Two amplitudes in the spectral action matrix model are
\begin{align*}
	\raisebox{-20pt}{
\begin{tikzpicture}[thick]
\draw (0.45,-0.7) to (1,0);
\draw (1.55,-0.7) to (1,0);
\draw (1,0) arc (-90:270:0.5cm);
\node at (1,-0.55) {$i$};
\node at (0.2,0.8) {$j$};
lldraw (1,0) circle (4pt);
lldraw[white] (1,0) circle (2pt)
	;
\end{tikzpicture}
}
\quad=\quad\sum_{k=1}^N
	\raisebox{-20pt}{
\begin{tikzpicture}[thick]
	\draw (0.45,-.7) to (1,0);
	\draw (1.55,-.7) to (1,0);
	\draw (1,0) arc (-90:270:0.5cm);
	\node at (1,-.55) {$i$};
	\node at (0.2,0.8) {$j$};
	lldraw (1,0) circle (4pt);
lldraw[white] (1,0) circle (2pt)
	;
        \node at (1,.575) {$k$};
\end{tikzpicture}
}\quad=\quad\sum_{k=1}^N\frac{f'[\lambda_j,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]},
\end{align*}
and
\begin{align*}
	\raisebox{-20pt}{
\begin{tikzpicture}[thick]
	\draw (0.45,-.7) to (1,0);
	\draw (1.55,-.7) to (1,0);
	\draw (1,0.054) arc (-90:270:0.3cm);
	\draw (1,-0.035) arc (-90:270:.7cm);
	\node at (1,-.55) {$i$};
	\node at (0,1) {$j$};
	lldraw (1,0) circle (4pt);
lldraw[white] (1,0) circle (2pt)
	;
\end{tikzpicture}
}
\quad=\quad\sum_{k,l=1}^N
	\raisebox{-20pt}{
\begin{tikzpicture}[thick]
	\draw (0.45,-.7) to (1,0);
	\draw (1.55,-.7) to (1,0);
	\draw (1,0.054) arc (-90:270:0.3cm);
	\draw (1,-0.035) arc (-90:270:.7cm);
	\node at (1,-.55) {$i$};
	\node at (0,1) {$j$};
	\node at (1,1) {$k$};
	\node at (1,0.4) {$l$};
	lldraw (1,0) circle (4pt);
lldraw[white] (1,0) circle (2pt)
	;
\end{tikzpicture}
}\quad=\quad\sum_{k,l=1}^N\frac{f'[\lambda_j,\lambda_k,\lambda_l,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]f'[\lambda_k,\lambda_l]}.
\end{align*}

\begin{tikzpicture}[thick]
\draw (-1.5,0) to (-1,0);
\draw (-1,0) to[out=90,in=90] (0,0);
\draw (-1,0) to[out=-90,in=-90] (0,0);
\draw (1,0) to[out=90,in=90] (0,0);
\draw (1,0) to[out=-90,in=-90] (0,0);
\draw (1,0) to (1.5,0);
\vertex{-1,0};
\vertex{0,0};
\vertex{1,0};
\node at (-1.65,0) {\footnotesize $1$};
\node at (1.65,0) {\footnotesize$2$};
\end{tikzpicture}$$
does not appear, because its two unbroken faces do not share an edge.
\end{rem}

\section{Power counting when allowing eigenvalues with vanishing derivative}\label{sct:main2}
In our first main theorem, Theorem \ref{thm:main}, we assumed that $f'[\lambda_{k_1},\ldots,\lambda_{k_n}]\neq0$. With $\{\lambda_k\}_{k\geq1}$ in general position, this assumption is typically satisfied. However, there will be at least one~${x\in\R}$ such that $f'(x)=0$, which means that the assumption of nonvanishing divided differences is sensitive to infinitesimal changes in $\{\lambda_k\}_{k\geq1}$. In fact, we shall show that the actual order of divergence of many graphs is sensitive to the question of whether or not $f'$ vanishes on $\{\lambda_k\}_{k\geq1}$.

We will refer to faces with an index $i_0$ such that $f'(\lambda_{i_0})=0$ as \emph{0-faces} or \emph{singular faces}, and similarly for indices and eigenvalues.
As in Section~\ref{sect: lower}, given a subset $\bfr\subseteq\mathcal U$ of the set of unbroken faces of a Feynman ribbon graph~$G$, we let~$G_{\bfr}$ be the graph obtained from $G$ by artificially declaring the faces in $\mathfrak{b}$ to be broken. Elements of $\mathfrak{b}$ are called 0-faces.\begin{thm}\label{thm:main2}
Let $f\in C^{\infty}(\mathbb{R})_\R$ be an even function with $f'$ of precise order $-p-1$ for some $p \in\mathbb{R}_{\ge0}$.
Let~$\{\lambda_k\}_{k=1}^\infty$ be a sequence of real numbers with constants $K,c_1,c_2>0$ such that $c_1k^{1/d}\leq |\lambda_k|\leq c_2k^{1/d}$ for all $k\geq K$. Assume that for all $k,l\in\N_{\geq1}$ we have $f'[\lambda_{k},\lambda_{l}]\neq0$. Then, for any Feynman ribbon graph $G=(G^0,n,G^1)$ whose vertices have valence $\geq3$, for all external indices $i_1,\ldots,i_n\in\N_{\geq1}$, there exist $M,c_4>0$ such that for all~$N\geq M$,
$$|\Ampl_{N,i_1,\ldots,i_n}(G)|\leq c_4 N^{\tilde\omega(G)},$$
$$\tilde \omega(G):=\max_{\bfr\subseteq\mathcal U}\omega_{\bfr}(G_{\bfr}):=\max_{\bfr\subseteq\mathcal U}(U^{\bfr}+\frac{p}{d}(E^{\bfr}_{\textnormal{fi}}-V^{\bfr}_{\textnormal{fi}})+\frac{p+1}{d}(E^{\bfr}_{10}-V^{\bfr}_{10})),$$
where $U^{\bfr}$ is the number of unbroken faces of $G_{\bfr}$, $E^{\bfr}_{\textnormal{fi}}$ is the number of fully internal edges of $G_{\bfr}$ (propagators bordered on both sides by unbroken faces) and $V^{\bfr}_{\textnormal{fi}}$ is the number of fully internal vertices of $G_{\bfr}$ (vertices bordered on all sides by unbroken faces). Respectively, $E^{\bfr}_{10}$ and $V^{\bfr}_{10}$ are the number of edges and vertices of~$G_{\bfr}$ that border exactly one 0-face and for the rest unbroken faces.
\end{thm}
\begin{proof}
Using the precise order of $f'$, there exists $R>0$ such that, for all $x\in\mathbb{R}\setminus[-R,R]$ and all~$0\le k\le \max_{v\in G^{0}}\;\deg(v)$, equation \eqref{eqn: precise_order} is satisfied and the conclusion of Lemma \ref{lem:zero-index bounds} holds. Let~$i_R$ be the lowest index such that $|\lambda_i| > R$ for $i\ge i_R$. For brevity we assume -- without loss of generality -- that $f'(\lambda_0)=0$ and $f'(\lambda_k)\neq0$ for~${k\geq 1}$. As in the proof of Lemma~\ref{lem: restricted_lower_sufficient}, a sum splitting argument yields
\begin{equation*}
    \Ampl_{N,i_1,\ldots,i_n}(G) = \sum_{\bfr\subseteq \mathcal{U}}\sum_{\gamma:\bfr\to\{0,\dotsc,i_R-1\}}\Ampl^{\ge i_R}_{N,{i_1},\dotsc,i_n,\gamma}(G_{\bfr}).
\end{equation*}
We claim that it suffices to show that \begin{align}\label{eqn: ampl_Gbgamma}
|\Ampl^{\ge i_R}_{N,{i_1},\dotsc,i_n,\gamma}(G_{\bfr})|&\lesssim N^{\omega_{\bfr,\gamma}(G_{\bfr})},\\
\omega_{\bfr,\gamma}(G_{\bfr})&:=(U^{\bfr}+\frac{p}{d}(E_{\textnormal{fi}}^b-V_{\textnormal{fi}}^b)+\frac{p+1}{d}(E_{10}^{\bfr,\gamma}-V_{10}^{\bfr,\gamma})),\nonumber
\end{align}
as the following graph-theoretical argument shows that $\omega_{\bfr,\gamma}(G_{\bfr})\le \omega_{\bfr}(G_{\bfr})$ for all such $\gamma$. In particular, if we let $\gamma_0$ denote the map that sends all artificially broken faces to the zero index, then $\omega_{\bfr,\gamma_0}(G_{\bfr})= \omega_{\bfr}(G_{\bfr})$, hence this bound is sharp.

More generally, we have $\omega_{\bfr}(G_{\bfr,\gamma})\leq\omega_{\bfr}(G_{\bfr,\gamma'})$ whenever $\gamma'$ is obtained from $\gamma$ by setting \begin{equation*}
    \gamma'(b)=\begin{cases}
        0,&b=b', \\
        \gamma(b),&\text{else},
    \end{cases}
\end{equation*}
for some $b'$ with $\gamma(b')\neq0$. Indeed, as the values $U^{\bfr}$, $\Efi^b$, and $\Vfi^b$ do not depend on $\gamma$, it suffices to show \begin{equation*}
E_{10}^{\bfr,\gamma}-V_{10}^{\bfr,\gamma}\leq E_{10}^{\bfr,\gamma'}-V_{10}^{\bfr,\gamma'}.
\end{equation*}
Towards this, first note that $E_{10}^{\bfr,\gamma'}-E_{10}^{\bfr,\gamma}$ equals the number of edges that border the face $b'$ on the one side and border an unbroken face (i.e.\ an element of $\mathcal{U}\setminus\bfr$) on the other. Moreover, $V_{10}^{\bfr,\gamma'}-V_{10}^{\bfr,\gamma}$ equals the number of vertices that border $b'$ and for the rest only unbroken faces. For each distinct vertex of the latter kind there is at least one distinct edge of the former kind, implying that $$ V_{10}^{\bfr,\gamma'}-V_{10}^{\bfr,\gamma}\leq E_{10}^{\bfr,\gamma'}-E_{10}^{\bfr,\gamma},$$which concludes the argument.

