitz
similarity}, introduced rigorously in Section
\ref{sec2a} also. For now, simply say that two linear operators are {\bf
 Lyapunov similar} if they (more precisely, the flows they generate) have the same
Lyapunov exponents, with matching multiplicities, and say that they
are {\bf Lipschitz similar} if they are Lyapunov similar and their
anti-diagonal parts are similar.

\begin{equation}\label{eq1_1}
  h \bigl( \varphi_{\R} (x) \bigr) = \psi_{\R} \bigl( h(x)\bigr)  \qquad
  \forall x \in X \, .
\end{equation}

\begin{equation}\label{eq1_2}
  h \bigl( \varphi  (t,x) \bigr) = \psi \bigl( t,h(x)\bigr)  \qquad
  \forall t\in \R , x \in X \, .
\end{equation}

\begin{theorem}\label{thm1x}
Let $\Phi$, $\Psi$ be linear flows on $X$. Then each of the following
four statements implies the other three:
\begin{enumerate}
\item $\Phi \stackrel{1}{\thicksim} \Psi$, i.e., $\Phi$, $\Psi$ are
  Lipschitz equivalent;
  \item there exists $\alpha
    \in \R \setminus \{0\}$ so that $\Phi \stackrel{1}{\cong}
    \Psi_{*\alpha}$, i.e., $\Phi$, $\Psi_{*\alpha}$ are Lipschitz conjugate;
\item there exists $\beta
    \in \R \setminus \{0\}$ so that $A^{\Phi}$, $\beta A^{\Psi}$ are
    Lyapunov similar while $A^{\Phi_{\sf  AD}}$, $\beta A^{\Psi_{\sf
        AD}}$ as well as $A^{\Phi_{\sf  C}}$, $\beta A^{\Psi_{\sf
        C}}$ are similar;
\item there exists $\gamma
    \in \R \setminus \{0\}$ so that $A^{\Phi}$, $\gamma A^{\Psi}$ are
    Lipschitz similar while $A^{\Phi_{\sf  C}}$, $\gamma A^{\Psi_{\sf
        C}}$ are similar.
  \end{enumerate}
  Moreover, $\Phi \stackrel{1}{\cong}\Psi$ if and
only if $A^{\Phi}$, $A^{\Psi}$ are Lipschitz similar while $A^{\Phi_{\sf  C}}$, $ A^{\Psi_{\sf
        C}}$ are similar.
\end{theorem}

Building on the classical literature briefly reviewed below, the
present article concludes the authors' earlier work \cite{BW, BW2}
by carrying out a comprehensive analysis of Lipschitz equivalence and conjugacy
for {\em linear\/} flows. Recall that a flow $\varphi$ on $X$ is {\bf linear} if the time-$t$
map $\varphi_t = \varphi (t,\cdot):X\to X$ is linear, or equivalently
if $\varphi_t = e^{t A^{\varphi}}$,
for every $t\in \R$, with a (unique) linear operator $A^{\varphi}$ on
$X$ called the {\bf generator} of $\varphi$. Henceforth, upper case Greek letters
$\Phi$, $\Psi$ are used exclusively to denote linear flows. The challenge, then, is to
characterize $\Phi \stackrel{1}{\thicksim} \Psi$ and $\Phi
\stackrel{1}{\cong} \Psi$ in terms of basic linear algebra properties
of $A^{\Phi}$, $A^{\Psi}$. Correspondingly the main result of this
article, Theorem \ref{thm1x} below, can be
viewed as a {\bf Lipschitz classification theorem}. To preview the result,
note that every linear flow $\Phi$ on $X$ determines unique
decompositions $\Phi \stackrel{{\sf lin}}{\cong} \Phi_{\sf D} \times \Phi_{\sf
 AD}
\stackrel{{\sf lin}}{\cong} \Phi_{\sf S} \times \Phi_{\sf C} \times
\Phi_{\sf U}$ into a {\em diagonal\/}
and {\em anti-diagonal\/} part, as well as into a
{\em stable}, {\em central}, and {\em unstable\/} part; see Section \ref{sec2a} below
for formal details. For convenience denote by
$\Phi_{*\alpha}$ the linear flow generated by $\alpha A^{\Phi}$, for
any $\alpha \in \R \setminus \{0\}$. Thus, $\Phi_{*\alpha}$ simply is
$\Phi$ with its first variable (``time'') rescaled by $\alpha$. Finally, the statement involves
the concepts of {\em Lyapunov\/} and {\em Lipschitz
similarity}, introduced rigorously in Section
\ref{sec2a} also. For now, simply say that two linear operators are {\bf
 Lyapunov similar} if they (more precisely, the flows they generate) have the same
Lyapunov exponents, with matching multiplicities, and say that they
are {\bf Lipschitz similar} if they are Lyapunov similar and their
anti-diagonal parts are similar.

To put Theorem \ref{thm1x} in context, it is instructive to compare it
to its differentiable (hence slightly more restrictive) and H\"{o}lder
(hence slightly less restrictive) counterparts; stated here without
proofs, these results have been presented by the authors in detail
(though sometimes couched in slightly different terminology and
notation) in \cite{BW} and \cite{BW2} respectively.

\begin{prop}\label{prop1zb}
Let $\Phi$, $\Psi$ be linear flows on $X$. Then each of the following
five statements implies the other four:
\begin{enumerate}
\item $\Phi \stackrel{\sf lin}{\thicksim} \Psi$, i.e., $\Phi$, $\Psi$ are
 linearly equivalent;
 \item $\Phi \stackrel{\sf diff}{\thicksim} \Psi$, i.e., $\Phi$, $\Psi$ are
 differentiably equivalent;
 \item there exists $\alpha
 \in \R \setminus \{0\}$ so that $\Phi \stackrel{\sf lin}{\cong}
 \Psi_{*\alpha}$, i.e., $\Phi$, $\Psi_{*\alpha}$ are linearly
 conjugate;
 \item there exists $\beta
 \in \R \setminus \{0\}$ so that $\Phi \stackrel{\sf diff}{\cong}
 \Psi_{*\beta}$, i.e., $\Phi$, $\Psi_{*\beta}$ are differentiably conjugate;
\item there exists $\gamma
 \in \R \setminus \{0\}$ so that $A^{\Phi}$, $\gamma A^{\Psi}$ are similar.
 \end{enumerate}
 Moreover, $\Phi \stackrel{\sf lin}{\cong} \Psi$ if and only if $\Phi
 \stackrel{\sf diff}{\cong} \Psi$ if and only if $A^{\Phi}$,
 $A^{\Psi}$ are similar.
\end{prop}

\begin{prop}\label{prop1za}
Let $\Phi$, $\Psi$ be linear flows on $X$. Then each of the following
three statements implies the other two:
\begin{enumerate}
\item $\Phi \stackrel{1^-}{\thicksim} \Psi$, i.e., $\Phi$, $\Psi$ are
 H\"{o}lder equivalent;
 \item there exists $\alpha
 \in \R \setminus \{0\}$ so that $\Phi \stackrel{1^-}{\cong}
 \Psi_{*\alpha}$, i.e., $\Phi$, $\Psi_{*\alpha}$ are H\"{o}lder conjugate;
\item there exists $\beta
 \in \R \setminus \{0\}$ so that $A^{\Phi}$, $\beta A^{\Psi}$ are
 Lyapunov similar while $A^{\Phi_{\sf C}}$, $\beta A^{\Psi_{\sf
 C}}$ are similar.
 \end{enumerate}
 Moreover, $\Phi \stackrel{1^-}{\cong} \Psi$ if and only if
 $A^{\Phi}$, $A^{\Psi}$ are Lyapunov similar while $A^{\Phi_{\sf C}}$,
 $ A^{\Psi_{\sf C}}$ are similar.
\end{prop}

\begin{lem}\label{lem4_2}
Given $m\in \N \setminus \{1\}$, $k,\ell\in \N$, and $b\in \R^k$, $c\in
\R^{\ell}$ with $m\, {\sf d}(b) = m\, {\sf d}(c) = d$, let $\Phi$, $\Psi$ be the flows on $\R^d$ generated by
$A$, $B$ in {\rm (\ref{eq4_1})} respectively. Then the following
statements are equivalent:
\begin{enumerate}
\item $\Phi \stackrel{{\sf lin}}{\cong}\Psi$;
\item $\Phi \stackrel{1}{\thicksim} \Psi$;
 \item $k=\ell$, and there exists a bijection $g:\{1, \ldots , k\}\to \{1, \ldots
 , \ell\}$ so that $|b_j| = |c_{g(j)}|$ for every $j$;
\item $A$, $B$ are similar;
\item $A$, $B$ are Lipschitz similar.
\end{enumerate}
\end{lem}

\begin{theorem}\label{thm71}
Let $\Phi$, $\Psi$ be $\K$-linear flows on $X$. Then the following
statements are equivalent:
\begin{enumerate}
\item $\Phi \stackrel{1}{\thicksim}\Psi$;
 \item there exists $\alpha\in \R\setminus \{0\}$ so that $\Phi \stackrel{1}{\cong}\Psi_{*\alpha}$;
 \item $\Phi^{\R} \stackrel{1}{\thicksim} \Psi^{\R}$;
 \item there exists $\beta\in \R\setminus \{0\}$ so that $\Phi^{\R} \stackrel{1}{\cong}\Psi^{\R}_{*\beta}$;
\item there exists $\gamma \in \R \setminus \{0\}$ so that
$A^{\Phi^{\R}}$, $\gamma A^{\Psi^{\R}}$ are Lipschitz similar while
$A^{\Phi_{\sf C}^{\R}}$, $\gamma A^{\Psi_{\sf C}^{\R}}$ are similar.
\end{enumerate}
Moreover, $\Phi \stackrel{1}{\cong}\Psi$ if and only if $\Phi^{\R} \stackrel{1}{\cong}\Psi^{\R}$
 if and only if $A^{\Phi^{\R}}$, $A^{\Psi^{\R}}$ are Lipschitz similar while
$A^{\Phi_{\sf C}^{\R}}$, $A^{\Psi_{\sf C}^{\R}}$ are similar.
\end{theorem}

\begin{theorem}\label{thm6_8}
Let $\Phi$, $\Psi$ be linear flows on $X$. Then the following
statements are equivalent:
\begin{enumerate}
\item $\Phi \stackrel{{\sf pw}1}{\cong}\Psi$;
 \item $\Phi$, $\Psi$ are kinematically similar while $\Phi_{\sf C}
 \stackrel{{\sf lin}}{\cong} \Psi_{\sf C}$;
 \item $A^{\Phi}$, $A^{\Psi}$ are kinematically similar while
 $A^{\Phi_{\sf C}}, A^{\Psi_{\sf C}}$ are similar.
\end{enumerate}
\end{theorem}