It remains to prove~\eqref{eqn: ampl_Gbgamma}. The strategy is similar to the proof of Theorem \ref{thm:upper}, but with different bounds for the edges and vertices, in which we distinguish three different types of indices and their corresponding eigenvalues.
We distinguish between (i) eigenvalues $\lambda_0$ such that $f'(\lambda_0)=0$, (ii) eigenvalues $\lambda_i$ for which $f'(\lambda_i)\neq0$ and the index $i<R$ is fixed, and (iii) eigenvalues $\lambda_k$ for which $f'(\lambda_k)\neq0$ and the index $k\geq R$ is running. Analogously to the proof of Theorem \ref{thm:upper}, noting Remark \ref{rem:apparent shortcut} and its resolution, the above estimate follows from the bounds on edges and vertices that are summarized and proved in Lemma \ref{lem:final bounds} below, where we stress that all constants that arise are independent of the running indices.
\end{proof}

\begin{lem}\label{lem:final bounds}
Let $n\geq3$. Let $f,\{\lambda_k\}_{k=1}^\infty$ be as in Theorem \ref{thm:main2}. Let $\lambda_0\in\R$ be such that $f'(\lambda_0)=0$. There exist $c_1,c_2>0$ such that we have
\begin{multicols}{2}
\begin{enumerate}
\item $|f'\{\lambda_{k_1},\lambda_{k_2}\}|>c_1|\lambda_{k_1}|^{-p}$,
\item $|f'\{\lambda_{i},\lambda_{k}\}|>c_1$,
\item $|f'\{\lambda_{i_1},\lambda_{i_2}\}|>c_1$,
\item $|f'\{\lambda_0,\lambda_k\}|> c_1|\lambda_k|^{-p-1}$,
\item $|f'\{\lambda_0,\lambda_i\}|>c_1$,
\item $|f'\{\lambda_0,\lambda_0\}|>c_1$,
\item $|f'\{\lambda_{k_1},\ldots,\lambda_{k_n}\}|\leq c_2|\lambda_{k_1}|^{-p}$,
\item $|f'\{\lambda_0,\lambda_{k_2},\ldots,\lambda_{k_n}\}|\leq c_2|\lambda_{k_2}|^{-p-1}$,
\item $|f'\{\lambda_{j_1},\ldots,\lambda_{j_n}\}|\leq c_2$,
ll}]
ll}]
ll}]
\end{enumerate}
\end{multicols}
\noindent
for all $i,i_1,i_2\leq i_R$, $k,k_1,\ldots,k_n\geq i_R$ such that $f'(\lambda_j)\neq0$ for $j\in\{i,i_1,i_2,k,k_1,\ldots,k_n\}$ and such that $|\lambda_{k_1}|\leq\cdots\leq|\lambda_{k_n}|$, and for all $j_1,\ldots,j_n\in\N$.
\end{lem}

\begin{proof}
The estimates 1.\ and 7.\ follow from Theorem~\ref{thm:weighted_divdiff_bound}, 4.\ and 8.\ follow from Lemma~\ref{lem:zero-index bounds}, and 9.\ was shown in Corollary~\ref{cor:wdivdifs are bounded}. Estimates 3., 5., and 6.\ follow from the fact that there is a finite amount of indices $i<i_R$, and for each $i_1,i_2<i_R$ the (weighted) divided difference is nonzero. {Estimate 2.\ follows from Theorem~\ref{thm:no_vanishing_divdifs}.}
\end{proof}
We conclude this section with some remarks and examples illustrating the influence of 0-faces on the behaviour of amplitudes.

\begin{rem}
The following explains the absence of a lower bound in Theorem \ref{thm:main2}. Suppose $i_1,i_2,i_3$ are such that $f'[i_1,i_2,i_3]=0$, $f'(\lambda_{i_1})\neq0$, and $f'(\lambda_{i_3})\neq0$. We then compute the amplitude
\begin{align*}
\raisebox{-19.5pt}{
\begin{tikzpicture}[thick]
\draw (-0.5,0.5) to (0,0);
\draw (-0.5,-.5) to (0,0);
\draw (0,0) to[out=60,in=120] (1,0);
\draw (0,0) to[out=-60,in=-120] (1,0);
\draw (1,0) to (1.5,0);
\node at (0,0.5) {$i_1$};
\node at (-.5,0) {$i_2$};
\node at (0,-.5) {$i_3$};
\vertex{0,0};
\vertex{1,0};
\end{tikzpicture}

\begin{align}\label{eq:power counting formula 2}
\tilde \omega(G)=\max(U^{\bfr}+\frac{p}{d}(E^{\bfr}_{\textnormal{fi}}-V^{\bfr}_{\textnormal{fi}})+\frac{p+1}{d}(E^{\bfr}_{10}-V^{\bfr}_{10})),
\end{align}
in which the maximum is taken over all subsets $\bfr$ of unbroken faces, $U^{\bfr},\Efi^b,\Vfi^b$ are as before when designating the elements of $\bfr$ as broken, and $V^{\bfr}_{10}/E^{\bfr}_{10}$ denotes the number of vertices/edges bordering only unbroken faces except for either exactly one element of $\bfr$ or exactly one external index $i_0$ satisfying~${f'(\lambda_{i_0})=0}$.

\paragraph{Consequences of \eqref{eq:power counting formula}}
Note that $\Efi-\Vfi\geq 0$ for any graph. A remarkable result is that a function~$f$ with \textit{faster} decay results in a \textit{higher} degree of divergence of the graph. This is not obvious from the definition of the Feynman rules, even for simple graphs like in Example 1, and even for concrete functions like $f(x)=x^{-p}$.
Because divided differences of such functions may well be negative, one might \textit{a priori} expect cancellations of terms damping the degree of divergence, but this does not happen.

Another corollary of our power counting formula is the following observation. Among the connected graphs with~$n\geq1$ external edges and $L$ loops, the maximal value of $U$ is $L-1$, and the maximal value of $\Efi-\Vfi$ is $L-1$. The maximal value $\omega(G)=L-1+\frac{p}{d}(L-1)$ is attained by a nonempty set of diagrams, by virtue of our lower bound. These diagrams with maximal divergence are precisely the planar diagrams that cannot be split into two connected components by removing one vertex and all external edges, and moreover have only one unbroken face -- cf.\ Remark \ref{rem:2-point 2-loop}.
\paragraph{Consequences of \eqref{eq:power counting formula 2}}
The situation where divided differences of $f'$ may vanish is more complicated (cf. Section \ref{sct:main2}), but \eqref{eq:power counting formula 2} leads us to conjecture that the diagrams with maximal divergence are of the same planar form as before -- cf.\ Remark \ref{rem:conjectures}. As we discuss in Remark \ref{rem:UV/IR}, the influence of modes $\lambda_i$ with $f'(\lambda_i)=0$ is reminiscent of the UV/IR behavior of scalar field theories on noncommutative spacetime, which underlines the need for a rigorous renormalization analysis of the spectral action beyond the weak field approximation.

\paragraph{Techniques} To prove the above power counting formulas, we introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them, which appear to be novel. A pivotal concept throughout the proof is the weighted divided difference
$$f'\{x_1,\ldots,x_n\}:=(-1)^n x_1\cdots x_n f'[x_1,\ldots,x_n].$$
Its asymptotic behavior is more easily understood, because sending any of its variables to infinity yields another weighted divided difference.
Furthermore, a key result is that weighted divided differences are positive on large enough subsets of $\R^n$ for the functions $f$ we are considering. We thus generalize and give a new proof for Hunter's positivity theorem \cite{Hunter1977}, which is recovered by taking $f(x)=x^{-p}$ ($p\in 2\N$). Moreover, we show that for functions bounded above or below by $x^{-p}$, with similar bounds on their derivatives, the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus, which is crucial in the proof of our main theorem.

Our main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.

\paragraph{Acknowledgements}
We are grateful to Martijn Caspers, Séverin Charbonnier, Harald Grosse, and Walter van Suijlekom for useful discussions. TvN thanks the Erwin Schr\"odinger Institute and the organizers and participants of the April 2023 conference 'Non-commutative Geometry meets Topological Recursion', where the motivation for this paper originated.
TvN was supported by NWO project ‘Noncommutative multi-linear harmonic analysis and higher order spectral shift’,
OCENW.M.22.070. EMH thanks the Max Planck Institute for Mathematics in Bonn for its financial support.

\section{Definitions}
\label{sct:Definitions}

Throughout we fix a positive real number $d\in\R_{>0}$, and a sequence $\{\lambda_k\}_{k=1}^\infty\subseteq\R$ with the property that there exist numbers $K\in\N$, $c_1,c_2\in\R_{>0}$ such that
\begin{align}\label{eq:eigenvalues asymptotics assumption}
    c_1 k^{1/d}\leq|\lambda_k|\leq c_2 k^{1/d}
\end{align}

\begin{align}\label{eq:power counting formula}
\omega(G)= U+\frac{p}{d}(E_{\f}-V_{\f}).
\end{align}

\begin{rem}\label{rem:UV/IR}
Smooth even functions $f$ satisfy $f'(0)=0$, and if $\{\lambda_k\}$ is the spectrum of a typical Dirac operator, these eigenvalues correspond to modes of zero momentum. The fact that UV-divergences can become higher as external momenta vanish is reminiscent of noncommutative quantum field theory, cf.\ \cite{MRS2000}. For instance, in the naive version of noncommutative $\phi^4_4$, such behavior was shown to lead to UV/IR mixing and, consequently, nonrenormalizability \cite{CR2000,CR2001,GW2005a}. This prompted the Grosse--Wulkenhaar model which solved the UV/IR problem \cite{GW2005b} and proved incredibly successful \cite{DGMR2007,GW2014}.

In light of this, it seems instructive to determine whether UV/IR-mixing is present in the spectral action matrix model and its relatives. This is a difficult question to answer,
requiring a careful renormalization analysis and the passage to continuous spectrum.
\end{rem}

\label{sect: intro}
Motivated by the quest for a QFT description of the spectral action~\cite{ChamseddineConnes1997} we would like to analyze the large $N$ behavior of the correlation functions
\begin{align}\label{eq:path integral}
\frac{\int_{H_N}V_{i_1j_1}\cdots V_{i_mj_m}e^{-\hbar^{-1}\Tr(f(D+V)-f(D))}d V}{\int_{H_N}e^{-\hbar^{-1}\Tr(f(D+V)-f(D))}d V},
\end{align}
where $\hbar$ is a formal parameter, $f:\mathbb{R}\to\mathbb{R}$ is a sufficiently regular function,
$D=\operatorname{diag}(\lambda_k)_{k=1}^\infty$ is a self-adjoint operator diagonalized with respect to some countable orthonormal basis, and $H_N$ is the space of hermitian matrices acting on the first $N$ basis elements\footnote{$H_N$ is equipped with the canonical measure coming from the real and imaginary parts of the components of $V$. At this point the integral is only formal.}.

Following \cite{vNvS21,vNvS22b,vNvS23}, we write
$$\Tr(f(D+V)-f(D))=\sum_{n=1}^\infty\sum_{i_1,\ldots,i_n=1}^N \frac{1}{n}f'[\lambda_{i_1},\ldots,\lambda_{i_n}]V_{i_1i_2}\cdots V_{i_n i_1},$$ in terms of the divided differences of the derivative of $f$, defined inductively
by $f'[x]:=f'(x)$, $f'[x,y]:=\frac{f'(x)-f'(y)}{x-y}$, $f'[x,y,z]:=\frac{f'[x,z]-f'[y,z]}{x-y}=\frac{f'(x)-f'(z)}{(x-y)(x-z)}-\frac{f'(y)-f'(z)}{(x-y)(y-z)}$, \textit{et cetera}. By designating the second order term $\frac12\sum_{k,l=1}^Nf'[\lambda_k,\lambda_l]|V_{kl}|^2$ as the free theory, we can apply standard methods of Gaussian integration in order to express \eqref{eq:path integral} as a combinatorial series whose terms are ribbon graph amplitudes \cite{Eynard2016,vNvS23}. The respective ribbon graphs are generalized Kontsevich graphs in the terminology of~\cite{BCEG}. Although \cite{BCEG} considers polynomials $f$ (the order of which limits the valency of the vertices), the corresponding amplitudes turn out to be formally the same as in \cite{vNvS22b}: these amplitudes are certain fractions of divided differences of $f'$.

A stunning feature of these graph amplitudes, noted independently by \cite{BCEG,vNvS22b}, is the Ward--Takahashi identity, which \cite{vNvS22b} showed to imply one-loop renormalizability of the spectral action matrix model in the Gomis--Weinberg sense, raising the question what algebraic relations govern the graphs at arbitrary loop order, and if their renormalization flow can be understood. These questions, besides being inherently interesting \cite{Azarfar2024,BG2016,HKPV2022,Perez2022,Perez2025,Steinacker2010,tHooft1982}, prepare for replacing the integration space~$H_N$ in \eqref{eq:path integral} by a space of fields incorporating physical content such as the spectral Standard Model \cite{ChamseddineConnes1997,CC2012,CIS2020}, and beyond \cite{AMST2016,CCM2015,CCS2013,DM2017,Suijlekom2015}. Moreover, their positive answer would bring the spectral action more in line with noncommutative quantum field theory \cite{GW2005a,GW2005b,GW2014,Riv2007}.