To put Theorem \ref{thm1x} in context, it is instructive to compare it
to its differentiable (hence slightly more restrictive) and H\"{o}lder
(hence slightly less restrictive) counterparts; stated here without
proofs, these results have been presented by the authors in detail
(though sometimes couched in slightly different terminology and
notation) in \cite{BW} and \cite{BW2} respectively.

\begin{prop}\label{prop1zb}
Let $\Phi$, $\Psi$ be linear flows on $X$. Then each of the following
five statements implies the other four:
\begin{enumerate}
\item $\Phi \stackrel{\sf lin}{\thicksim} \Psi$, i.e., $\Phi$, $\Psi$ are
  linearly equivalent;
 \item $\Phi \stackrel{\sf diff}{\thicksim} \Psi$, i.e., $\Phi$, $\Psi$ are
  differentiably equivalent;
  \item there exists $\alpha
    \in \mathbb{R} \setminus \{0\}$ so that $\Phi \stackrel{\sf lin}{\cong}
    \Psi_{*\alpha}$, i.e., $\Phi$, $\Psi_{*\alpha}$ are linearly
    conjugate;
     \item there exists $\beta
    \in \mathbb{R} \setminus \{0\}$ so that $\Phi \stackrel{\sf diff}{\cong}
    \Psi_{*\beta}$, i.e., $\Phi$, $\Psi_{*\beta}$ are differentiably conjugate;
\item there exists $\gamma
    \in \mathbb{R} \setminus \{0\}$ so that $A^{\Phi}$, $\gamma A^{\Psi}$ are similar.
  \end{enumerate}
  Moreover, $\Phi \stackrel{\sf lin}{\cong} \Psi$ if and only if $\Phi
  \stackrel{\sf diff}{\cong} \Psi$ if and only if $A^{\Phi}$,
  $A^{\Psi}$ are similar.
\end{prop}

\begin{prop}\label{prop1za}
Let $\Phi$, $\Psi$ be linear flows on $X$. Then each of the following
three statements implies the other two:
\begin{enumerate}
\item $\Phi \stackrel{1^-}{\thicksim} \Psi$, i.e., $\Phi$, $\Psi$ are
  H\"{o}lder equivalent;
  \item there exists $\alpha
    \in \mathbb{R} \setminus \{0\}$ so that $\Phi \stackrel{1^-}{\cong}
    \Psi_{*\alpha}$, i.e., $\Phi$, $\Psi_{*\alpha}$ are H\"{o}lder conjugate;
\item there exists $\beta
    \in \mathbb{R} \setminus \{0\}$ so that $A^{\Phi}$, $\beta A^{\Psi}$ are
    Lyapunov similar while $A^{\Phi_{\sf C}}$, $\beta A^{\Psi_{\sf
        C}}$ are similar.
  \end{enumerate}
  Moreover, $\Phi \stackrel{1^-}{\cong} \Psi$ if and only if
  $A^{\Phi}$, $A^{\Psi}$ are Lyapunov similar while $A^{\Phi_{\sf C}}$,
  $ A^{\Psi_{\sf C}}$ are similar.
\end{prop}

Classifications of linear flows have long been studied in the
literature, notably for {\em hyperbolic\/} flows, that is, for
$\Phi_{\sf C}$, $\Psi_{\sf C}$ being trivial; see, e.g., \cite{Amann, BW,
  Irwin, R} for broad context, as well as \cite{ACK1, AK, DSS, He, LZ, Willems} for
specific studies.
One striking aspect of Theorem
\ref{thm1x} is the fact that (i)$\Rightarrow$(ii). Thus, for {\em linear\/}
flows $\varphi$, $\psi$ validity of (\ref{eq1_1}) always entails
validity of (\ref{eq1_2}), up to a linear, orbit-independent rescaling
of time. This remarkable property, which does not
seem to be shared by any wider class of flows on $X$, is indicative of
the extraordinary coherence between individual orbits of linear
flows. As far as the authors have been able to ascertain, the
property has not been stated, let alone proved rigorously before,
though it appears to have been part of linear systems folklore for
quite some time; see, e.g., \cite[Rem.\ 7.4]{Willems} as well as
\cite{ACK1, CK, KS}.

With Lipschitz continuity being a fundamental concept throughout
analysis \cite{Hei}, another key aspect of Theorem \ref{thm1x} is its
relation to the differentiable and H\"{o}lder counterparts,
Propositions \ref{prop1zb} and \ref{prop1za} respectively. While
\cite[Rem.\ 2.5]{ACK1} seems to suggest, somewhat misleadingly, that the equivalence
relation $\stackrel{1}{\cong}$ for linear flows simply coincides with $\stackrel{{\sf
    diff}}{\cong}$, the characterization of Lipschitz {\em
  conjugacy\/} contained in Theorem \ref{thm1x} has, in essence, been
established in \cite{KS}; see also \cite{MM} for a related (albeit quite
informal) discussion. Specifically, \cite{KS} argues that
$\stackrel{1}{\cong}$ is ``very close'' to $\stackrel{{\sf
    diff}}{\cong}$, and a crucial role in the argument is subsequently
played by a well-known theorem due to Rademacher which asserts that
every Lipschitz map $h:X\to X$ is differentiable almost
everywhere. However, in
the general setting of Theorem \ref{thm1x}, that is, for mere Lipschitz {\em equivalence}, no
assumptions whatsoever are made regarding the (orientation and
regularity of) re-parametrizations of individual orbits, and hence it appears
doubtful whether Rademacher's theorem can be applied
fruitfully (or at all). Instead, the present
article utilizes a simple geometric idea gleaned from \cite{MM} which
it develops into the basic but consequential concept of {\em
  distortion\/} ({\em points\/}) for stable flows. Aided by this
concept, the article then proceeds to prove, in an entirely elementary
manner, that (i)$\Rightarrow$(iii) in Theorem \ref{thm1x}. From this,
validity of the entire theorem follows rather straightforwardly, as
(iii)$\Leftrightarrow$(iv)$\Rightarrow$(ii), and obviously
(ii)$\Rightarrow$(i).  As indicated where appropriate below, the
elementary approach developed here also helps to address other gaps and
inaccuracies in the existing literature.

\medskip

\begin{rem}
An analogous classification problem presents itself in discrete time,
i.e., for linear operators $A, B:X\to X$ which are {\bf conjugate} (or
{\bf nonlinearly similar} \cite{CappSh, CSSW}), in symbols
$A\stackrel{0}{\cong} B$, if $h(Ax) = Bh(x)$ for some homeomorphism
$h:X\to X$ and all $x\in X$. As in continuous time, it is
natural to consider $\stackrel{\bigstar}{\cong}$ for $\bigstar \in
\{0,1^-,1,{\sf diff}, {\sf lin}\}$, each of which yields an
equivalence relation on all linear operators on $X$. In
straightforward analogy to Proposition \ref{prop1zb}, notice that
$A\stackrel{{\sf lin}}{\cong} B$ if and only if $A\stackrel{{\sf
    diff}}{\cong} B$ if and only if $A$, $B$ are similar. By contrast,
the problem of characterizing $A\stackrel{\bigstar}{\cong} B$ for
$\bigstar \in \{0,1^-, 1\}$, in terms of similarity invariants akin to
Theorem \ref{thm1x}, has turned out to be significantly more challenging than in
continuous time; see, e.g., \cite{CappSh, CSSW, Cruz, HP, KR} for the
long history of the problem and its many ramifications, with
\cite{Cruz} specifically addressing the Lipschitz case.
\end{rem}

The remainder of this article is organized as follows: Section
\ref{sec2} properly introduces various forms of equivalence and
conjugacy to be studied in subsequent sections, together with some
tailor-made analytical notation. Section \ref{sec2a} briefly reviews a
few basic concepts pertinent to linear flows, notably irreducibility
and Lyapunov exponents, and then discusses the novel concept of
Lipschitz similarity. Section \ref{sec3} defines distortion points and
refined Lyapunov spaces for stable flows which in turn yield crucial Lipschitz
invariants for such flows. Section
\ref{sec4} presents a proof of the main result, Theorem \ref{thm1x},
via a series of preparatory lemmas addressing important special cases
thereof. It also outlines the straightforward extension of Theorem
\ref{thm1x} to {\em complex\/} spaces. Finally, Section \ref{sec6}
highlights several subtle aspects of Lipschitz equivalence by briefly
discussing a novel variant of the concept, referred to as {\em
  pointwise\/} Lipschitz equivalence at $0$.

The article explains that, rather than using Rademacher’s theorem as in \cite{KS}, it “utilizes a simple geometric idea gleaned from \cite{MM} which it develops into the basic but consequential concept of {\em distortion\/} ({\em points\/}) for stable flows.” Using this concept, it “then proceeds to prove, in an entirely elementary manner, that (i)$\Rightarrow$(iii) in Theorem \ref{thm1x}.” From there, “validity of the entire theorem follows rather straightforwardly, as (iii)$\Leftrightarrow$(iv)$\Rightarrow$(ii), and obviously (ii)$\Rightarrow$(i).”

The article explains that, rather than using Rademacher’s theorem as in \cite{KS}, it “utilizes a simple geometric idea gleaned from \cite{MM} which it develops into the basic but consequential concept of {\em distortion\/} ({\em points\/}) for stable flows.” Using this concept, it “then proceeds to prove, in an entirely elementary manner,” one implication needed to establish the main theorem. From there, “validity of the entire theorem follows rather straightforwardly, as (iii)$\Leftrightarrow$(iv)$\Rightarrow$(ii), and obviously (ii)$\Rightarrow$(i).”

Biconditional or Equivalence
Implication

Question: Let $X\neq\{0\}$ be a finite-dimensional normed space over $\mathbb{R}$, and let $\Phi,\Psi$ be linear flows on $X$, meaning $\Phi_t=e^{tA^{\Phi}}$ and $\Psi_t=e^{tA^{\Psi}}$ for linear generators $A^{\Phi},A^{\Psi}$. For a flow $\Theta$, write $\Theta_{\mathbb{R}}(x)=\{\Theta(t,x):t\in\mathbb{R}\}$. Say that $\Phi$ and $\Psi$ are Lipschitz equivalent if there is a bi-Lipschitz homeomorphism $h:X\to X$ with $h(0)=0$ such that $h(\Phi_{\mathbb{R}}(x))=\Psi_{\mathbb{R}}(h(x))$ for all $x\in X$. Say that they are Lipschitz conjugate if there is a bi-Lipschitz homeomorphism $h:X\to X$ with $h(0)=0$ such that $h(\Phi(t,x))=\Psi(t,h(x))$ for all $t\in\mathbb{R}$ and $x\in X$. For $\alpha\neq 0$, let $\Psi_{*\alpha}$ denote the time-rescaled flow generated by $\alpha A^{\Psi}$, equivalently $\Psi_{*\alpha}(t,x)=\Psi(\alpha t,x)$. Also, let $\Phi_{\sf AD},\Psi_{\sf AD}$ and $\Phi_{\sf C},\Psi_{\sf C}$ denote the anti-diagonal and central parts in the canonical decomposition of linear flows; “similar” means similar as linear operators. Finally, two linear operators are called Lyapunov similar if the flows they generate have the same Lyapunov exponents with matching multiplicities, and Lipschitz similar if they are Lyapunov similar and their anti-diagonal parts are similar. Under these assumptions, which statement about $\Phi$ and $\Psi$ holds?

The following are equivalent: (i) $\Phi$ and $\Psi$ are Lipschitz equivalent; (ii) there exists $\alpha\in\mathbb{R}\setminus\{0\}$ such that $\Phi$ and $\Psi_{*\alpha}$ are Lipschitz conjugate; (iii) there exists $\beta\in\mathbb{R}\setminus\{0\}$ such that $A^{\Phi}$ and $\beta A^{\Psi}$ are Lyapunov similar, while $A^{\Phi_{\sf AD}}$ and $\beta A^{\Psi_{\sf AD}}$ are similar and $A^{\Phi_{\sf C}}$ and $\beta A^{\Psi_{\sf C}}$ are similar; (iv) there exists $\gamma\in\mathbb{R}\setminus\{0\}$ such that $A^{\Phi}$ and $\gamma A^{\Psi}$ are Lipschitz similar, while $A^{\Phi_{\sf C}}$ and $\gamma A^{\Psi_{\sf C}}$ are similar. Moreover, $\Phi$ and $\Psi$ are Lipschitz conjugate if and only if $A^{\Phi}$ and $A^{\Psi}$ are Lipschitz similar and $A^{\Phi_{\sf C}}$ and $A^{\Psi_{\sf C}}$ are similar.

Choices:
B. The following are equivalent: (i) $\Phi$ and $\Psi$ are Lipschitz equivalent; (ii) there exists $\alpha\in\mathbb{R}$ such that $\Phi$ and $\Psi_{*\alpha}$ are Lipschitz conjugate; (iii) there exists $\beta\in\mathbb{R}$ such that $A^{\Phi}$ and $\beta A^{\Psi}$ are Lyapunov similar, while $A^{\Phi_{\sf AD}}$ and $\beta A^{\Psi_{\sf AD}}$ are similar and $A^{\Phi_{\sf C}}$ and $\beta A^{\Psi_{\sf C}}$ are similar; (iv) there exists $\gamma\in\mathbb{R}$ such that $A^{\Phi}$ and $\gamma A^{\Psi}$ are Lipschitz similar, while $A^{\Phi_{\sf C}}$ and $\gamma A^{\Psi_{\sf C}}$ are similar. Moreover, $\Phi$ and $\Psi$ are Lipschitz conjugate if and only if $A^{\Phi}$ and $A^{\Psi}$ are Lipschitz similar and $A^{\Phi_{\sf C}}$ and $A^{\Psi_{\sf C}}$ are similar.
C. If $\Phi$ and $\Psi$ are Lipschitz equivalent, then there exists $\alpha\in\mathbb{R}\setminus\{0\}$ such that $\Phi$ and $\Psi_{*\alpha}$ are Lipschitz conjugate.
D. The following are equivalent: (i) $\Phi$ and $\Psi$ are Lipschitz equivalent; (ii) there exists $\alpha\in\mathbb{R}\setminus\{0\}$ such that $\Phi$ and $\Psi_{*\alpha}$ are Lipschitz conjugate; (iii) there exists $\beta\in\mathbb{R}\setminus\{0\}$ such that $A^{\Phi}$ and $\beta A^{\Psi}$ are Lyapunov similar and $A^{\Phi_{\sf C}}$ and $\beta A^{\Psi_{\sf C}}$ are similar; (iv) there exists $\gamma\in\mathbb{R}\setminus\{0\}$ such that $A^{\Phi}$ and $\gamma A^{\Psi}$ are Lipschitz similar. Moreover, $\Phi$ and $\Psi$ are Lipschitz conjugate if and only if $A^{\Phi}$ and $A^{\Psi}$ are Lipschitz similar.
E. The following are equivalent: (i) $\Phi$ and $\Psi$ are Lipschitz equivalent; (ii) $\Phi$ and $\Psi$ are Lipschitz conjugate; (iii) there exists $\beta\in\mathbb{R}\setminus\{0\}$ such that $A^{\Phi}$ and $\beta A^{\Psi}$ are Lyapunov similar, while $A^{\Phi_{\sf AD}}$ and $\beta A^{\Psi_{\sf AD}}$ are similar and $A^{\Phi_{\sf C}}$ and $\beta A^{\Psi_{\sf C}}$ are similar; (iv) there exists $\gamma\in\mathbb{R}\setminus\{0\}$ such that $A^{\Phi}$ and $\gamma A^{\Psi}$ are Lipschitz similar, while $A^{\Phi_{\sf C}}$ and $\gamma A^{\Psi_{\sf C}}$ are similar. Moreover, $\Phi$ and $\Psi$ are Lipschitz conjugate if and only if $A^{\Phi}$ and $A^{\Psi}$ are Lipschitz similar and $A^{\Phi_{\sf C}}$ and $A^{\Psi_{\sf C}}$ are similar.

full equivalence of all four conditions and moreover-clause

necessary anti-diagonal and central similarity clauses in the algebraic characterization

need for time-rescaling in passing from equivalence to conjugacy

The stem does not explicitly state the correct option, but it heavily signals the structure of the answer by introducing exactly the notions used in the correct choice (time rescaling, Lipschitz similarity, and central/anti-diagonal decomposition). This creates strong thematic hints.

The item is close to a theorem-recall question: it asks for the statement equivalent to the given conditions, and the correct choice is essentially the precise classification statement. However, the alternatives are not identical restatements, so it is not fully tautological.

There is some reasoning pressure because the options differ in subtle but important ways: existential vs universal quantification, allowing gamma = 0, dropping central similarity, or weakening Lipschitz similarity to Lyapunov similarity. Still, the question mainly tests recognition of the exact theorem statement rather than deeper derivation.

The distractors are strong and mathematically meaningful. Each reflects a plausible failure mode: boundary case on gamma, omission of a necessary condition, weakening a key equivalence notion, or quantifier strengthening. They are distinct and well-targeted.

A solid theorem-recognition MCQ with strong distractors, but it leans toward recalling the precise statement rather than generating a conclusion from substantial reasoning.


Question: Let $X\neq\{0\}$ be a finite-dimensional normed space over $\mathbb{R}$, and let $\Phi,\Psi$ be linear flows on $X$, meaning $\Phi_t=e^{tA^{\Phi}}$ and $\Psi_t=e^{tA^{\Psi}}$ for linear generators $A^{\Phi},A^{\Psi}$. For a flow $\Theta$, write $\Theta_{\mathbb{R}}(x)=\{\Theta(t,x):t\in\mathbb{R}\}$. Say that $\Phi$ and $\Psi$ are Lipschitz equivalent if there is a bi-Lipschitz homeomorphism $h:X\to X$ with $h(0)=0$ such that $h(\Phi_{\mathbb{R}}(x))=\Psi_{\mathbb{R}}(h(x))$ for all $x\in X$. Say that they are Lipschitz conjugate if there is a bi-Lipschitz homeomorphism $h:X\to X$ with $h(0)=0$ such that $h(\Phi(t,x))=\Psi(t,h(x))$ for all $t\in\mathbb{R}$ and $x\in X$. For $\alpha\neq 0$, let $\Psi_{*\alpha}$ denote the time-rescaled flow generated by $\alpha A^{\Psi}$, equivalently $\Psi_{*\alpha}(t,x)=\Psi(\alpha t,x)$. Also, let $\Phi_{\sf AD},\Psi_{\sf AD}$ and $\Phi_{\sf C},\Psi_{\sf C}$ denote the anti-diagonal and central parts in the canonical decomposition of linear flows; “similar” means similar as linear operators. Finally, two linear operators are called Lyapunov similar if the flows they generate have the same Lyapunov exponents with matching multiplicities, and Lipschitz similar if they are Lyapunov similar and their anti-diagonal parts are similar. Under these assumptions, which statement about $\Phi$ and $\Psi$ holds?