\paragraph{Feynman rules}
We summarize the Feynman rules referred to above --
precise definitions are in Section \ref{sct:Definitions}. The Feynman diagrams are ribbon graphs for which not the edges but the faces are labeled by indices.
For each vertex bordered by faces with indices $i_1,\ldots,i_n$, we multiply by a factor $f'[\lambda_{i_1},\ldots,\lambda_{i_n}]$, e.g.,
\begin{align*}
\raisebox{-18pt}{
\begin{tikzpicture}[thick]
 \draw (-0.5,-0.5) to (0,0);
 \draw (-0.5,0.5) to (0,0);
 \draw (0.5,0.5) to (0,0);
 \draw (0.5,-0.5) to (0,0);
 \node at (-0.5,0) {$i$};
 \node at (0,0.5) {$j$};
 \node at (0.5,0) {$k$};
 \node at (0,-0.5) {$l$};
 lldraw (0,0) circle (4pt);
lldraw[white] (0,0) circle (2pt)
 ;
\end{tikzpicture}}
\,\quad&=\quad f'[\lambda_i,\lambda_j,\lambda_k,\lambda_l],\\
\intertext{for each internal edge bordered by $i$ and $j$, we divide by a factor $f'[\lambda_i,\lambda_j]$, i.e.,}
\raisebox{-14pt}{
\begin{tikzpicture}[thick]
 \draw (-0.5,-0.5) to (0.5,0.5);
 \node at (-0.25,0.25) {$i$};
 \node at (0.25,-0.25) {$j$};
\end{tikzpicture}}
\,\quad&=\quad\frac{1}{f'[\lambda_i,\lambda_j]},\\
\intertext{and, finally, we sum over each unbroken face (face without external edges), e.g.,}
\raisebox{-18pt}{
\begin{tikzpicture}[thick]
\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);
\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);
\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);
\draw (-0.5,0) to (-0.8,0);
\draw (0.5,0.5) to (0.7,0.7);
\draw (0.5,-0.5) to (0.7,-0.7);
lldraw (-0.5,0) circle (4pt);
lldraw[white] (-0.5,0) circle (2pt)
 ;
lldraw (0.5,0.5) circle (4pt);
lldraw[white] (0.5,0.5) circle (2pt)
 ;
lldraw (0.5,-0.5) circle (4pt);
lldraw[white] (0.5,-0.5) circle (2pt)
 ;
\end{tikzpicture}}
\,\quad&=\quad\sum_{k=1}^N
\raisebox{-18pt}{
\begin{tikzpicture}[thick]
\draw (-0.5,0) to[out=90,in=180] (0.5,0.5);
\draw (-0.5,0) to[out=-90,in=180] (0.5,-0.5);
\draw (0.5,0.5) to[out=-45,in=45] (0.5,-0.5);
\draw (-0.5,0) to (-0.8,0);
\draw (0.5,0.5) to (0.7,0.7);
\draw (0.5,-0.5) to (0.7,-0.7);
\node at (0.15,0) {$k$};
lldraw (-0.5,0) circle (4pt);
lldraw[white] (-0.5,0) circle (2pt)
 ;
lldraw (0.5,0.5) circle (4pt);
lldraw[white] (0.5,0.5) circle (2pt)
 ;
lldraw (0.5,-0.5) circle (4pt);
lldraw[white] (0.5,-0.5) circle (2pt)
 ;
\end{tikzpicture}}\,\,.
\end{align*}

For renormalization purposes, it is relevant \cite{CR2000,CR2001,GW2005a,GW2005b,ILV2012,KLV2014,LOR2015,Riv2007,RVW} to know the asymptotic behavior of these amplitudes as $N\to\infty$. The main objective of this paper is to prove the following formulas describing this asymptotic behavior.

\begin{rem}\label{rem:apparent shortcut}
We explain here a problem with an apparent shortcut to the above proof. Indeed, given positive~${n_1,\ldots,n_U}$ it is not hard to show (using $\lambda_j\sim j^{1/d}$) that
\begin{align}\label{eq:sums like integrals}
 \sum_{j_1=1}^N\cdots\sum_{j_U=1}^N \lambda_{j_1}^{n_1}\cdots\lambda_{j_U}^{n_U}=\O(N^{U+(n_1+\ldots+n_U)/d}),
\end{align}
exactly as in the case with integrals instead of sums. If one skips the graph-theoretical Lemma \ref{lem:injection fi vertices to fi edges} and applies the estimates of Theorem \ref{thm:weighted_divdiff_bound} for arbitrary bordering indices of the fully internal vertices and edges, one can estimate the amplitude by the sum on the left-hand side of \eqref{eq:sums like integrals} where indeed the powers automatically add up to $n_1+\ldots+n_U=p(\Efi-\Vfi)$ as required! However, the $n_1,\ldots,n_U$ may not be positive, which invalidates \eqref{eq:sums like integrals}. The following example shows that such an estimate of the amplitude by \eqref{eq:sums like integrals} really is too coarse in general. We surely have
\begin{align}\label{eq:example diagram}
\raisebox{-20pt}{
\begin{tikzpicture}[thick]
 \draw (0.45,-.7) to (1,0);
 \draw (1.55,-.7) to (1,0);
 \draw (1,0.054) arc (-90:270:0.3cm);
 \draw (1,-0.035) arc (-90:270:.7cm);
 \node at (1,-.55) {$i_1$};
 \node at (0,1) {$i_2$};
 \vertex{1,0};
\end{tikzpicture}
}
=\lambda_{i_1}^{-1}\lambda_{i_2}^{-1}\sum_{k=1}^N\sum_{l=1}^N\frac{f'\{i_1,k,l,k,i_1,i_2\}}{f'\{i_1,k\}f'\{k,l\}}\lesssim
\sum_{k=1}^N\sum_{l=1}^N\frac{l^{-p/d}}{k^{-p/d}k^{-p/d}}=\sum_{k=1}^N k^{2p/d}\sum_{l=1}^N l^{-p/d}.
\end{align}
Naively adding up the orders gives the correct result, $\O(N^{2+\frac{p}{d}})$. But because $-p/d$ is negative, we cannot add up the powers: Assuming $p/d>1$, the sequence $(\sum_{l=1}^N l^{-p/d})_{N\in\N}$ is convergent with nonzero limit. Hence, $\sum_{l=1}^N l^{-p/d}$ is of order $N^0$, not of order $N^{1-p/d}$. The right-hand side of \eqref{eq:example diagram} is therefore $\O(N^{1+2p/d})$, and as we now know we have a better bound for the left-hand side. The trick is simply to choose the indices so that each vertex contribution is canceled by an edge contribution.
\end{rem}

\begin{rem}
The following explains the absence of a lower bound in Theorem \ref{thm:main2}. Suppose $i_1,i_2,i_3$ are such that $f'[i_1,i_2,i_3]=0$, $f'(\lambda_{i_1})\neq0$, and $f'(\lambda_{i_3})\neq0$. We then compute the amplitude
\begin{align*}
\raisebox{-19.5pt}{
\begin{tikzpicture}[thick]
\draw (-0.5,0.5) to (0,0);
\draw (-0.5,-.5) to (0,0);
\draw (0,0) to[out=60,in=120] (1,0);
\draw (0,0) to[out=-60,in=-120] (1,0);
\draw (1,0) to (1.5,0);
\node at (0,0.5) {$i_1$};
\node at (-.5,0) {$i_2$};
\node at (0,-.5) {$i_3$};
\vertex{0,0};
\vertex{1,0};
\end{tikzpicture}
}
=-\lambda_{i_1}^{-1}\lambda_{i_2}^{-1}\lambda_{i_3}^{-1}\sum_{k=1}^N\frac{f'\{i_1,i_2,i_3,k\}f'\{i_1,i_3,k\}}{f'\{i_1,k\}f'\{i_3,k\}}.
\end{align*}
Contrary to the situation where $f'[i_1,i_2,i_3]\neq0$, the factor $|f'\{i_1,i_2,i_3,k\}|$ obtained from the 4-vertex is proportional to $|\lambda_k|^{-1}$ as $k\to\infty$.
The other factors are proportional to $1$ as usual. We obtain
\begin{align*}
\raisebox{-19.5pt}{
\begin{tikzpicture}[thick]
\draw (-0.5,0.5) to (0,0);
\draw (-0.5,-.5) to (0,0);
\draw (0,0) to[out=60,in=120] (1,0);
\draw (0,0) to[out=-60,in=-120] (1,0);
\draw (1,0) to (1.5,0);
\node at (0,0.5) {$i_1$};
\node at (-.5,0) {$i_2$};
\node at (0,-.5) {$i_3$};
\vertex{0,0};
\vertex{1,0};
\end{tikzpicture}}
\sim\sum_{k=1}^N|\lambda_k|^{-1}\sim\sum_{k=1}^Nk^{-\frac{1}{d}}<\O(N).
\end{align*}
The order depends on $d$ but (since $d\geq0$) at least it is smaller than $\O(N)$ in the sense that there exists no $c$ such that the lower bound is $\geq cN$. For $d<1$ the graph is in fact finite.
\end{rem}

\begin{rem}
Even though there exist only finitely many eigenmodes $\lambda_k$ with $f'(\lambda_k)=0$, these singular modes can boost the order of divergence not only when occurring as external indices. For instance, assuming non-singular external indices, we have the divergences
\begin{align*}
\raisebox{-20pt}{
\begin{tikzpicture}[thick]
\draw (-1,0) to (1,0);
\draw (-.7,.7) arc (180:0:.7cm);
\draw (0,0) to[out=160,in=-90] (-.7,.7);
\draw (0,0) to[out=20,in=-90] (.7,.7);
\draw (-.3,.6) arc (180:0:.3cm);
\draw (0,0) to[out=110,in=-90] (-.3,.6);
\draw (0,0) to[out=70,in=-90] (.3,.6);
\vertex{0,0};
\end{tikzpicture}}
=\O(N^{2+\frac{p}{d}})
\quad\text{and}\quad
\raisebox{-20pt}{
\begin{tikzpicture}[thick]
\draw (-1,0) to (1,0);
\draw (-.7,.7) arc (180:0:.7cm);
\draw (0,0) to[out=160,in=-90] (-.7,.7);
\draw (0,0) to[out=20,in=-90] (.7,.7);
\draw (-.3,.6) arc (180:0:.3cm);
\draw (0,0) to[out=110,in=-90] (-.3,.6);
\draw (0,0) to[out=70,in=-90] (.3,.6);
\vertex{0,0};
\node at (0,.5) {$0$};
\end{tikzpicture}}
=\O(N^{1+\frac{p+1}{d}}).
\end{align*}
The latter is larger than the former precisely if $d<1$. More generally, at second loop order, the boost of UV-divergence by internal singular indices is not apparent for $d\geq1$. Indeed, if $d\geq1$, then the diagrams of Remark \ref{rem:2-point 2-loop} remain precisely those of maximal order, also in the more general setting of Theorem \ref{thm:main2}.
However, for any $d\in\N$, at loop order $L=d+2$ the maximal diagrams become those where one of the faces is artificially broken by a singular index, such as
\begin{align}
\raisebox{-35pt}{
\begin{tikzpicture}[thick]
\draw (-1.5,0) to (-1,0);
\draw (-1,0) arc (-180:180:1cm);
\draw (-1,0) to (0.5,.866);
\draw (0.5,.866) to (0.5,-.866);
\draw (-1,0) to (0.5,-.866);
\draw (0.5,.866) to (.75,1.3);
\draw (0.5,-.866) to (.75,-1.3);
\vertex{-1,0};
\vertex{0.5,.866};
\vertex{0.5,-.866};
\end{tikzpicture}}
~~
=\O(N^{4+3p/d})\quad\text{and}\quad
\raisebox{-35pt}{
\begin{tikzpicture}[thick]
\draw (-1.5,0) to (-1,0);
\draw (-1,0) arc (-180:180:1cm);
\draw (-1,0) to (0.5,.866);
\draw (0.5,.866) to (0.5,-.866);
\draw (-1,0) to (0.5,-.866);
\draw (0.5,.866) to (.75,1.3);
\draw (0.5,-.866) to (.75,-1.3);
\node at (0,0) {$0$};
\vertex{-1,0};
\vertex{0.5,.866};
\vertex{0.5,-.866};
\end{tikzpicture}}
=\O(N^{3+3(p+1)/d}).\label{eq:3-point 4-loop}
\end{align}
The latter is larger than the former precisely if $d<3$. Graphs similar to the examples above yield divergences increased by breaking faces to singular modes for any $d<L-1$.
\end{rem}
\begin{rem}\label{rem:conjectures}
In the setting of Theorem \ref{thm:main2} and for $d<3$, the second graph of \eqref{eq:3-point 4-loop} has maximal order among the 3-point 4-loop diagrams, but not every 3-point 4-loop diagram that has maximal order in the setting of Theorem \ref{thm:main} has maximal order in the setting of Theorem \ref{thm:main2}. That being said, we conjecture that, among the $n$-points $L$-loop graphs, every graph with maximal order in the setting of Theorem \ref{thm:main2} has maximal order in the setting of Theorem \ref{thm:main}. We moreover conjecture that the Ward identity, in the sense of \cite{vNvS22b}, holds in both cases when restricting to the graphs of maximal order of divergence.
This might help generalize the results of \cite{vNvS22b} to higher loop, which is a pressing open problem.
\end{rem}

For renormalization purposes, it is relevant \cite{CR2000,CR2001,GW2005a,GW2005b,ILV2012,KLV2014,LOR2015,Riv2007,RVW} to know the asymptotic behavior of these amplitudes as $N\to\infty$. The main objective of this paper is to prove the following formulas describing this asymptotic behavior.