Choices:
B. The following are equivalent: (i) $\Phi$ and $\Psi$ are Lipschitz equivalent; (ii) there exists $\alpha\in\mathbb{R}$ such that $\Phi$ and $\Psi_{*\alpha}$ are Lipschitz conjugate; (iii) there exists $\beta\in\mathbb{R}$ such that $A^{\Phi}$ and $\beta A^{\Psi}$ are Lyapunov similar, while $A^{\Phi_{\sf AD}}$ and $\beta A^{\Psi_{\sf AD}}$ are similar and $A^{\Phi_{\sf C}}$ and $\beta A^{\Psi_{\sf C}}$ are similar; (iv) there exists $\gamma\in\mathbb{R}$ such that $A^{\Phi}$ and $\gamma A^{\Psi}$ are Lipschitz similar, while $A^{\Phi_{\sf C}}$ and $\gamma A^{\Psi_{\sf C}}$ are similar. Moreover, $\Phi$ and $\Psi$ are Lipschitz conjugate if and only if $A^{\Phi}$ and $A^{\Psi}$ are Lipschitz similar and $A^{\Phi_{\sf C}}$ and $A^{\Psi_{\sf C}}$ are similar.
C. If $\Phi$ and $\Psi$ are Lipschitz equivalent, then there exists $\alpha\in\mathbb{R}\setminus\{0\}$ such that $\Phi$ and $\Psi_{*\alpha}$ are Lipschitz conjugate.
D. The following are equivalent: (i) $\Phi$ and $\Psi$ are Lipschitz equivalent; (ii) there exists $\alpha\in\mathbb{R}\setminus\{0\}$ such that $\Phi$ and $\Psi_{*\alpha}$ are Lipschitz conjugate; (iii) there exists $\beta\in\mathbb{R}\setminus\{0\}$ such that $A^{\Phi}$ and $\beta A^{\Psi}$ are Lyapunov similar and $A^{\Phi_{\sf C}}$ and $\beta A^{\Psi_{\sf C}}$ are similar; (iv) there exists $\gamma\in\mathbb{R}\setminus\{0\}$ such that $A^{\Phi}$ and $\gamma A^{\Psi}$ are Lipschitz similar. Moreover, $\Phi$ and $\Psi$ are Lipschitz conjugate if and only if $A^{\Phi}$ and $A^{\Psi}$ are Lipschitz similar.
E. The following are equivalent: (i) $\Phi$ and $\Psi$ are Lipschitz equivalent; (ii) $\Phi$ and $\Psi$ are Lipschitz conjugate; (iii) there exists $\beta\in\mathbb{R}\setminus\{0\}$ such that $A^{\Phi}$ and $\beta A^{\Psi}$ are Lyapunov similar, while $A^{\Phi_{\sf AD}}$ and $\beta A^{\Psi_{\sf AD}}$ are similar and $A^{\Phi_{\sf C}}$ and $\beta A^{\Psi_{\sf C}}$ are similar; (iv) there exists $\gamma\in\mathbb{R}\setminus\{0\}$ such that $A^{\Phi}$ and $\gamma A^{\Psi}$ are Lipschitz similar, while $A^{\Phi_{\sf C}}$ and $\gamma A^{\Psi_{\sf C}}$ are similar. Moreover, $\Phi$ and $\Psi$ are Lipschitz conjugate if and only if $A^{\Phi}$ and $A^{\Psi}$ are Lipschitz similar and $A^{\Phi_{\sf C}}$ and $A^{\Psi_{\sf C}}$ are similar.

The stem defines the relevant notions but does not explicitly reveal the theorem statement or point to option A. There are no obvious lexical cues or direct hints indicating which choice is correct.

The item is essentially asking for the exact theorem characterizing Lipschitz equivalence/conjugacy of linear flows. The correct option is a near-verbatim theorem statement rather than a problem requiring an independently derived conclusion.

Some comparison-based reasoning is needed to distinguish subtle alterations in the options (e.g., allowing 0 for the rescaling parameter, dropping anti-diagonal conditions, confusing equivalence with conjugacy). However, the question primarily tests precise theorem recall rather than generative mathematical reasoning from the definitions.

The distractors are plausible and mathematically meaningful: they weaken equivalence to implication, omit necessary similarity conditions, or mishandle the nonzero time-rescaling parameter. These reflect realistic failure modes and are distinct from one another.

A technically well-constructed but theorem-recall-heavy MCQ: little answer leakage and strong distractors, but low tautology avoidance and only moderate pressure for genuine reasoning.


\begin{thm} \label{thm:intro-cap-comp}
Let $\Om \subset \subset \bC^n$ be a strictly pseudoconvex domain.
\begin{itemize}
\item[(a)] There exists a constant $A>0$ such that for every Borel set $E\subset \Om$,
$$
	\tc(E) \leq A [cap(E,\Om)]^\frac{1}{n}.
$$
\item[(b)]
Assume $D \subset \subset \Om$ be a subdomain. There exists a constant $A'$ such that for every Borel set $E\subset D$,
$$
	\frac{1}{A'} cap(E,\Om) \leq \tc (E).
$$
\end{itemize}
Moreover, the above inequalities are sharp as far as the exponents are concerned.
\end{thm}

\begin{thm} \label{thm:subextension}
Let $f \in W^*(B(0,1))$ be such that $f\leq -1$. Assume $\|f\|_*\leq 1$. Let $\veps>0$  and $0<s<1$. Then, there exists $u \in \cL(\bC^n)$ such that $u\leq -|f^\star|^{\frac{2}{n}-\veps}$ on $B(0,s)$ everywhere.
\end{thm}

\begin{lem}\label{lem:cap-sublevel-set} Let $f\in W^*(\Om) \cap C^0(\Om)$.  Assume either $f\geq 0$ or $f\leq 0$ on $\Om$. Denote $E_s = \{z\in \Om: |f(z)| > s\}$ for $s > 0$. Then,
\[\label{eq:c-sublevel-set-est}
	\tc (E_s) \leq \frac{1}{s^2} \|f\|_*^2.
\]
\end{lem}

\begin{thm} \label{thm:VV} Let $\Om \subset \subset \bC^n$ be an open subset and let $f\in W^*(\Om)$. There exists a Borel subset $E\subset \Om$ such that $\tc (E) =0$ and it satisfies
\begin{itemize}
\item
[(i)] for each $x\in \Om\setminus E$,
$$
	\lim_{r\to 0} \intavg_{B(x,r)} f dy = f^\star(x).
$$
\item
[(ii)] Moreover, for each $x\in \Om\setminus E$,
$$
	\lim_{r\to 0} \intavg_{B(x,r)} |f-f^\star(x)|^2 dy =0.
$$
\end{itemize}
Consequently, the precise representative $f^\star$ is quasi-continuous.
\end{thm}

\begin{thm}\label{thm:intro-choquet-c} Let $\Om$ be a bounded open set in $\bC^n$. The set function on Borel subsets $E\mapsto \tc (E)= \tc (E,\Om)$ is a Choquet capacity.
\end{thm}

This result is the local analogue of \cite{Vigny}. On compact K\"ahler manifolds the proof of  \cite[Theorem~30]{Vigny} employed the capacity notion in the Dirichlet spaces \cite[Lemma~23]{Vigny} and the characterization of pluripolar sets via a family of capacity associated with closed positive current \cite{FO84}.  Here we use a similar strategy. Notably, we are able to give a simpler proof of the characterization in Corollary~\ref{cor:FO-polar} by using only pluripotential theory.

Another remarkable property of functions in the complex Sobolev space is the uniformly exponential integrability due to Dinh, Marinescu and Vu  \cite{DMV}, which is very close to the one of psh functions.  Namely,
there exist positive constants $\al$ and $A$  such that for every $f \in W^*(B(0,1))$ with $\|f\|_*\leq 1$,
\[\label{eq:DMV-intro}
	\int_{\bar B(0,\frac{1}{8})} e^{\al |f|^2} d x \leq A.
\]
This result should be compared with the well-known exponential integrability of psh functions due to H\"ormander \cite[Proposition~4.2.9]{Ho07} and Skoda \cite{Sk72}. In the proof of  \cite[Theorem~1.2]{DMV}  the authors used the induction argument in dimension and slicing theory for positive currents. We provide a new and simpler one in Proposition~\ref{prop:DMV} by reducing the inequality to one dimensional case. In this way, it is enough  to  work with smooth functions after taking  the standard convolution with smooth kernels. The constants can be explicitly computed though they  are suboptimal.  Additionally, we point out in Remark~\ref{rmk:sharp-DMV} that the exponent 2 of $|f|$ in \eqref{eq:DMV-intro} is the optimal for all $n\geq 1$, which was only known for $n=1$ in \cite{DMV}.

The third main result fully characterizes the inequality \eqref{eq:DMV-intro} by the volume-capacity inequality of sublevel sets.

\begin{thm} \label{prop:equiv-est-intro} Let $K\subset\subset \Om$ be a compact subset.  The following statements hold and they are equivalent to each other.
\begin{itemize}
\item[(a)]
There exist constants $A_1>0$ and $\al>0$ depending only on $K, \Om$ such that for every  Borel  set $E\subset K$ $$V_{2n}(E) \leq A_1 e^{-\al/\tc (E)}.$$

\item[(b)] There exist uniform constants $A_1>0$ and $\al>0$ depending only on $K, \Om$ such that for every  $f\in W^*(\Om)$ and $f\leq 0$,
 $$V_{2n}(\{f< -1\} \cap K) \leq A_1 e^{\frac{-\al}{\|f\|_*^{2}}}.$$
\item
[(c)]   There exist uniform constants $A_1>0$ and $\al>0$ depending only on $K, \Om$ such that for every $f\in W^*(\Om)$ and $f\leq 0$ whose norm $\|f\|_* \leq
\ka$,
$$V_{2n} (\{f < -s\} \cap K) \leq A_1 e^{-\al s^2/\ka^2}, \quad \forall s>0.$$

\item
[(d)]  There exist uniform constants $A_1>0$ and $\al>0$ depending only on $K, \Om$ such that for every $f\in W^*(\Om)$ and $f\leq 0$ with $\|f\|_* \leq 1$,
$$V_{2n} (\{f < -s\} \cap K) \leq A_1 e^{-\al s^2}, \quad \forall s>0.$$
\end{itemize}
\end{thm}

\begin{prop} \label{prop:ae-qe}  Let $D\subset \Om$ be an open set and $f$ is quasi-continuous on $D$. If $f\geq 0$ a.e. on $D$, then $f\geq 0$ quasi-everywhere on $D$.
\end{prop}

\begin{prop} \label{prop:DMV} Let $B(0,1) \subset \bC^n$ be the unit ball. There exist positive constants $\al$ and $A$  such that for all $f \in W^*(B(0,1))$ with $\|f\|_*\leq 1$,
$$
	\int_{\bar B(0,\frac{1}{8})} e^{\al |f|^2} dx \leq A.
$$
\end{prop}

\begin{lem}[outer regularity] \label{lem:outer-reg} For a Borel subset $E\subset \Om$,
\[\notag	\tc(E) = \inf\{ \tc(G) : E \subset G,\; G \text{ is open}\}.
\]
\end{lem}

\begin{remark}\label{rmk:DMV-holder} Let $\Om$ be a bounded strictly psedocovex domain and $K\subset \subset \Om$ be a compact subset. Let $u_1,...,u_n$ be H\"older continuous psh functions in $\Om$. It follows from \cite[Proposition~2.4]{DKN} that the Radon measure
$$\mu = {\bf 1}_K \, dd^c u_1 \wed \cdots \wed dd^c u_n$$
is $W^*(\Om)$-H\"older continuous. That means there exist positive constants $c>0$ and $\al>0$  such that for $f\in W^*(\Om)$ and $\|f\|_* \leq 1$,
$$
	\left| \int_\Om f d\mu \right| \leq  c \|f\|_{*}^\al.
$$
Combining this inequality and the arguments in the proof of \cite[Proposition~2.9]{Ng18} we get that there exist $\al_1>0$ and $c_1>0$ such that   for $f \in W^*(\Om)$, $f\leq 0$ and $\|f\|_* \leq 1$,
$$
	\mu (f < -s) \leq c_1 e^{-\al_1 s^2},  \quad  \forall s>0.
$$
In other words, the equivalent inequalities in Proposition~\ref{prop:equiv-est} holds for a very large family of Monge-Amp\`ere measures associated with  H\"older continuous psh functions. This result is also equivalent to the statement of \cite[Theorem~1.2]{DMV}.
\end{remark}

\begin{defn}
\label{defn:L2-ma} Let $\phi \in PSH\cap L^\infty(\Om)$.
Define $\mu: = (dd^c\phi)^n$. Let $g \in W^*(\Om)$ and $K \subset \subset \Om$ be a compact subset.  If $g\geq 0$, then
$$
	\int_K g d\mu := \int_K \wt g d\mu,
$$
where $\wt g$ is a quasi-continuous modification.
Generally,
$$
	\int_K g d\mu := \int_K g^+ d\mu - \int_K g^- d\mu.
$$
\end{defn}

\begin{cor} \label{cor:FO-polar}
Let $E\subset \Om$ be a Borel subset. Then, $cap (E) =0$ if  and only if $\tc_\te(E) =0$ for all $\te $ of the form \eqref{eq:current-te} with $0<\de \leq 1$.
\end{cor}