\paragraph{Main results}
Suppose that $\{\lambda_k\}_{k=1}^\infty\subseteq\mathbb{R}$ is a sequence of asymptotic behavior $\lambda_k\asymp k^{1/d}$ for $d\in\R_{>0}$. Suppose that $f$ is smooth, even, and satisfies, for $x\to\infty$, $f^{(n)}(x)\asymp (-1)^nx^{-p-n}$ ($n\in\mathbb{N}$) for some~$p>0$. Suppose moreover that the divided differences of $f$ do not vanish on $\{\lambda_k\}_{k=1}^\infty$. For a graph $G$ with $U$ unbroken faces, $E_{\textnormal{fi}}$ edges which do not border a broken face, and $V_{\textnormal{fi}}$ vertices which do not border a broken face, the amplitude of $G$ is bounded from above and below by a constant times $N^{\omega(G)}$, where
\begin{align}\label{eq:power counting formula}
\omega(G)= U+\frac{p}{d}(E_{\textnormal{fi}}-V_{\textnormal{fi}}).
\end{align}

However, when the assumption of vanishing divided differences is not satisfied, the divergences become (for certain graphs \textit{strictly}) larger. In this case, the amplitude is bounded from above by $N^{\tilde\omega(G)}$, where
\begin{align}\label{eq:power counting formula 2}
\tilde \omega(G)=\max(U^{\mathfrak{b}}+\frac{p}{d}(E^{\mathfrak{b}}_{\textnormal{fi}}-V^{\mathfrak{b}}_{\textnormal{fi}})+\frac{p+1}{d}(E^{\mathfrak{b}}_{10}-V^{\mathfrak{b}}_{10})),
\end{align}
in which the maximum is taken over all subsets $\mathfrak{b}$ of unbroken faces, $U^{\mathfrak{b}},E_\textnormal{fi}^b,V_\textnormal{fi}^b$ are as before when designating the elements of $\mathfrak{b}$ as broken, and $V^{\mathfrak{b}}_{10}/E^{\mathfrak{b}}_{10}$ denotes the number of vertices/edges bordering only unbroken faces except for either exactly one element of $\mathfrak{b}$ or exactly one external index $i_0$ satisfying~${f'(\lambda_{i_0})=0}$.

\paragraph{Consequences of \eqref{eq:power counting formula}}
Note that $E_\textnormal{fi}-V_\textnormal{fi}\geq 0$ for any graph. A remarkable result is that a function~$f$ with \textit{faster} decay results in a \textit{higher} degree of divergence of the graph. This is not obvious from the definition of the Feynman rules, even for simple graphs like in Example 1, and even for concrete functions like $f(x)=x^{-p}$.
Because divided differences of such functions may well be negative, one might \textit{a priori} expect cancellations of terms damping the degree of divergence, but this does not happen.

Another corollary of our power counting formula is the following observation. Among the connected graphs with~$n\geq1$ external edges and $L$ loops, the maximal value of $U$ is $L-1$, and the maximal value of $E_\textnormal{fi}-V_\textnormal{fi}$ is $L-1$. The maximal value $\omega(G)=L-1+\frac{p}{d}(L-1)$ is attained by a nonempty set of diagrams, by virtue of our lower bound. These diagrams with maximal divergence are precisely the planar diagrams that cannot be split into two connected components by removing one vertex and all external edges, and moreover have only one unbroken face -- cf.\ Remark \ref{rem:2-point 2-loop}.
\paragraph{Consequences of \eqref{eq:power counting formula 2}}
The situation where divided differences of $f'$ may vanish is more complicated (cf. Section \ref{sct:main2}), but \eqref{eq:power counting formula 2} leads us to conjecture that the diagrams with maximal divergence are of the same planar form as before -- cf.\ Remark \ref{rem:conjectures}. As we discuss in Remark \ref{rem:UV/IR}, the influence of modes $\lambda_i$ with $f'(\lambda_i)=0$ is reminiscent of the UV/IR behavior of scalar field theories on noncommutative spacetime, which underlines the need for a rigorous renormalization analysis of the spectral action beyond the weak field approximation.

\paragraph{Techniques} To prove the above power counting formulas, we introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them, which appear to be novel. A pivotal concept throughout the proof is the weighted divided difference
$$f'\{x_1,\ldots,x_n\}:=(-1)^n x_1\cdots x_n f'[x_1,\ldots,x_n].$$
Its asymptotic behavior is more easily understood, because sending any of its variables to infinity yields another weighted divided difference.
Furthermore, a key result is that weighted divided differences are positive on large enough subsets of $\mathbb{R}^n$ for the functions $f$ we are considering. We thus generalize and give a new proof for Hunter's positivity theorem \cite{Hunter1977}, which is recovered by taking $f(x)=x^{-p}$ ($p\in 2\mathbb{N}$). Moreover, we show that for functions bounded above or below by $x^{-p}$, with similar bounds on their derivatives, the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus, which is crucial in the proof of our main theorem.

Our main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.

\paragraph{Acknowledgements}
We are grateful to Martijn Caspers, Séverin Charbonnier, Harald Grosse, and Walter van Suijlekom for useful discussions. TvN thanks the Erwin Schr\"odinger Institute and the organizers and participants of the April 2023 conference 'Non-commutative Geometry meets Topological Recursion', where the motivation for this paper originated.
TvN was supported by NWO project ‘Noncommutative multi-linear harmonic analysis and higher order spectral shift’,
OCENW.M.22.070. EMH thanks the Max Planck Institute for Mathematics in Bonn for its financial support.

To prove the above power counting formulas, the authors “introduce several techniques in the asymptotic analysis of divided differences, and prove upper and lower bounds for them.” A “pivotal concept throughout the proof is the weighted divided difference
$$f'\{x_1,\ldots,x_n\}:=(-1)^n x_1\cdots x_n f'[x_1,\ldots,x_n],$$
whose asymptotic behavior is “more easily understood, because sending any of its variables to infinity yields another weighted divided difference.”

A “key result” used in the proof is that “weighted divided differences are positive on large enough subsets of $\mathbb{R}^n$ for the functions $f$ we are considering,” which generalizes Hunter’s positivity theorem. Moreover, for functions “bounded above or below by $x^{-p}$, with similar bounds on their derivatives,” they show that “the asymptotic behavior of the weighed divided differences away from the origin is determined by the variable with the smallest modulus,” and this is stated to be “crucial in the proof of our main theorem.”

Finally, “our main theorem is proved by combining this positivity and the mentioned upper and lower bounds with an interesting graph-theoretical argument.”,

Question: Consider the spectral action matrix model determined by a self-adjoint diagonal operator $D=\operatorname{diag}(\lambda_k)_{k=1}^\infty$ and a sufficiently regular function $f:\mathbb R\to\mathbb R$. For ribbon-graph amplitudes, use the Feynman rules from the spectral action model: a vertex bordered by faces with labels $i_1,\dots,i_n$ contributes $f'[\lambda_{i_1},\dots,\lambda_{i_n}]$, an internal edge bordered by face labels $a,b$ contributes $1/f'[\lambda_a,\lambda_b]$, and each unbroken internal face is summed over its label from $1$ to $N$. For the two one-vertex 2-point ribbon graphs with external face labels $j$ and $i$—the first having one internal loop face, and the second having two nested internal loop faces— which statement gives their amplitudes for all choices of the external labels?

Their amplitudes are
\[
\sum_{k=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]}
\]
for the graph with one internal loop face, and
\[
\sum_{k,l=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_l,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]f'[\lambda_k,\lambda_l]}
\]
for the graph with two nested internal loop faces.

Choices:
B.
Their amplitudes are
\[
\sum_{k=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_i]}{f'[\lambda_j,\lambda_k]}
\]
for the graph with one internal loop face, and
\[
\sum_{k,l=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_l,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]f'[\lambda_k,\lambda_l]}
\]
for the graph with two nested internal loop faces.
C. Their amplitudes are obtained by summing over the internal face labels and taking the product of the vertex factor with the reciprocal edge factors. In particular, the first graph is a single sum over one internal face label and the second graph is a double sum over two internal face labels, with denominators \(f'[\lambda_j,\lambda_k]\) and \(f'[\lambda_j,\lambda_k]f'[\lambda_k,\lambda_l]\), respectively.
D.
Their amplitudes are
\[
\sum_{k=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_i]}
\]
for the graph with one internal loop face, and
\[
\sum_{k,l=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_l,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_i]f'[\lambda_k,\lambda_l]}
\]
for the graph with two nested internal loop faces.
E.
Their amplitudes are
\[
\sum_{k=1}^N \frac{f'[\lambda_i,\lambda_j,\lambda_k,\lambda_j]}{f'[\lambda_j,\lambda_k]}
\]
for the graph with one internal loop face, and
\[
\sum_{k,l=1}^N \frac{f'[\lambda_i,\lambda_j,\lambda_k,\lambda_l,\lambda_k,\lambda_j]}{f'[\lambda_j,\lambda_k]f'[\lambda_k,\lambda_l]}
\]
for the graph with two nested internal loop faces.

cyclic face-label repetition at the vertex

explicit vertex numerators were dropped while retaining the summation/edge-factor structure

edge denominator depends on adjacent face labels, not the external pair

ordered incidence data of face labels around the vertex

The stem does not explicitly reveal the correct option. It gives the general Feynman-style rules, but the student still has to apply them to the two specific ribbon graphs and track the face labels correctly.

This is not a pure theorem restatement, but it is very close to a direct instantiation of the stated rules. The task is essentially to translate the graph data into the prescribed amplitude formula.

Some reasoning is required to determine the cyclic order in the divided differences and the correct edge denominators, especially for the nested-loop graph. However, the problem is largely procedural rather than strongly generative.

Several distractors reflect plausible mistakes: using the wrong adjacent labels in edge factors, omitting repeated labels in the vertex term, or inserting incorrect denominator powers. But choice C is a weaker true statement, which reduces answer exclusivity and weakens distractor quality.

A reasonably solid application-style MCQ with no direct answer leakage and some mathematical checking required, but it is fairly close to rule-following and is weakened by the presence of a partially correct distractor.