Let $(X,\omega)$ be a compact K\"ahler manifold.  A natural subspace of the Sobolev space $W^{1,2}(X)$, called the complex Sobolev space $W^*$, was introduced by Dinh and Sibony \cite{DS} in their work on complex dynamics.
This new space turned out to be a good complex version of the former one as it takes into account the complex structure of manifolds. Many important families of functions which carry useful properties of underlying manifolds belong to the space. Among them are Lipschitz functions, bounded quasi plurisubharmonic (psh) functions or more general bounded delta quasi-psh function studied in  \cite{CW05}. Basic functional aspects of this space were studied by Vigny \cite{Vigny} where he showed that it is a Banach space with a corresponding norm $\| \cdot\|_*$, this norm  is stronger than the Sobolev norm. However, it is not reflexive and smooth functions are not dense in $W^*$ with respect to the strong topology of that norm.  He also defined a functional capacity for this space and showed that it is a capacity in sense of Choquet. Moreover, the functional capacity is qualitative comparable to the relative capacity of Bedford and Taylor in global pluripotential theory defined in  \cite{ko03}.

Thanks to flexible properties this space has found many of applications in complex dynamic in high dimensions, complex Monge-Amp\`ere equations and other areas, see e.g., \cite{BiD23}, \cite{DKN}, \cite{DKW}, \cite{DNV25}, \cite{Vig15}, \cite{Vu20, Vu24, Vu26} and \cite{WZ}. Also a higher complex Sobolev space is proposed in \cite{DoN25}.

An element in a Sobolev space is defined almost everywhere up to a set of Lebesgue measure zero which is unlike psh or quasi-psh functions. Its fine properties with respect to the Sobolev capacity are classical ones \cite{EG92}, \cite{KLV-book}. However, this capacity is not good enough to characterize pluripolar sets, therefore it is not suitable for studying functions in $W^*$. One needs to work with the corresponding capacity of $W^*$ which dominates the Sobolev one. However, the new capacity in complex Sobolev spaces is harder to analyze. A crucial difference is that we can no longer rely on maximal function techniques, which are very powerful, in these spaces (see, e.g \cite{KLV-book}).

Recently, it has been showed in \cite{DMV}, \cite{Vigny} and \cite{VV24} that pluripotential theory is a suitable and powerful tool for studying $W^*$. As a result they made major  progresses on study local properties of functions in such spaces.

We continue to study local complex Sobolev spaces which has been considered partially in \cite{DMV} and  \cite{Vigny}. Our goal is to develop further this approach by focusing on new and optimal inequalities between the functional capacity and classical capacities of Bedford and Taylor \cite{BT82} and of Alexander and Taylor \cite{AT84} in pluripotential theory.

Let $\Omega \subset \mathbb{C}^n$ be a bounded domain  and $\omega = dd^c |z|^2$  the standard K\"ahler form in $\mathbb{C}^n$.  Denote by $W^{1,2}(\Omega, \mathbb{R})$ the usual Sobolev space. For  $f\in W^{1,2}(\Omega, \mathbb{R})$ we define $\Ga_f$  to be the set of all positive closed $(1,1)$-current $T$ satisfying
\[\label{eq:test-currents}\notag
	df \wedge d^c f \leq T \quad\text{weakly in } \Omega.
\]
Here we use the normalization
$$d^c = \frac{\sqrt{-1}}{2\pi} (\overline{\d} -\partial), \quad dd^c = \frac{\sqrt{-1}}{\pi}\partial\overline{\d}.
$$
Consider the subspace
\[ \label{eq:sobolev-space} \notag
W^*(\Omega)= \left\{f \in W^{1,2}(\Omega,\mathbb{R}):   \text{there exists $T\in \Ga_f$ with } \|T\|_\Omega <+\infty
\right\},
\]
where
$
	 \|T \|_\Omega := \int_\Omega T \wedge \omega^{n-1}.
$
For $f \in W^*(\Omega)$ one defines the $W^*$-norm
\[\label{eq:norm} \notag
	\|f\|_*^2 = \| f\|_{L^2(\Omega)}^2 + \inf_{T\in \Ga_f} \|T\|_{\Omega}.
\]
We then reprove basic functional results obtained in \cite{Vigny}. Namely, $(W^*(\Omega), \|\cdot\|_*)$ is a Banach space (Theorem~\ref{thm:star-norm}) and  continuous functions are not dense in $W^*(\Omega)$ with respect to strong topology of the norm $\|\cdot\|_*$ (Proposition~\ref{prop:density}). If $n=1$, then  $W^*(\Omega) = W^{1,2}(\Omega)$, otherwise $W^*(\Omega)$ is not reflexive (Corollary~\ref{cor:reflexive}). It should be pointed out that there are some similarity of these properties with the ones of the space of delta psh functions studied by Cegrell and Wiklund \cite{CW05}.

Inspired by the functional capacity in \cite{Vigny} we define for a Borel subset $E\subset \Omega$ the local capacity ${\tt c}(E):= {\tt c}(E,\Omega)$  as follows:
\[\label{eq:intro-cap-defn}\notag
	{\tt c}(E) = \inf
	\left\{ \| v \|_*^2 \;\rvert\;  v\in \cali{K}(E)
	\right\},
\]
where
\[\label{eq:intro-ke-cap}\notag
 	\cali{K}(E) = \left\{ v \in W^*(\Omega) \;\rvert\; \{v \leq  -1\}^o \supset E \text{ and } v\leq 0 \right\}.
\]
Here $ \{v \leq  -1\}^o \supset E$ means $v\leq -1$ a.e in a neighborhood of $E$.

The functional capacity in the Sobolev spaces is very well-understood which is also an effective tool to study fine properties of its element, see e.g. \cite[Chapters~4.7-4.8]{EG92}. In contrast, the use of ${\tt c}(\cdot)$ for the complex Sobolev space is not effective so far.
Our goal is to investigate systematically local properties of functions in $W^*(\Omega)$ by using this capacity. This is a different perspective compared to \cite{Vigny} and \cite{DMV} who used the relative capacity for psh functions due to Bedford and Taylor \cite{BT82}. Namely, for a Borel set $E\subset \Omega$,
\[\label{eq:intro-BT-cap}\notag
 cap(E,\Omega) = \sup\left\{ \int_E (dd^c u)^n :  u\in PSH(\Omega), \;-1 \leq u \leq 0\right\}.
\]
Our first main result explains the reason why $cap(\cdot, \Omega)$ could be effectively used in previous works. Roughly speaking they are equivalent to each other.

Let $(X,\omega)$ be a compact K\"ahler manifold. A natural subspace of the Sobolev space $W^{1,2}(X)$, called the complex Sobolev space $W^*$, was introduced by Dinh and Sibony \cite{DS} in their work on complex dynamics.
This new space turned out to be a good complex version of the former one as it takes into account the complex structure of manifolds. Many important families of functions which carry useful properties of underlying manifolds belong to the space. Among them are Lipschitz functions, bounded quasi plurisubharmonic (psh) functions or more general bounded delta quasi-psh function studied in \cite{CW05}. Basic functional aspects of this space were studied by Vigny \cite{Vigny} where he showed that it is a Banach space with a corresponding norm $\| \cdot\|_*$, this norm is stronger than the Sobolev norm. However, it is not reflexive and smooth functions are not dense in $W^*$ with respect to the strong topology of that norm. He also defined a functional capacity for this space and showed that it is a capacity in sense of Choquet. Moreover, the functional capacity is qualitative comparable to the relative capacity of Bedford and Taylor in global pluripotential theory defined in \cite{ko03}.

An element in a Sobolev space is defined almost everywhere up to a set of Lebesgue measure zero which is unlike psh or quasi-psh functions. Its fine properties with respect to the Sobolev capacity are classical ones \cite{EG92}, \cite{KLV-book}. However, this capacity is not good enough to characterize pluripolar sets, therefore it is not suitable for studying functions in $W^*$. One needs to work with the corresponding capacity of $W^*$ which dominates the Sobolev one. However, the new capacity in complex Sobolev spaces is harder to analyze. A crucial difference is that we can no longer rely on maximal function techniques, which are very powerful, in these spaces (see, e.g \cite{KLV-book}).

Recently, it has been showed in \cite{DMV}, \cite{Vigny} and \cite{VV24} that pluripotential theory is a suitable and powerful tool for studying $W^*$. As a result they made major progresses on study local properties of functions in such spaces.