Question: Consider the spectral action matrix model determined by a self-adjoint diagonal operator $D=\operatorname{diag}(\lambda_k)_{k=1}^\infty$ and a sufficiently regular function $f:\mathbb R\to\mathbb R$. For ribbon-graph amplitudes, use the Feynman rules from the spectral action model: a vertex bordered by faces with labels $i_1,\dots,i_n$ contributes $f'[\lambda_{i_1},\dots,\lambda_{i_n}]$, an internal edge bordered by face labels $a,b$ contributes $1/f'[\lambda_a,\lambda_b]$, and each unbroken internal face is summed over its label from $1$ to $N$. For the two one-vertex 2-point ribbon graphs with external face labels $j$ and $i$—the first having one internal loop face, and the second having two nested internal loop faces— which statement gives their amplitudes for all choices of the external labels?

Choices:
B.
Their amplitudes are
\[
\sum_{k=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_i]}{f'[\lambda_j,\lambda_k]}
\]
for the graph with one internal loop face, and
\[
\sum_{k,l=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_l,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_k]f'[\lambda_k,\lambda_l]}
\]
for the graph with two nested internal loop faces.
C. Their amplitudes are obtained by summing over the internal face labels and taking the product of the vertex factor with the reciprocal edge factors. In particular, the first graph is a single sum over one internal face label and the second graph is a double sum over two internal face labels, with denominators \(f'[\lambda_j,\lambda_k]\) and \(f'[\lambda_j,\lambda_k]f'[\lambda_k,\lambda_l]\), respectively.
D.
Their amplitudes are
\[
\sum_{k=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_i]}
\]
for the graph with one internal loop face, and
\[
\sum_{k,l=1}^N \frac{f'[\lambda_j,\lambda_k,\lambda_l,\lambda_k,\lambda_j,\lambda_i]}{f'[\lambda_j,\lambda_i]f'[\lambda_k,\lambda_l]}
\]
for the graph with two nested internal loop faces.
E.
Their amplitudes are
\[
\sum_{k=1}^N \frac{f'[\lambda_i,\lambda_j,\lambda_k,\lambda_j]}{f'[\lambda_j,\lambda_k]}
\]
for the graph with one internal loop face, and
\[
\sum_{k,l=1}^N \frac{f'[\lambda_i,\lambda_j,\lambda_k,\lambda_l,\lambda_k,\lambda_j]}{f'[\lambda_j,\lambda_k]f'[\lambda_k,\lambda_l]}
\]
for the graph with two nested internal loop faces.

The stem gives the general Feynman rules but does not reveal the final amplitudes for the two specific graphs. The correct choice must still be derived by applying those rules to the stated ribbon-graph configurations.

This is not a mere restatement of the rules/theorem. The test-taker must instantiate the vertex and edge rules on concrete graph topologies and track the face-label sequence around the vertex.

Some genuine reasoning is required: one must determine the correct summation indices, adjacent-face edge denominators, and cyclic label order at the vertex. However, once the rules are known, the task is largely procedural rather than deeply generative, and the missing diagrams may make it more about reconstruction than conceptual choice.

Several distractors target plausible errors: missing repeated face labels at the vertex (B), wrong edge-label dependence (D), and altered cyclic ordering (E). But choice C is a weaker true statement rather than a clearly false alternative, which undermines single-best-answer quality and makes the distractor set mixed rather than strong.

A reasonably solid application-based MCQ with no answer leakage and little tautology, but its distractor set is weakened by an arguably true-but-incomplete option and by reliance on verbally described graph structure rather than explicit diagrams.


\label{main1}
Let $\Om$ be a bounded Lipschitz domain in $\mathbb{R}^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\in L^{2,\infty}(\Omega)^2$ and $\mathop{\rm div}\nolimits b=0$.
Then there exists $p_0(\|b\|_{L^{2,\infty}})>2$ such that for any $p_0'<q<p_0$ and $\mathbb{G}\in L^q(\Om)^{2\times 2}$, there exists a  weak solution pair $u\in W^{1,q}_{0,\si}(\Omega)$ and $\pi\in L^q_0(\Omega)$ of \eqref{drifteq} such that
\EQ{\label{main1-eq}
\|u\|_{ W^{1,q}(\Omega)}+\|\pi\|_{L^q(\Omega)}\le C(\Omega,\|b\|_{L^{2,\infty}})\|\mathbb{G}\|_{L^q(\Omega)}.
}
For $p_0'<q<p_0$, the above pair $(u,\pi)$ is the unique weak solution pair  in $W^{1,q}_0\times L^q_0(\Om)$ without assuming \eqref{main1-eq}.

\begin{proof}
Similar to \cite{CT1} we will use the Gehring's lemma approach \cite{Gehring, Gia}. Denote $\Om_{x,R}=\Om \cap B(x,R)$ for $x\in \overline \Om$ and $R>0$. Since $\pd \Om$ is Lipschitz and compact, there
is a radius $R_0\in (0,1]$ such that for every $x_0 \in\pd \Om$,
\EQ{ \label{R0.def}
\Om_{x_0,2R_0}= \{ (x',x_n) \in B_{2R_0}(0) \subset \R^2,\ x_n>\ga(x')\},
}
after suitable coordinate rotation and translation, where $\ga(x')$ is a Lipschitz function defined for $x'\in B_{2R_0}'(0) \subset \R$ with Lipschitz constant $L$ and $\ga(0)=0$.

To prove the a priori estimate \eqref{eq5.1}, we first consider the interior case with $x_0\in\Omega$ and $B_{\rho}=B_{\rho}(x_0)\subset\Omega$. We rescale it to the unit ball $B$ and denote $u_{\rho}(x)=\rho u(\rho x+x_0),b_{\rho}(x)=\rho b(\rho x+x_0),\pi_{\rho}(x)=\rho^2\pi(\rho x+x_0)$ and $\GG_{\rho}(x)=\rho^2 \GG(\rho x+x_0)$. Let $\bar{u}_{\rho}=u_{\rho}-k$ where $k$ is a real constant to be choose later.  By Remark \ref{Wolfpdecom}, we can decompose $\pi_{\rho} - (\pi_{\rho})_{B}=\pi_1+\pi_2+\pi_3$ in $B$, where $\pi_1,\pi_2,\pi_3$ are given by
\EQ{
\nabla\pi_1=\mathcal{W}_{4/3,B}(\Delta \bar{u}_{\rho}),\quad \nabla\pi_2=\mathcal{W}_{4/3,B}(-\div(b_{\rho}\otimes \bar{u}_{\rho})),\quad \nabla\pi_3=\mathcal{W}_{4/3,B}(\div \GG_{\rho}),
}
with $\int_B \pi_i=0$.
Therefore, by Theorem \ref{theoremL2wolf} we have the following pressure bound
\EQ{\label{wolfbound}
\|\pi_1\|_{4/3,B}\le c_1\|\nabla \bar{u}_{\rho}\|_{4/3,B},\quad\|\pi_2\|_{4/3,B}\le c_1 \|b_{\rho}\bar{u}_{\rho}\|_{4/3,B},\quad \|\pi_3\|_{4/3,B}\le c_1\|\GG_{\rho}\|_{4/3,B},
}
where $c_1$ is a global constant.
For simplicity of notations let us drop the subscript $\rho$ for the further calculations, until we need to scale back to ball $B_{\rho}(x_0)$.

Take a smooth cutoff function $\eta$ on $B$ with $\eta=1$ in $B_{\frac{1}{2}}$. Use the test function $\zeta=\eta^4 \bar{u}$ in the weak form \eqref{soln-pair} with $(u,\pi)$ replaced by $(\bar u, \pi - (\pi)_B)$ to get the energy estimate (using \eqref{5.1} due to $b \in L^{2+\de}$)
\EQS{\label{3.1}
\int_{B} |\nabla (\eta^2\bar{u})| ^2
\le &\int_{B} |\bar{u}|^2(|\nabla \eta^2|^2) +\int_{B} |\bar{u}|^2|b||\nabla\eta|\eta^3
\\
&+\int_{B} (|\pi_1|+|\pi_2|+|\pi_3|)|\bar{u}||\nabla\eta|\eta^3+\bka{\div \GG_{\rho}, \bar{u}\eta^4}_B.
}

We will estimate each of the terms on the right hand side separately, using Lemmas \ref{Holder} and \ref{Sobolev} in ball $B$,
\EQS{
\int_{B} |\bar{u}|^2|b||\nabla \eta|\eta^3
&\le \|b\|_{L^{2,\infty}(B)}\|\bar{u}^2\eta^3\|_{L^{2,1}(B)}\le \|b\|_{L^{2,\infty}(B)}\|\bar{u}\eta^{3/2}\|_{L^{4,2}(B)}^2
\\
&\le C \|b\|_{L^{2,\infty}(B)}\|\bar{u}\eta^{3/2}\|^2_{W^{1,4/3}(B)}.
}
For pressure terms we use \eqref{wolfbound} and get the following
\EQN{
\int_{B}|\pi_1||\bar{u}||\nabla\eta|\eta^3 &\le \|\pi_1\|_{4/3,B}\|\eta^2 \bar{u}\|_{L^4(B)}
\\
&\le
c\|\nabla \bar{u}\|_{4/3,B}\|\eta^2 \bar{u}\|_{W^{1,4/3}(B)}\le c\|\nabla \bar{u}\|_{4/3,B}^2+c\|\eta \bar{u}\|^2_{4/3,B},
}
\EQN{
\int_{B}|\pi_2||\bar{u}||\nabla\eta|\eta^3 &\le \|\pi_2\|_{L^{4/3}(B)}\|\eta^2\bar{u}\|_{L^4(B)}\le c\|\bar{u}\|_{L^{4,4/3}(B)}\|b\|_{L^{2,\infty}(B)}\|\eta^2\bar{u}\|_{L^{4}(B)}
\\
&\le c(b)\|\nabla \bar{u}\|_{4/3,B}^2+c(b)\| \bar{u}\|^2_{4/3,B}
}
\EQN{
\int_{B}|\pi_3||\bar{u}||\nabla\eta|\eta^3 &\le \|\pi_3\|_{4/3,B}\|\eta^2 \bar{u}\|_{L^4(B)}\le  c\|\GG\|_{2,B}\|\eta^2 \bar{u}\|_{W^{1,4/3}(B)}
\\
&\le \frac 14 \|\GG\|_{2,B}^2+c\|\eta^2 \bar{u}\|_{W^{1,4/3}(B)}^2.
}
Lastly we estimate the term containing $\GG$,
\EQN{
\bka{\div \GG_{\rho}, \bar{u}\eta^4}_B
&=-\int_{B} \GG :\nabla(\bar{u}\eta^4)=-\int_{B} \GG: \nabla(\eta^2\bar{u})\eta^2-2\int_{B} \GG :\bar{u}\nabla\eta\eta^3
\\
&\le \frac{1}{16}\|\nabla(\eta^2\bar{u})\|^2_{2,B}+c\|\eta^2 \bar{u}\|_{2,B}^2+ \frac 14\|\GG\|_{2,B}^2.
}
Combining these estimates with \eqref{3.1} we get the following
\EQ{
\int |\nabla (\eta^2\bar{u})|^2\le c(b)\|\bar{u}\|^2_{W^{1,4/3}(B)}+c(b)(\|\bar{u}\|_{2,B}^2+\|\bar{u}\|_{4/3,B}^2)+\|\GG\|_{2,B}^2,
}
Here $c(b)$ depends on $b$ only through $\norm{b}_{L^{2,\infty}(\Om)}$. Finally, since $\eta=1$ on $B_{1/2}$ we get that
\EQ{
\int_{B_{1/2}} |\nabla \bar{u}|^2\le c(b)\|\nabla \bar{u}\|^2_{4/3,B}+c(b)\|\bar{u}\|_{2,B}^2+\|\GG\|_{2,B}^2.
}
We now choose $k=(u_{\rho})_{B}$ as the constant in $\bar{u}=u_{\rho}-k$ and apply Poincar\'e inequality to get
\EQ{
\int_{B_{1/2}} |\nabla u_{\rho}|^2\le c(b)\|\nabla u_{\rho}\|^2_{4/3,B}+\|\GG\|_{2,B}^2.
}
Lastly we scale back to $B_{\rho}(x_0)$ and get that
\EQ{\label{5.12}
\frac{1}{|B_{\rho/2}|}\int_{B_{\rho/2}} |\nabla u|^2
\le c(b) \Big(\frac{1}{|B_{\rho}|}\int_{B_{\rho}}|\nabla u|^{4/3}\Big)^{3/2}+\frac{c}{|B_{\rho}|}\|\GG\|_{2,B_{\rho}}^2.
}

\medskip

Next we consider the boundary case, $\Omega_{\rho}=\Omega\cap B_{\rho}(x_0)$ with $x_0\in\pd\Om$, and $0<\rho<R_0$. The significant difference from the internal case is the pressure estimate: Unlike the internal case we can not use Wolf decomposition estimate in a uniform ball, therefore we need to track the dependence of constant $c_1$ when we apply Theorem \ref{theoremL2wolf} to get estimates similar to \eqref{pressure_bound_est}
 and make sure it is uniform across all  $\Omega_{\rho}(x_0)$.
 Rescale the domain as the interior case so that $x_0=0$ and $\rho=1$.
 We will use change of coordinates
 \EQ{\label{change-var}
 y_1=x_1, \quad y_2=x_2-\gamma(x_1),
 }
 to straighten the boundary and map $\Om_{5/8}\subset\Om_1$ into domains $\Om_{5/8}'\subset\Om_1'$ with flat boundary on $y_2=0$. Since $\gamma$ was a Lipschitz curve with sufficiently small Lipschitz constant we can assume that $\Om_{5/8}'\subset B_{6/8}^+\subset B_{7/8}^+\subset\Om_1'$.
  (This is the first place that specifies the smallness of the Lipschitz constant.) By Lemma \ref{geometry} we can fix a smooth $V$ such that $B_{6/8}^+\subset V\subset B_{7/8}^+$. After rewriting the equation \eqref{drifteq} in new variables we get the following system (with all derivatives in $y$) in $\Om_1'$:
\EQS{\label{changed_var}
-\Delta_y u+\div (\gamma' e_1\otimes\partial_{y_2}u)+
\gamma' \pd_1\pd_2 u - (\gamma')^2\partial_{y_2}^2u+
\\
+\div_y(b\otimes u)- \gamma'\partial_{y_2}(b\otimes u)_1+\nabla_y \pi-\gamma'\partial_{y_2}\pi e_1&=\div\GG-\gamma'\partial_2 \GG_1 ,
\\
\div_y u-\gamma'\partial_2 u_1&=0.
}
Above, $(b\otimes u)_1=(b_1u_1,b_1u_2)$ and $\GG_1=(G_{11},G_{12})$ are their first rows. In the second term, $\ga'$ is hidden behind $\div$ to avoid $\ga''$ which we do not assume any bound.

We apply Wolf's local pressure projection $\mathcal{W}_{4/3,V}$ on \eqref{changed_var} in the smooth domain $V$ and get that  $\pi-(\pi)_{V}=\pi_1+\pi_2+\pi_3+\pi_4$ where
\EQS{\label{pressure_decomp_b}
\nabla\pi_1&=\mathcal{W}_{4/3,V}\bkt{\Delta_y u-\div (\gamma' e_1\otimes\partial_{y_2}u)-\gamma' \pd_1\pd_2 u +(\gamma')^2\partial_{y_2}^2u}
\\
\nabla\pi_2&=\mathcal{W}_{4/3,V}\bkt{-\div_y(b\otimes u)+\gamma'\partial_{y_2}(b\otimes u)_1}
\\
 \nabla\pi_3&=\mathcal{W}_{4/3,V}(\gamma' \partial_{y_2}\pi e_1)
 \\
 \nabla\pi_4&= \mathcal{W}_{4/3,V}(\div \GG-\gamma'\partial_2 \GG_1).
}
From Theorem \ref{theoremL2wolf} the local pressure projection  $\mathcal{W}_{4/3,V}$ is well defined, and we have the bound \eqref{pWolfbound} with global constant $c$. Also using $\ga'\pd_2 = \pd_2 \ga'$, and Lemmas \ref{Holder} and \ref{Sobolev} in $V$,
\EQS{\label{pressure_bound_est}
\|\pi_1\|_{4/3,V}&\le c(1+\|\gamma'\|_{\infty}+\|\gamma'\|_{\infty}^2)\|\nabla u\|_{4/3,V}
\\
\|\pi_2\|_{4/3,V}&\le  c(1+\|\gamma'\|_{\infty})\|u\otimes b\|_{4/3,V}
\\
\|\pi_3\|_{4/3,V}&\le  c\|\gamma'\|_{\infty}\|\pi\|_{4/3,V}\le c\|\gamma'\|_{\infty}\|\pi_3\|_{4/3,V}+c\|\gamma'\|_{\infty}\|\pi_1+\pi_2+\pi_4\|_{4/3,V}
\\
\|\pi_4\|_{4/3,V}&\le  c(1+\|\gamma'\|_{\infty})\|\GG\|_{4/3,V}
}
Here we apply our choice of $\gamma$ so that $\|\gamma'\|_{\infty}<1$ and
$ c\|\gamma'\|_{\infty}<1/2$. (This is the second place that specifies the smallness of the Lipschitz constant.) After applying it to \eqref{pressure_bound_est} we get the following
\EQS{\label{pressure_3}
\|\pi_3\|_{4/3,V}\le 2c\|\gamma'\|_{\infty}\|\pi_1+\pi_2+\pi_4\|_{4/3,V}.
}

Let $\Om^\sharp$ be the pre-image of $V$ under the map \eqref{change-var} in $x$-variables.
The pressure component $\pi_i$ can be considered functions on $\Om^\sharp$ and since \eqref{change-var} has determinant 1 and $\|\nabla_y u\|_{4/3,V}\le 2\|\nabla_x u\|_{4/3,\Omega^\sharp}$, by assumption $\|\gamma'\|_{\infty}\le 1$ we have
\EQS{\label{pressure-est}
\norm{\pi-(\pi)_{V}}_{4/3, \Om^\sharp} &\le {\textstyle\sum}_{i=1}^4 \norm{\pi_i}_{4/3, \Om^\sharp}
={\textstyle\sum}_{i=1}^4 \norm{\pi_i}_{4/3, V}
\\
&\le c\|\nabla _y u\|_{4/3,V} +c \|u\otimes b\|_{4/3,V} +c \|\GG\|_{4/3,V}
\\
&\le c(1+ \| b\|_{L^{2,\infty}(V)}) (\|\nabla_y u\|_{4/3,V} +\| u\|_{4/3,V} )+ c \|\GG\|_{4/3,V}
\\
&\le c(1+ \| b\|_{L^{2,\infty}(\Omega)}) (\|\nabla_x u\|_{4/3,\Omega^\sharp} +\| u\|_{4/3,\Omega^\sharp} )+ c \|\GG\|_{4/3,\Omega^\sharp}.
}
This pressure estimate is the only purpose of our change of variables \eqref{change-var}. Noting that $\Om_{5/8}\subset \Om^\sharp\subset \Om_1$, we then proceed the following.

 Let $\eta$ be a cutoff function supported in $B_{5/8}(x_0)$ such that $\eta=1$ on $B_{1/2}(x_0)$. Due to our boundary condition $u|_{\partial\Omega}=0$ we can use function $\eta^4u$ as an admissible test function in \eqref{drifteq}. Therefore we get the following local energy inequality
\EQS{\label{eq5.19}
\int_{\Om_1} |\nabla (\eta^2u)|^2
&\le \int_{\Om_1} |u|^2|\nabla \eta^2|^2+\int_{\Om_1} |u|^2|b||\nabla \eta|\eta^3\\
&\quad +\int_{\Om_1} |\pi-(\pi)_{V}||u||\nabla\eta|\eta^3-\int_{\Om_1} \GG :\nabla(u\eta^4).
}
Using \eqref{pressure-est} to bound the pressure term and similarly to internal case
using Lemmas \ref{Holder} and \ref{Sobolev} in $\Om_1$, with uniform constant $C_2$ in Lemma \ref{Sobolev} using $\eta^4u\in W^{1,2}_0(\Om_1)$, we get
\EQS{\label{5.16}
\int_{\Om_{1/2}} |\nabla u|^2 &\le C(b)(\|\nabla u\|^2_{4/3,\Om_1}+\|u\|_{2,\Om_1}^2)+c\|\GG\|_{2,\Om_1}^2.
}
Notice that here the constant $C(b)$ does not depend on $\Om_1$.
Finally we extend function $u,\GG$ by zero to  $B_1(x_0)\setminus \Omega$. Since $u=0$ on the set $E:=\Om^c\cap B_1(x_0)$ of non zero measure with the ratio $\frac{|E|}{|B_{1}|}$ bounded from below, we can use Poincar\'e inequality to bound $\|u\|_{2,\Om_1}^2 \le c \|\nabla u\|^2_{4/3,\Om_1}$, with a constant independent of $\Om_1$ with the given lower bound of the ratio.
(This version of Poincar\'e inequality follows from Proposition 6.14 of \cite[page 116]{Lieberman},  with $f=\al 1_E$, nt_{B_1} f=1$. It gives $\norm{u}_{4/3,B_1} \le C\al  \norm{\nabla u}_{4/3,B_1} $.
We also have
$\norm{u}_{4,B_1} \le C \norm{u}_{W^{1,4/3}(B_1)}$.)

We also re-scale the inequality for arbitrary $B_r(x_0)$ and get \eqref{5.12}.

\medskip
Lastly we need to consider the case of arbitrary $x_0,r$ such that $\Om^c\cap B_{2r}(x_0)\neq\emptyset$. For any such $x_0$ we can find $y_0\in\partial\Om\cap B_{2r}(x_0)$ such that
\EQ{
B_r(x_0)\subset B_{3r}(y_0)\subset B_{6r}(y_0)\subset B_{8r}(x_0).
}
Using \eqref{5.12} for $B_{3r}(y_0)$ (assuming $6r<R_0$) we get
\EQS{\label{5.17}
\frac{1}{|B_r|}\int_{B_r(x_0)} |\nabla u|^2
&\le \frac{C}{|B_{3r}|}\int_{B_{3r}(y_0)} |\nabla u|^2
\\
&\le c(b) \Big(\frac{1}{|B_{6r}|}\int_{B_{6r}(y_0)}|\nabla u|^{\frac{4}{3}}\Big)^{\frac{3}{2}}+\frac{C}{|B_{6r}|}\|\GG\|_{L^2(B_{6r}(y_0))}^2
\\
& \le c(b) \Big(\frac{1}{|B_{8r}|}\int_{B_{8r}(x_0)}|\nabla u|^{\frac{4}{3}}\Big)^{\frac{3}{2}}+\frac{C}{|B_{8r}|}\|\GG\|_{L^2(B_{8r}(x_0))}^2.
}

For Gehring's lemma we combine the internal estimate \eqref{5.12} and the boundary estimate \eqref{5.17} and conclude that
\EQ{\label{5.19}
\frac{1}{|B_r|}\int_{B_r(x_0)} |\nabla u|^2 \le
c(b) \Big(\frac{1}{|B_{8r}|}\int_{B_{8r}(x_0)}|\nabla u|^{\frac{4}{3}}\Big)^{\frac{3}{2}}+\frac{C}{|B_{8r}|}\|\GG\|_{L^2(B_{8r}(x_0))}^2,
}
for any $x_0$ in a big cube containing $\overline \Om$, and any $r\le R_0/6$.
We can now apply Proposition 1.1 of Giaquinta \cite[page 122]{Gia} (also see \cite[Proposition 3.7]{DongKim}) to get that $\nb u \in L^p_\loc$ for $p \in (2,p_0)$ for some $p_0 >2$, and
\EQ{\label{0619}
nt_{B_r(x_0)} |\nabla u|^p}^{\frac 1p} \le
nt_{B_{8r}(x_0)} |\GG|^p}^{\frac 1p},
}
for all $x_0 \in \overline \Om$ and $r\le R_0/6$, with
$p_0$ and $c_1$ depending only on $c(b)$. Summing \eqref{0619} over a finite cover of $\overline \Om$ of balls of radius $R_0/6$,
we get \eqref{eq5.1}.
\end{proof}

\begin{theorem}\label{main1}
Let $\Om$ be a bounded Lipschitz domain in $\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\in L^{2,\infty}(\Omega)^2$ and $\div b=0$.
Then there exists $p_0(\|b\|_{L^{2,\infty}})>2$ such that for any $p_0'<q<p_0$ and $\GG\in L^q(\Om)^{2\times 2}$, there exists a  weak solution pair $u\in W^{1,q}_{0,\si}(\Omega)$ and $\pi\in L^q_0(\Omega)$ of \eqref{drifteq} such that
\EQ{\label{main1-eq}
\|u\|_{ W^{1,q}(\Omega)}+\|\pi\|_{L^q(\Omega)}\le C(\Omega,\|b\|_{L^{2,\infty}})\|\GG\|_{L^q(\Omega)}.
}
For $p_0'<q<p_0$, the above pair $(u,\pi)$ is the unique weak solution pair  in $W^{1,q}_0\times L^q_0(\Om)$ without assuming \eqref{main1-eq}.
\end{theorem}

\emph{Comments on Theorem \ref{main1}:}
\EN{
\item This theorem extends the higher dimensional result \cite[Theorem 1.5]{CT1} to the dimension 2 case.
In \cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \eqref{change-var} and after \eqref{pressure_bound_est}, and is computable.
We skip in this paper the analogous results for small $b$ case (but full range $1<q<\infty$) as in \cite[Proposition 1.3, Theorem 1.4]{CT1}.