Let $\Omega \subset \mathbb{C}^n$ be a bounded domain and $\omega = dd^c |z|^2$ the standard K\"ahler form in $\mathbb{C}^n$. Denote by $W^{1,2}(\Omega, \mathbb{R})$ the usual Sobolev space. For $f\in W^{1,2}(\Omega, \mathbb{R})$ we define $\Ga_f$ to be the set of all positive closed $(1,1)$-current $T$ satisfying
\[\label{eq:test-currents}\notag
 df \wedge d^c f \leq T \quad\text{weakly in } \Omega.
\]
Here we use the normalization
$$d^c = \frac{\sqrt{-1}}{2\pi} (\overline{\d} -\partial), \quad dd^c = \frac{\sqrt{-1}}{\pi}\partial\overline{\d}.
$$
Consider the subspace
\[ \label{eq:sobolev-space} \notag
W^*(\Omega)= \left\{f \in W^{1,2}(\Omega,\mathbb{R}): \text{there exists $T\in \Ga_f$ with } \|T\|_\Omega <+\infty
\right\},
\]
where
$
 \|T \|_\Omega := \int_\Omega T \wedge \omega^{n-1}.
$
For $f \in W^*(\Omega)$ one defines the $W^*$-norm
\[\label{eq:norm} \notag
 \|f\|_*^2 = \| f\|_{L^2(\Omega)}^2 + \inf_{T\in \Ga_f} \|T\|_{\Omega}.
\]
We then reprove basic functional results obtained in \cite{Vigny}. Namely, $(W^*(\Omega), \|\cdot\|_*)$ is a Banach space (Theorem~\ref{thm:star-norm}) and continuous functions are not dense in $W^*(\Omega)$ with respect to strong topology of the norm $\|\cdot\|_*$ (Proposition~\ref{prop:density}). If $n=1$, then $W^*(\Omega) = W^{1,2}(\Omega)$, otherwise $W^*(\Omega)$ is not reflexive (Corollary~\ref{cor:reflexive}). It should be pointed out that there are some similarity of these properties with the ones of the space of delta psh functions studied by Cegrell and Wiklund \cite{CW05}.

Inspired by the functional capacity in \cite{Vigny} we define for a Borel subset $E\subset \Omega$ the local capacity ${\tt c}(E):= {\tt c}(E,\Omega)$ as follows:
\[\label{eq:intro-cap-defn}\notag
 {\tt c}(E) = \inf
 \left\{ \| v \|_*^2 \;\rvert\; v\in \cali{K}(E)
 \right\},
\]
where
\[\label{eq:intro-ke-cap}\notag
 \cali{K}(E) = \left\{ v \in W^*(\Omega) \;\rvert\; \{v \leq -1\}^o \supset E \text{ and } v\leq 0 \right\}.
\]
Here $ \{v \leq -1\}^o \supset E$ means $v\leq -1$ a.e in a neighborhood of $E$.

The functional capacity in the Sobolev spaces is very well-understood which is also an effective tool to study fine properties of its element, see e.g. \cite[Chapters~4.7-4.8]{EG92}. In contrast, the use of ${\tt c}(\cdot)$ for the complex Sobolev space is not effective so far.
Our goal is to investigate systematically local properties of functions in $W^*(\Omega)$ by using this capacity. This is a different perspective compared to \cite{Vigny} and \cite{DMV} who used the relative capacity for psh functions due to Bedford and Taylor \cite{BT82}. Namely, for a Borel set $E\subset \Omega$,
\[\label{eq:intro-BT-cap}\notag
 cap(E,\Omega) = \sup\left\{ \int_E (dd^c u)^n : u\in PSH(\Omega), \;-1 \leq u \leq 0\right\}.
\]
Our first main result explains the reason why $cap(\cdot, \Omega)$ could be effectively used in previous works. Roughly speaking they are equivalent to each other.

\begin{cor} \label{cor:reflexive} Let $\Om$ be an open set in $\bC^n$,  $n\geq 2$. Then, $W^*(\Om)$ is not reflexive.
\end{cor}

\begin{prop} \label{prop:density} Let $\Om\subset \bC^n$ be a bounded domain.
\begin{itemize}
\item[(a)] If $n=1$, then $W^*(\Om) = W^{1,2}(\Om)$ together with its Sobolev norm.
\item[(b)] If $n\geq 2$, then the continuous functions in $C^0(\Om) \cap W^*(\Om)$ are not dense in $W^*(\Om)$ with respect to $W^*$-norm.
\end{itemize}
\end{prop}

\begin{thm}\label{thm:star-norm} Let $\Om$ be a bounded domain in $\bC^n$.
\begin{itemize}
\item[(a)]  $\| \cdot\|_*$ is a norm in $W^*(\Om)$ and $\| \cdot \|_{W^{1,2}(\Om)} \leq \|\cdot \|_*$.
\item[(b)] $W^*(\Om)$ and $W^*_0(\Om)$ are Banach spaces with this norm.
\end{itemize}
\end{thm}

The functional capacity in the Sobolev spaces is very well-understood which is also an effective tool to study fine properties of its element, see e.g. \cite[Chapters~4.7-4.8]{EG92}. In contrast, the use of $\tc(\cdot)$ for the complex Sobolev space is not effective so far.
Our goal is to investigate systematically local properties of functions in $W^*(\Om)$ by using this capacity. This is a different perspective compared to \cite{Vigny} and \cite{DMV} who used the relative capacity for psh functions due to Bedford and Taylor \cite{BT82}. Namely, for a Borel set $E\subset \Om$,
\[\label{eq:intro-BT-cap}\notag
 cap(E,\Om) = \sup\left\{ \int_E (dd^c u)^n : u\in PSH(\Om), \;-1 \leq u \leq 0\right\}.
\]
Our first main result explains the reason why $cap(\cdot, \Om)$ could be effectively used in previous works. Roughly speaking they are equivalent to each other.

In the local setting the proofs are different. The key idea in the proof of the first inequality is making use of Cegrell's inequality \cite{Ce04} and the one for the second inequality comes from the comparison principle. On the other hand, the optimality of the exponents is extracted from the Alexander-Taylor type inequality which is proved in Lemma~\ref{lem:AT-type}. The latter inequality is of independent interest. One of its applications is the existence of an entire psh subextension for a given $f \in W^*(B(0,1))$ in Theorem~\ref{thm:subextension}. This psh subextension result is an important ingredient to derive the property that the set Lebesgue points of such functions are of capacity zero. This result is due to Vigny and Vu \cite{VV24} where the bounded case was proved earlier in \cite{Vigny}. We improve the statement of that result and its proof is also simplified in Theorem~\ref{thm:VV}.

\begin{thm} \label{prop:equiv-est-intro} Let $K\subset\subset \Om$ be a compact subset. The following statements hold and they are equivalent to each other.
\begin{itemize}
\item[(a)]
There exist constants $A_1>0$ and $\al>0$ depending only on $K, \Om$ such that for every Borel set $E\subset K$ $$V_{2n}(E) \leq A_1 e^{-\al/\tc (E)}.$$

\begin{remark} \label{rmk:c-equiv} \mbox{}
\begin{enumerate}
\item[(a)] Let $E \subset E'$ be Borel sets in $\Om$. If $V_{2n}(E'\setminus E)=0$, then $\tc(E) = \tc(E')$. It is non-decreasing under holomorphic maps, i.e., if $h: G \to \Om$ is a holomorphic map, then
$$
 \tc (E,G) \leq \tc (h(E), \Om).
$$
\item[(b)] Let $D\subset\subset \Om' \subset \Om$.
We do not know if there exists a constant $A>0$ depending only on $\Om$ and $\Om'$ such that for every Borel set $ E\subset D$,
\[\label{eq:c-comparablity}
 \tc(E,\Om) \leq A\; \tc(E, \Om').
\]
However, we will have a weaker version with some exponent on the right hand side later
in Corollary~\ref{cor:dom-equiv-c} for the new capacity $\tc (\cdot, \Om)$.
\end{enumerate}
\end{remark}

\begin{lem}[dominated by Bedford-Taylor capacity] \label{lem:c-cap} Assume $\Om$ is strictly pseudoconvex. There exists a constant $A>0$ such that for every Borel set $E\subset\subset \Om$,
$$
 \tc(E) \leq A \left[cap(E,\Om)\right]^\frac{1}{n}.
$$
\end{lem}

\begin{proof} Let $u_E^*$ denote the relative extremal for $E$ in \eqref{eq:rel-ext-fct}.
Since both capacities are outer regular (Lemma~\ref{lem:outer-reg} and \cite{BT82}), we may assume that $E=G$ is an open set and hence $u_G = u_G^*$. Then, $-1\leq u_G \leq 0$ and $u_G =-1$ on $G$. It follows that
\[\label{eq:BT-dom-a}
 \tc(G) \leq \|u_G\|_*^2.
\]
Let $\rho$ be a strictly psh defining function for $\Om$. We may assume that $dd^c \rho \geq \om$. Since $G\subset \subset \Om$, it implies that $u_G =0$ on $\d\Om$ (see, e.g., \cite[Properties (13.11)]{De89}). Using $-1 \leq u_G \leq 0$ and integration by parts,
$$
 \|u_G\|_{L^2}^2 = \int (u_G)^2 (dd^c \rho)^n \leq \int - u_G (dd^c\rho)^n = \int -\rho dd^c u_G \wed (dd^c\rho)^{n-1}.
$$
Here and in the proof below we skip writing $\Om$ in the integral symbols
for simplicity. The last integral is less than
$$
 \|\rho\|_{L^\infty(\Om)} \int dd^c u_G \wed (dd^c\rho)^{n-1}.
$$
Since $0\leq u_G+1 \in PSH(\Om)$, we have
$d u_G \wed d^c u_G \leq \frac{1}{2} dd^c (1+u_G)^2.$
Hence,
\[\label{eq:BT-dom-b}\begin{aligned}
 \|u_G\|_*^2
&\leq \int (u_G)^2 (dd^c \rho)^n + \frac{1}{2}\int dd^c (1+u_G)^2 \wed (dd^c \rho)^{n-1} \\
&\leq A_1 \int dd^c u_G \wed (dd^c\rho)^{n-1} + \int dd^c u_G \wed (dd^c \rho)^{n-1},
\end{aligned}\]
where $A_1 = \|\rho\|_{L^\infty(\Om)}$ and we used the fact that
$$\begin{aligned}
& \frac{1}{2}\int dd^c (1+u_G)^2 \wed (dd^c \rho)^{n-1} \\
&=\int du_G \wed d^c u_G \wed (dd^c\rho)^{n-1} + \int (1+u_G) dd^c u_G \wed (dd^c \rho)^{n-1} \\
 &= \int dd^c u_G \wed (dd^c \rho)^{n-1}.
\end{aligned}
$$
Invoking an inequality of Cegrell \cite{Ce04} we have
\[\label{eq:BT-dom-c}\begin{aligned}
 \int dd^c u_G \wed (dd^c \rho)^{n-1}& \leq \left[\int (dd^c\rho)^n\right]^\frac{n-1}{n} \left[\int (dd^c u_G)^n\right]^\frac{1}{n} \\ &= A_0 [cap(G,\Om)]^\frac{1}{n},
\end{aligned}\]
where $A_0$ is a uniform constant as $\rho$ is smooth on $\bar\Om$ and we used the formula \eqref{eq:BT-cap-id} for the last identity.