\item  Since $\Om$ is Lipschitz we can assume that there exists $R_0>0$ such that for any $x_0\in \partial\Om$ we have
\EQ{
\Om\cap B_{2R_0}(x_0)=\{(x',x_n)\in B_{2R_0}(0),x_n>\gamma(x')\}.
}
The constant $C$ in \eqref{main1-eq} will actually depend only on $R_0$, $\text{diam}~\Om$, and $\|b\|_{L^{2,\infty}}$.

\item If we further assume $b \in L^2(\Om)$
and $b$ has a divergence-free extension to $L^2(\R^2)$,
then there is another way to prove uniqueness of the weak solution in $W^{1,2}_{0,\si}(\Om)$ by using Hardy-BMO duality, even when the Lipschitz constant of $\Om$ is large.
See Theorem \ref{energy_2} in Section \ref{sec3}.

\item We may simply assume $\Om$ is a bounded $C^1$ domain, as for
a bounded $C^1$ domain and any constant $\lambda>0$, each point $x$ on $\pd\Om$ has a neighborhood $U_x$ such that $\pd \Om \cap U_x$ is the graph of a Lipschitz function with a Lipschitz constant less than $\lambda$.

}

For corresponding results on scalar equations
\EQ{\label{scalar.eq}
-\Delta u +b\cdot\nabla u=\div  G, \quad u|_{\pd \Om}=0,
}
with critical drift coefficient $b \in L^n$, see
Kim \& Kim \cite{KK}. When $b \in L^{n,\infty}(\Om)$ for $n \ge 3$, see \cite{KiTs,KPT} which also cover more general assumptions on drift $b$ as well as \cite{CS20,CS24} and Kwon \cite{Kwon} for $n=2$ case or drift in Morrey spaces which include $L^{n,\infty}$.
There are also results on H\"older continuity of the solutions, for example with $b\in BMO^{-1}(\R^3)$, $\div b=0$, see Seregin-Sverak-Silvestre-Zlatos \cite{SSSZ}.

Using the same proof scheme we can also prove the following theorem for the  scalar equation \eqref{scalar.eq}.
\begin{theorem}\label{th1.2}
  Let $\Om$ be a bounded Lipschtiz domain in $\RR^2$.  Assume $b\in L^{2,\infty}(\Omega)$ and $\div b=0$.
Then there exists $p_0(\|b\|_{L^{2,\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\in L^q(\Om)$, there exists a  weak solution pair $u\in W^{1,q}_{0,\si}(\Omega)$ of \eqref{scalar.eq} such that
\EQ{\label{main2-eq}
\|u\|_{ W^{1,q}(\Omega)}\le C(\Omega,\|b\|_{L^{2,\infty}})\|G\|_{L^q(\Omega)}.
}
For $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\Om)$ without assuming \eqref{main2-eq}.
\end{theorem}

\label{1.2}
\De u = \div (\nb u),\quad
b\cdot\nabla u = \div (b\otimes u), \quad \int_\Om  \div \GG \cdot \zeta = -
\int_\Om  \GG:\nb \zeta.
}
For zero divergence spaces or spaces with zero average we

\label{soln-pair}
\int_{\Om} (\nb u - b \otimes u): \nb \zeta - \pi \div \zeta  = -\int_\Om \GG : \nb \zeta,\quad
\forall \zeta\in C_c^{\infty}(\Om)^2.
}
Here we used \eqref{1.2} to integrate by parts

\label{sec1}

For a bounded Lipschitz domain $\Om$ in $\mathbb{R}^2$ we consider the solutions of the perturbed Stokes system $(u,\pi):\Om \to \mathbb{R}^2 \times \mathbb{R}$ that satisfy
\EQ{\label{drifteq}
-\Delta u +b\cdot\nabla u+\nabla \pi=\mathop{\rm div}\nolimits \mathbb{G},\quad \mathop{\rm div}\nolimits u=0, \quad u|_{\partial \Om}=0,
}
with given drift $b$ in the critical space, $b\in L^{2,\infty}(\Omega)^2$, $\mathop{\rm div}\nolimits b=0$, and force $\mathbb{G}\in L^q(\Om)^{2\times 2}$, $1 <q<\infty$. We denote by
$L^{q,r}(\Om)$ the Lorentz spaces. For derivatives of vectors and matrixes we use the following notations
$(\mathop{\rm div}\nolimits \mathbb{G})_i=\pd_j \GG_{ji}$ and $(\nabla \zeta)_{ji} = \pd_j \zeta_i$. Since $\mathop{\rm div}\nolimits b=0$ we can integrate by parts in the following terms
\EQ{\label{1.2}
\De u = \mathop{\rm div}\nolimits (\nabla u),\quad
b\cdot\nabla u = \mathop{\rm div}\nolimits (b\otimes u), \quad \int_\Om \mathop{\rm div}\nolimits \mathbb{G} \cdot \zeta = -
\int_\Om \mathbb{G}:\nabla \zeta.
}
For zero divergence spaces or spaces with zero average we use the following notations
(for $\Om\subset\mathbb{R}^n$, $n\ge2$)
\EQS{
C^{\infty}_{c}(\Omega) &= \big\{ u \in C^{\infty}(\Omega):~ \supp\{u\}\Subset \Omega\big\},\\
C^{\infty}_{c,\si}(\Omega) &= \big\{ u \in C^{\infty}_c(\Omega)^n:\ \mathop{\rm div}\nolimits u=0\big\},\\
W^{1,q}_{0,\si}(\Omega) &= \big\{ u \in W^{1,q}_0(\Omega)^n:\ \mathop{\rm div}\nolimits u=0\big\},\\ \quad L^q_0(\Om)&=\left \{ \pi \in L^q(\Om):\ \textstyle \int_\Om \pi \,dx=0 \right \}.
}

Next we consider $(u,\pi)$ as a \emph{weak solution pair} of \eqref{drifteq} if $u\in W^{1,q}_{0,\si}(\Omega)$, $\pi\in L^q_0(\Omega)$, and $(u,\pi)$ satisfies \eqref{drifteq} in distributional sense,\EQ{\label{soln-pair}
\int_{\Om} (\nabla u - b \otimes u): \nabla \zeta - \pi \mathop{\rm div}\nolimits \zeta = -\int_\Om \mathbb{G} : \nabla \zeta,\quad
\forall \zeta\in C_c^{\infty}(\Om)^2.
}
Here we used \eqref{1.2} to integrate by parts.
We say $u$ is a \emph{$q$-weak solution} of \eqref{drifteq} if $u\in W^{1,q}_{0,\si}(\Omega)$ and $u$ satisfies \eqref{drifteq} in weak sense, i.e.,
\EQ{\label{weak-soln}
\int_{\Om} (\nabla u - b \otimes u): \nabla \zeta = -\int_\Om \mathbb{G} : \nabla \zeta,\quad
\forall \zeta\in C_{c,\sigma}^{\infty}(\Omega).
}
In case $q=2$ we call $u$ a \emph{weak solution}. In this paper we also consider a dual system of \eqref{drifteq}
\EQ{\label{drifteq_dual}
-\Delta v -\nabla\cdot(b\otimes v)+\nabla \pi=\mathop{\rm div}\nolimits \mathbb{F},\quad \mathop{\rm div}\nolimits v=0, \quad v|_{\partial \Om}=0.
}
This system is equivalent to \eqref{drifteq} when the drift is divergence free, $\mathop{\rm div}\nolimits b=0$, but in general case it is different.

The boundary value problem of the perturbed Stokes system \eqref{drifteq} and the corresponding scalar equation \eqref{scalar.eq}
has a huge amount of literature when we consider subcritical drift $b\in L^p(\Om)$, $\Om \subset \mathbb{R}^n$, with $p> n \ge2$. In the subcritical case we may treat the drift term $b\cdot \nabla u$ as a perturbation to $\De u$. The critical case is when $b\in L^n(\Om)$ or $b\in L^{n,\infty}(\Om)$. System \eqref{drifteq} in $\Om \subset \mathbb{R}^3$ with critical drift $b\in L^3(\Om)$ has previously been studied;
See Section 4 of Kim \cite{Kim09}, Theorem 18 in Choe \& Kim \cite{ChoeKim}, and Section 5 of
Amrouche \& Rodríguez-Bellido \cite{AmRB}. They contain estimates similar to \eqref{main1-eq}. A similar result for small $b\in L^{n,\infty}(\Om)$, $n \ge3$, is proved by the authors \cite[Theorem 1.4]{CT1}.

In this paper we extend our previous work for higher dimensions $n \ge 3$ \cite{CT1} to $n=2$. Our main theorem below extends \cite[Theorem 1.5]{CT1} to the dimension 2 case and proves existence of the solutions with higher integrability of the gradient.

The boundary value problem of the perturbed Stokes system \eqref{drifteq} and the corresponding scalar equation \eqref{scalar.eq}
has a huge amount of literature when we consider subcritical drift $b\in L^p(\Om)$, $\Om \subset \R^n$, with $p> n \ge2$. In the subcritical case we may treat the drift term $b\cdot \nb u$ as a perturbation to $\De u$. The critical case is when $b\in L^n(\Om)$ or $b\in L^{n,\infty}(\Om)$. System \eqref{drifteq} in $\Om \subset \R^3$ with critical drift $b\in L^3(\Om)$ has previously been studied;
See Section 4 of Kim \cite{Kim09}, Theorem 18 in Choe \& Kim \cite{ChoeKim}, and Section 5 of
Amrouche \& Rodríguez-Bellido \cite{AmRB}. They contain estimates similar to \eqref{main1-eq}. A similar result for small $b\in L^{n,\infty}(\Om)$, $n \ge3$, is proved by the authors \cite[Theorem 1.4]{CT1}.

\emph{Comments on Theorem \ref{main1}:}
\EN{
\item This theorem extends the higher dimensional result \cite[Theorem 1.5]{CT1} to the dimension 2 case.
In \cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \eqref{change-var} and after \eqref{pressure_bound_est}, and is computable.
We skip in this paper the analogous results for small $b$ case (but full range $1<q<\infty$) as in \cite[Proposition 1.3, Theorem 1.4]{CT1}.