By \cite[Th\'eor\`em~1]{Okada82} we have a Poincar\'e type inequality, for $f\in C^\infty_c(\Om)$,
$$
 \int f^2 d\mu = \int f^2 \te \wed dd^c |z|^2 \leq 8 \sup_\Om |z| \int df \wed d^c f \wed \te.
$$
It follows that $\cE_1^\te(f) = \| f \|^2_{L^2(d\mu)} + \cE^\te(f)$ is equivalent to $\cE^\te(f)$. As noted in \cite[page 211]{FO84} that potential theory of \cite[Chapter~2]{FOT94} can be formulated in terms of $\cE^\te$ instead of $\cE^\te_1$. For an open subset $E\subset \Om$ its capacity is given by
\[\label{eq:cap-te}
 \tc_\te (E) = \inf\{\cE(v) : v\in \cF, \; v\geq 1 \quad \mu-\text{\rm a.e.} \text{ on } E\}.
\]
For a general Borel set we let
$$
 \tc_\te(E) = \inf \{\tc_\te(G): E \subset G, \quad G \text{ is open subset in }\Om\}.
$$
In particular, for a compact subset $K\subset \Om$,
$$
 \tc_\te(K) = \inf\{ \cE^\te(v): v\in \cF\cap C^0(\Om), \; v\geq 1 \text{ on }K\}.
$$
Thank to special feature of the functional capacity in a Hilbert space it is a Choquet capcity \cite[Theorem~2.1.1]{FOT94}.
Moreover, we have a comparison between this capacity and the Bedford-Taylor capacity as follows.
\begin{lem}\label{lem:FO-polar} Let $\Om \subset \subset \bC^n$ be a strictly pseudoconvex domain and $0<\de \leq 1$.
\begin{itemize}
\item[(a)] For every compact subset $K\subset \Om$,
$$
 cap (K) \leq 8 \tc_\te (K),
$$
where $\te= (dd^c u_K^* +\de \om)^{n-1}$.
\item[(b)] Let $D\subset \subset \Om$ be a subdomain. There exists a constant $A_0= A(D,\Om)$ such that for every open subset $G\subset D$,
$$
 \tc_\te{(G)} \leq A_0 \left[cap (G)\right]^\frac{1}{n}.
$$
\end{itemize}
\end{lem}

\begin{remark}\label{rmk:sharp-DMV} The inequalities in Proposition~\ref{prop:equiv-est} are sharp as far as the exponents are concerned. In fact, it follows from the first inequality (a) and Lemma~\ref{lem:c-cap} that for a Borel set $E\subset \Om$,
$$
 V_{2n} (E) \leq A_1 e^{-\al/ \tc (E)} \leq A_1 e^{-\al/ [cap(E, \Om)]^\frac{1}{n}}.
$$
We know from \cite[Theorem~A]{ACKPZ09} that the exponent $1/n$ in the inequality between the volume and Bedford-Taylor capacity is sharp. So are the exponents in the inequalities of the proposition. Consequently, the exponent 2 in Proposition~\ref{prop:DMV} and \cite[Theorem~1.2]{DMV} is the optimal one for all $n\geq 1$.
\end{remark}

The statements of the theorem consist of Lemmas~\ref{lem:c-cap}, ~\ref{lem:cap-c} and Remarks~\ref{rmk:inner-reg},~\ref{rmk:sharp-exp}.
On compact K\"ahler manifolds both inequalities  was obtained in \cite[Proposition~5.1]{DKN} where ${\tt c}(E)$ is defined by another equivalent norm with $W^*$-norm, but the optimality was unknown.  A qualitative version  of the first inequality on such manifolds was obtained early in \cite{Vigny} and the second inequality was also proved there without showing that the exponents are sharp.

In the local setting the proofs are different. The key idea in the proof of the first inequality is making use of Cegrell's inequality \cite{Ce04} and the one for the second inequality comes from the comparison principle. On the other hand, the optimality of the exponents is extracted from the Alexander-Taylor type inequality which is proved in Lemma~\ref{lem:AT-type}. The latter inequality is of independent interest. One of its applications is the existence of an entire psh subextension for a given $f \in W^*(B(0,1))$ in Theorem~\ref{thm:subextension}. This psh subextension result is an important ingredient to derive the property that  the set Lebesgue points of such functions are of capacity zero. This result is due to Vigny and Vu \cite{VV24} where the bounded case was proved earlier in \cite{Vigny}. We improve the statement of that result and its  proof is also simplified in Theorem~\ref{thm:VV}.

An immediate consequence of Theorem~\ref{thm:intro-cap-comp} and \cite{BT82} is that a Borel set $P\subset \Omega$ is pluripolar if and only if ${\tt c}(P)=0$. Provided the equivalence of capacities  the results in \cite{DMV} can be restated in terms of ${\tt c} (\cdot)$ where they are obtained previously for $cap(\cdot, \Omega)$. However, as emphasized above we would take another route by using advantages of the functional capacity ${\tt c}(\cdot)$ (see e.g., Lemma~\ref{lem:outer-reg}, Proposition~\ref{prop:ae-qe}, Lemma~\ref{lem:cap-sublevel-set}) and hence providing alternative  proofs for the ones obtained in \cite{DMV}. In particular,  using the ideas in \cite{DMV} and \cite{Vigny},  we  simplify  the proof of quasi-continuous representative result proved earlier in \cite[Theorem~2.10]{DMV} (see also \cite[Theorem~22]{Vigny}).

\begin{cor} \label{cor:intro-quasi-mod} Let  $f\in W^*(\Omega)$. There exists a quasi-continuous modification $\wt f$ such that $\wt f = f$ a.e. Moreover, if $g$ is another quasi-continuous modification of $f$, then $g =\wt f$ outside a pluripolar set.
\end{cor}

The most technical part of the proof  lies in delicate integral estimates. Along the way  we refine the ones obtained in \cite{DMV}. It turns out in  Theorem~\ref{thm:VV}  that the precise representative, in the sense of  Sobolev spaces \cite[Chapter~4]{EG92},
\[\label{eq:rep} f^*(x) \equiv
\begin{cases}
\lim_{r\to 0} \intavg_{B(x,r)} f \; d y &\quad \text{if this limit exits,} \\
=0 &\quad \text{otherwise},
\end{cases}
\]
is a quasi-continuous modification, where $d y$ is the Lebesgue measure on $\mathbb{C}^n = \mathbb{R}^{2n}$.

Thanks to the quasi-continuity result
 we can define the integration of a function $g\in W^*(\Omega)$ against  Monge-Amp\`ere measures associated with a bounded psh function (Definition~\ref{defn:L2-ma}). This result is pushed much further in Section~9 where we show that $W^*(\Omega)$ is  naturally embedded  into a Dirichlet space associated with a closed positive  $(n-1,n-1)$ current $
\theta = [dd^c (\phi + \delta |z|^2)]^{n-1}$ where $\phi \in PSH \cap L^\infty(\Omega)$ and $\delta>0$ are given.
Therefore, the integral  $\int_K df \wedge d^c g\wedge \theta$ for $f,g\in W^*(\Omega)$ is well-defined on every compact set $K\subset \Omega$. This aspect has been considered recently by Do, Nguyen and Vu \cite{DNV25}.

The second main result is as follows.
\begin{thm}\label{thm:intro-choquet-c} Let $\Omega$ be a bounded open set in $\mathbb{C}^n$. The set function on Borel subsets $E\mapsto {\tt c} (E)= {\tt c} (E,\Omega)$ is a Choquet capacity.
\end{thm}

This result is the local analogue of \cite{Vigny}. On compact K\"ahler manifolds the proof of  \cite[Theorem~30]{Vigny} employed the capacity notion in the Dirichlet spaces \cite[Lemma~23]{Vigny} and the characterization of pluripolar sets via a family of capacity associated with closed positive current \cite{FO84}.  Here we use a similar strategy. Notably, we are able to give a simpler proof of the characterization in Corollary~\ref{cor:FO-polar} by using only pluripotential theory.

Another remarkable property of functions in the complex Sobolev space is the uniformly exponential integrability due to Dinh, Marinescu and Vu  \cite{DMV}, which is very close to the one of psh functions.  Namely,
there exist positive constants $\alpha$ and $A$  such that for every $f \in W^*(B(0,1))$ with $\|f\|_*\leq 1$,
\[\label{eq:DMV-intro}
	\int_{\bar B(0,\frac{1}{8})} e^{\alpha |f|^2} d x \leq A.
\]
This result should be compared with the well-known exponential integrability of psh functions due to H\"ormander \cite[Proposition~4.2.9]{Ho07} and Skoda \cite{Sk72}. In the proof of  \cite[Theorem~1.2]{DMV}  the authors used the induction argument in dimension and slicing theory for positive currents. We provide a new and simpler one in Proposition~\ref{prop:DMV} by reducing the inequality to one dimensional case. In this way, it is enough  to  work with smooth functions after taking  the standard convolution with smooth kernels. The constants can be explicitly computed though they  are suboptimal.  Additionally, we point out in Remark~\ref{rmk:sharp-DMV} that the exponent 2 of $|f|$ in \eqref{eq:DMV-intro} is the optimal for all $n\geq 1$, which was only known for $n=1$ in \cite{DMV}.

The third main result fully characterizes the inequality \eqref{eq:DMV-intro} by the volume-capacity inequality of sublevel sets.

\begin{thm} \label{prop:equiv-est-intro} Let $K\subset\subset \Omega$ be a compact subset.  The following statements hold and they are equivalent to each other.
\begin{itemize}
\item[(a)]
There exist constants $A_1>0$ and $\alpha>0$ depending only on $K, \Omega$ such that for every  Borel  set $E\subset K$ $$V_{2n}(E) \leq A_1 e^{-\alpha/{\tt c} (E)}.$$

\item[(b)] There exist uniform constants $A_1>0$ and $\alpha>0$ depending only on $K, \Omega$ such that for every  $f\in W^*(\Omega)$ and $f\leq 0$,
 $$V_{2n}(\{f< -1\} \cap K) \leq A_1 e^{\frac{-\alpha}{\|f\|_*^{2}}}.$$
\item
[(c)]   There exist uniform constants $A_1>0$ and $\alpha>0$ depending only on $K, \Omega$ such that for every $f\in W^*(\Omega)$ and $f\leq 0$ whose norm $\|f\|_* \leq
\kappa$,
$$V_{2n} (\{f < -s\} \cap K) \leq A_1 e^{-\alpha s^2/\kappa^2}, \quad \forall s>0.$$

\item
[(d)]  There exist uniform constants $A_1>0$ and $\alpha>0$ depending only on $K, \Omega$ such that for every $f\in W^*(\Omega)$ and $f\leq 0$ with $\|f\|_* \leq 1$,
$$V_{2n} (\{f < -s\} \cap K) \leq A_1 e^{-\alpha s^2}, \quad \forall s>0.$$
\end{itemize}
\end{thm}

The characterization is partially inspired by the work of \AA hag, Cegrell, Ko\l odziej, Pham and Zeriahi \cite{ACKPZ09}. Lastly, we point out in Remark~\ref{rmk:DMV-holder} that the inequalities holds for a very large family  of Monge-Amp\`ere measures associated with  H\"older continuous psh functions.

\bigskip

{\em Acknowledgement.} I would like to thank G. Marinescu and D.-V. Vu for a kind invitation to visit University of Cologne in September 2025 and to whom I had many helpful discussions on the results of \cite{DMV}.  I am grateful to S. Ko\l odziej and D.-V. Vu for useful comments on the draft of the paper. It was written while the author  visited the Center for Complex Geometry (Daejeon). He would like to thank Jun-Muk Hwang and Yongnam Lee for their kind support and exceptional hospitality. He is also grateful to the institution for providing perfect working conditions.

For Theorem~\ref{thm:intro-cap-comp}, the text says that its statements are provided by Lemmas~\ref{lem:c-cap}, \ref{lem:cap-c} and Remarks~\ref{rmk:inner-reg}, \ref{rmk:sharp-exp}. In the local setting, “the key idea in the proof of the first inequality is making use of Cegrell's inequality \cite{Ce04}” while “the one for the second inequality comes from the comparison principle.” The “optimality of the exponents is extracted from the Alexander-Taylor type inequality which is proved in Lemma~\ref{lem:AT-type}.”

To prove the main theorem, the text says that its statements are provided by the following lemmas and remarks.

\begin{lem}[dominated by Bedford-Taylor capacity]  \label{lem:c-cap} Assume $\Om$ is strictly pseudoconvex. There exists a constant $A>0$ such that for every Borel set $E\subset\subset \Om$,
$$
	\tc(E) \leq A \left[cap(E,\Om)\right]^\frac{1}{n}.
$$
\end{lem}

\begin{lem} \label{lem:cap-c} Let $D \subset \subset \Om$. There exists  a constant $A'$ depending  only on $D$ and $\Om$ such that for every Borel set $E \subset D$,
$$	cap(E,\Om) \leq A'\, \tc(E)
$$
\end{lem}

\begin{remark}\label{rmk:inner-reg} The relative compactness condition of $E$ will be removed later after proving that $\tc(\cdot,\Om)$ is a capacity in the sense of Choquet (Theorem~\ref{thm:choquet-c}).
\end{remark}

\begin{remark}[Proof of the last statement in Theorem~\ref{thm:intro-cap-comp}]
\label{rmk:sharp-exp} The Alexander-Taylor inequality \cite{AT84} reads
$$
	\exp (-A_r/cap(K,\Om)) \leq T_R(K) \leq \exp(-2\pi/[cap(K,\Om)]^\frac{1}{n}),
$$
where the exponents in both inequalities are sharp (\cite[Remark~2]{AT84}).
Together with $cap(K) \leq A_r \tc(K)$ in Lemma~\ref{lem:cap-c}
and the first inequality in Lemma~\ref{lem:AT-type} we derive
$$
	\exp (-A/cap(K,\Om)) \leq \exp (-A/\tc(K,\Om))    \leq \exp(-2\pi/[cap(K,\Om)]^\frac{1}{n})
$$
with the sharp exponents. So are the ones in Lemma~\ref{lem:c-cap} and Lemma~\ref{lem:cap-c}.
\end{remark}

In the local setting, “the key idea in the proof of the first inequality is making use of Cegrell's inequality \cite{Ce04}” while “the one for the second inequality comes from the comparison principle.” The “optimality of the exponents is extracted from the Alexander-Taylor type inequality which is proved in the following lemma.”

\begin{lem} \label{lem:AT-type} There exist positive constants $A_r$ and $A_R$ depending on $r,R$ respectively such that for every compact  set $K\subset B_r$,
$$ \exp(- A_R \; \tc (K,B_R)^{-1}) \leq  T_R(K) \leq 	\exp(- A_r\;\tc (K, B_R)^{-1/n}).
$$
Equivalently,
\[\label{eq:AT}
	\frac{1}{A_r} \frac{1}{M_K^n} \leq \tc (K,B_R) \leq \frac{A_R }{M_K}.
\]
\end{lem}

\label{thm:intro-cap-comp}
Let $\Omega \subset \subset \mathbb{C}^n$ be a strictly pseudoconvex domain.
\begin{itemize}
\item[(a)] There exists a constant $A>0$ such that for every Borel set $E\subset \Omega$,
$$
	{\tt c}(E) \leq A [cap(E,\Omega)]^\frac{1}{n}.
$$
\item[(b)]
Assume $D \subset \subset \Omega$ be a subdomain. There exists a constant $A'$ such that for every Borel set $E\subset D$,
$$
	\frac{1}{A'} cap(E,\Omega) \leq {\tt c} (E).
$$
\end{itemize}
Moreover, the above inequalities are sharp as far as the exponents are concerned.,

Existential–Universal
Inequality or Bound

Question:
Let \(\Omega\Subset \mathbb C^n\) be a strictly pseudoconvex domain. For a Borel set \(E\subset \Omega\), define the Bedford--Taylor capacity by
\[
\operatorname{cap}(E,\Omega)=\sup\left\{\int_E (dd^c u)^n:\ u\in PSH(\Omega),\ -1\le u\le 0\right\}.
\]
Also define the local functional capacity
\[
\mathbf c(E)=\inf\{\|v\|_*^2:\ v\in \mathcal K(E)\},
\]
where
\[
\mathcal K(E)=\left\{v\in W^*(\Omega): v\le 0\text{ on }\Omega,\ \{v\le -1\}^o\supset E\right\},
\]
with \(\{v\le -1\}^o\supset E\) meaning that \(v\le -1\) almost everywhere in a neighborhood of \(E\), and where
\[
W^*(\Omega)=\left\{f\in W^{1,2}(\Omega,\mathbb R): \exists\text{ a positive closed }(1,1)\text{-current }T\text{ with }df\wedge d^cf\le T\text{ weakly and }\|T\|_\Omega<\infty\right\},
\]
\[
\|f\|_*^2=\|f\|_{L^2(\Omega)}^2+\inf_{T\in \Gamma_f}\|T\|_\Omega,
\qquad \|T\|_\Omega=\int_\Omega T\wedge \omega^{n-1}.
\]
Under these assumptions, which quantitative comparison between \(\mathbf c(E)\) and \(\operatorname{cap}(E,\Omega)\) holds?

There exists a constant \(A>0\) such that for every Borel set \(E\subset \Omega\),
\[
\mathbf c(E)\le A\,[\operatorname{cap}(E,\Omega)]^{1/n}.
\]
Moreover, if \(D\Subset \Omega\) is any subdomain, then there exists a constant \(A'>0\) such that for every Borel set \(E\subset D\),
\[
\frac{1}{A'}\operatorname{cap}(E,\Omega)\le \mathbf c(E).
\]
In addition, these inequalities are sharp with respect to the exponents.

Choices:
B.
There exists a constant \(A>0\) such that for every Borel set \(E\subset \Omega\),
\[
\mathbf c(E)\le A\,\operatorname{cap}(E,\Omega).
\]
Moreover, if \(D\Subset \Omega\) is any subdomain, then there exists a constant \(A'>0\) such that for every Borel set \(E\subset D\),
\[
\frac{1}{A'}\,[\operatorname{cap}(E,\Omega)]^{1/n}\le \mathbf c(E).
\]
In addition, these inequalities are sharp with respect to the exponents.
C.
There exists a constant \(A>0\) such that for every Borel set \(E\subset \Omega\),
\[
\mathbf c(E)\le A\,[\operatorname{cap}(E,\Omega)]^{1/n}.
\]
Moreover, if \(D\Subset \Omega\) is any subdomain, then there exists a constant \(A'>0\) such that for every Borel set \(E\subset D\),
\[
\frac{1}{A'}\operatorname{cap}(E,\Omega)\le \mathbf c(E).
\]
D.
There exists a constant \(A>0\), depending only on \(\Omega\), such that for every subdomain \(D\Subset \Omega\) and every Borel set \(E\subset D\),
\[
\mathbf c(E)\le A\,[\operatorname{cap}(E,\Omega)]^{1/n}
\quad\text{and}\quad
\frac{1}{A}\operatorname{cap}(E,\Omega)\le \mathbf c(E).
\]
In particular, the same constant works uniformly for all choices of \(D\Subset \Omega\), and the exponents are sharp.
E.
There exists a constant \(A>0\) such that for every Borel set \(E\subset \Omega\),
\[
\mathbf c(E)\le A\,[\operatorname{cap}(E,\Omega)]^{1/n}.
\]
Moreover, there exists a constant \(A'>0\) such that for every Borel set \(E\subset \Omega\),
\[
\frac{1}{A'}\operatorname{cap}(E,\Omega)\le \mathbf c(E).
\]
In addition, these inequalities are sharp with respect to the exponents.

dependence of lower-bound constant on the subdomain D

restriction of the reverse inequality to E\subset D\Subset\Omega

The stem gives only the definitions of the two capacities and asks for the valid comparison statement; it does not explicitly state the correct inequalities, exponents, or quantifier structure.

The item is close to a theorem-identification question: it asks for the exact comparison result rather than an application or derivation. However, it is not a pure restatement because the options differ in exponents, constant dependence, and sharpness claims.

Solving it requires moderate reasoning or precise theorem recall to distinguish subtle alternatives, especially the exponent swap, the need to restrict the lower bound to subdomains, and the sharpness clause. It does not strongly force original generative reasoning beyond discriminating among variants.

The distractors are plausible and mathematically meaningful: they target common failure modes such as swapped exponents, omission of one inequality, incorrect uniformity in constants, and replacing comparison by exact equality.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise recall/recognition than deep generative reasoning.