Using the same proof scheme we can also prove the following theorem for the scalar equation \eqref{scalar.eq}.
\begin{theorem}\label{th1.2}
 Let $\Om$ be a bounded Lipschtiz domain in $\RR^2$. Assume $b\in L^{2,\infty}(\Omega)$ and $\div b=0$.
Then there exists $p_0(\|b\|_{L^{2,\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\in L^q(\Om)$, there exists a weak solution pair $u\in W^{1,q}_{0,\si}(\Omega)$ of \eqref{scalar.eq} such that
\EQ{\label{main2-eq}
\|u\|_{ W^{1,q}(\Omega)}\le C(\Omega,\|b\|_{L^{2,\infty}})\|G\|_{L^q(\Omega)}.
}
For $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\Om)$ without assuming \eqref{main2-eq}.
\end{theorem}

\section{Existence of weak solutions}\label{sec3}
In this section we will prove existence of weak solutions to the perturbed Stokes system \eqref{drifteq} for $b\in L^2$ or $L^{2,\infty}$, $\div b=0$,
\EQ{ \label{eq3.1}
-\Delta u+ b\cdot\nabla u + \nb \pi =f,\quad \div u=0, \quad u|_{\pd \Om}=0,
}
in $\Om \subset \R^2$. When $b\in L^2$, we will further show uniqueness.
Recall that a weak solution $u$ belongs to $ W^{1,2}_{0,\sigma}(\Omega)$ and satisfies a weak form of \eqref{drifteq} similar to \eqref{weak-soln}.
\begin{theorem}[Weak solutions for weak $L^2$ drift]\label{energy}
Let $\Om$ be a bounded Lipschitz domain in $\R^2$.
Assume $b\in L^{2,\infty}(\Omega)^2$ and $\div b=0$.
Then for any $f \in W^{-1,2}(\Omega)^2$,
there exists a weak solution $u\in W^{1,2}_{0,\sigma}(\Omega)$
of \eqref{drifteq}. Moreover, there is $\pi \in L^2(\Om)$ so that $(u,\pi)$ solves \eqref{drifteq} in distributional sense, $\int_\Om \pi=0$, and
\EQ{\label{eq6.1}
\|u\|_{ W^{1,2}(\Omega)}+\|\pi\|_{L^2(\Omega)}\le C\|f\|_{W^{-1,2}(\Om)},
}
for some constant $C=C(\Om)$ independent of $b$.
\end{theorem}

\begin{theorem}[Unique weak solution for $L^2$ drift]\label{energy_2}
Let $\Om$ be a bounded Lipschitz domain in $\R^2$.
Assume $b\in L^{2}(\Om)^2$ has an extension
$\tilde{b}\in L^{2}(\R^2)^2$ with $\div \tilde{b}=0$ in $\R^2$.
Then for any $f \in W^{-1,2}(\Omega)^2$,
there exists a unique weak solution $u\in W^{1,2}_{0,\sigma}(\Omega)$
of \eqref{drifteq}. Furthermore, there is $\pi \in L^2(\Om)$ so that $(u,\pi)$ solves \eqref{drifteq} in distributional sense, $\int_\Om \pi=0$, and
\EQ{\label{eq6.1b}
\|u\|_{ W^{1,2}(\Omega)}+\|\pi\|_{L^2(\Omega)}\le C\|f\|_{W^{-1,2}(\Om)},
}
for some constant $C=C(\Om)$ independent of $b$.
\end{theorem}

Approximating $u$ in $W^{1,2}_{0,\si}(\Om)$ by smooth $u_j$
and using $\div b=0$ conclude that
\EQ{\label{eq3.4}
\int_\Om (b\cdot \nb) u\cdot u = \lim_{j \to \infty} \int_\Om (b\cdot \nb) u_j\cdot u_j = \int_\Om b\cdot \nb(\frac12|u_j|^2) = 0.
}
 Thus we get the coercivity of this bilinear form, indeed,
\EQ{
\mathcal{B}(u,u)=\|\nabla u\|_{L^2(\Om)}^2, \quad \forall u\in W_{0,\sigma}^{1,2}(\Om).
}
Therefore, by Lax-Milgram theorem, for any $f \in W^{-1,2}(\Omega)^n\subset (W_{0,\sigma}^{1,2}(\Om)^n)^*$, there exists a unique $u\in W_{0,\sigma}^{1,2}(\Om)$ such that
\EQ{
\mathcal{B}(u,v)=\langle f,v\rangle,\quad \forall v\in W_{0,\sigma}^{1,2}(\Om).
}
This is the weak form of \eqref{drifteq} and hence
$u$ is the unique weak solution. By taking $v=u$, we get $\norm{u}_{W^{1,2}_0(\Om)} \le C\norm{f}_{W^{-1,2}(\Om)}$.
Lastly, we apply Lemma \ref{pressure} with $q=2$ (and using that $\Om$ is bounded and Lipschitz) to get an associated pressure $\pi\in L^2_0(\Om)$ that satisfies the bound.
\end{proof}

\begin{theorem}\label{theoremL2wolf}
Assume $p>1$ and $V\subset \R^n$, $n\ge2$, is a bounded Lipschitz domain with Lipschitz constant less than a sufficiently small $L_0(p,n)>0$.
Then the problem \eqref{Stokes} has a unique weak solution pair $(v,\pi)$ with $v \in W^{1,p}_{0,\si}(V)$ and $\pi \in L^p_0(V)$ for given $f=\div \FF$, $ \FF \in L^p(\Om)^{n\times n}$, and the local pressure projection in $V$,
\[
\mathcal{W}_{p,V}(\div \FF)=\nabla\pi,
\]
satisfies the following inequality
\EQ{\label{pWolfbound}
\|\pi\|_{L^p(V)}\le c_1\|\FF\|_{L^p(V)},
}
where $c_1=c_1(n,V,p)$.
\end{theorem}

\begin{theorem}\label{Capriori}
Let $\Om$ be a bounded Lipschitz domain in $\R^2$ with Lipschitz constant $L \le L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\in L^{2,\infty}(\Omega)^2$ and $\div b=0$.
Assume further $b \in L^{2+\delta}$ for some $\delta>0$.
Then there exists $p_0(\|b\|_{L^{2,\infty}})>2$ such that for any $2<p<p_0$,
any weak solution $u\in W^{1,2}_{0,\si}(\Omega)$ of \eqref{drifteq} with force $\GG\in L^p(\Om)^{2\times 2}$
is in $ W^{1,p}(\Omega)$ and
\EQ{\label{eq5.1}
\|u\|_{W^{1,p}(\Omega)}\le c\|u\|_{W^{1,2}(\Omega)}+ c\|\GG\|_{L^p(\Omega)}
}
for some constant $c(\Omega,\|b\|_{L^{2,\infty}})>0$.
\end{theorem}

\emph{Comments on Theorem \ref{main1}:}
\EN{
\item This theorem extends the higher dimensional result \cite[Theorem 1.5]{CT1} to the dimension 2 case.
In \cite[Theorem 1.5]{CT1} we take $L_0=1/2$. In this paper $L_0$ is specified after \eqref{change-var} and after \eqref{pressure_bound_est}, and is computable.
We skip in this paper the analogous results for small $b$ case (but full range $1<q<\infty$) as in \cite[Proposition 1.3, Theorem 1.4]{CT1}.

\item  Since $\Om$ is Lipschitz we can assume that there exists $R_0>0$ such that for any $x_0\in \partial\Om$ we have
\EQ{
\Om\cap B_{2R_0}(x_0)=\{(x',x_n)\in B_{2R_0}(0),x_n>\gamma(x')\}.
}
The constant $C$ in \eqref{main1-eq} will actually depend only on $R_0$, $\text{diam}~\Om$, and $\|b\|_{L^{2,\infty}}$.

\item If we further assume $b \in L^2(\Om)$
and $b$ has a divergence-free extension to $L^2(\mathbb{R}^2)$,
then there is another way to prove uniqueness of the weak solution in $W^{1,2}_{0,\si}(\Om)$ by using Hardy-BMO duality, even when the Lipschitz constant of $\Om$ is large.
See Theorem \ref{energy_2} in Section \ref{sec3}.

\item We may simply assume $\Om$ is a bounded $C^1$ domain, as for
a bounded $C^1$ domain and any constant $\lambda>0$, each point $x$ on $\partial\Om$ has a neighborhood $U_x$ such that $\partial \Om \cap U_x$ is the graph of a Lipschitz function with a Lipschitz constant less than $\lambda$.

}

For corresponding results on scalar equations
\EQ{\label{scalar.eq}
-\Delta u +b\cdot\nabla u=\mathop{\rm div}\nolimits  G, \quad u|_{\partial \Om}=0,
}
with critical drift coefficient $b \in L^n$, see
Kim \& Kim \cite{KK}. When $b \in L^{n,\infty}(\Om)$ for $n \ge 3$, see \cite{KiTs,KPT} which also cover more general assumptions on drift $b$ as well as \cite{CS20,CS24} and Kwon \cite{Kwon} for $n=2$ case or drift in Morrey spaces which include $L^{n,\infty}$.
There are also results on H\"older continuity of the solutions, for example with $b\in BMO^{-1}(\mathbb{R}^3)$, $\mathop{\rm div}\nolimits b=0$, see Seregin-Sverak-Silvestre-Zlatos \cite{SSSZ}.

Using the same proof scheme we can also prove the following theorem for the  scalar equation \eqref{scalar.eq}.
\begin{theorem}\label{th1.2}
  Let $\Om$ be a bounded Lipschtiz domain in $\mathbb{R}^2$.  Assume $b\in L^{2,\infty}(\Omega)$ and $\mathop{\rm div}\nolimits b=0$.
Then there exists $p_0(\|b\|_{L^{2,\infty}})>2$ such that for any $p_0'<q<p_0$ and $G\in L^q(\Om)$, there exists a  weak solution pair $u\in W^{1,q}_{0,\si}(\Omega)$ of \eqref{scalar.eq} such that
\EQ{\label{main2-eq}
\|u\|_{ W^{1,q}(\Omega)}\le C(\Omega,\|b\|_{L^{2,\infty}})\|G\|_{L^q(\Omega)}.
}
For $p_0'<q<p_0$, the above solution $u$ is the unique weak solution in $W^{1,q}_0(\Om)$ without assuming \eqref{main2-eq}.
\end{theorem}

This theorem improves $\alpha=0$ case of \cite[Theorem 1.1]{CS20} and $n=2$ case of Kwon \cite[Theorem 1.1]{Kwon} which show that for any $q>2$, there is $p \in (2,q)$ depending on $q$, $\left  \| b  \right \|_{L^{2,\infty}}$ and $\Om$ such that for any $G \in L^q(\Om)$, there is a unique $p$-weak solution $u$ of \eqref{scalar.eq} with force $G$. Moreover,
\[
\left  \| \nabla u  \right \|_{L^p(\Om)} \le C \left  \| G  \right \|_{L^q(\Om)}.
\]
Theorem \ref{th1.2} improves this result by asserting that $p=q$ when $2<q<p_0$ is sufficiently close to $2$. This improvement further allows us to prove a priori bounds for $q$-weak solutions for $q \in (p_0',2]$ by duality, which yields existence and uniqueness
 for $q \in (p_0',2]$.

We skip the proof of Theoem \ref{th1.2} since it is similar to Theoem \ref{main1} and is easier, without the need of pressure estimate. Because it has no pressure estimate, the Lipschitz constant of the boundary $\partial \Om$ is allowed to be arbitrarily large.

\medskip

We now explain the new difficulties and ideas.
Theorem \ref{main1}
extends the higher dimensional result \cite[Theorem 1.5]{CT1} to the 2D case, and
we will still use Gehring's approach. In this approach, we need to do local energy estimate in every ball and prove a reversed H\"older inequality in a general ball with a uniform constant.

The first new difficulty is that we cannot use $\zeta = u \eta^4$ for some cut-off function $\eta$ as the test function in the weak form \eqref{soln-pair}, because $b \otimes u$ is no longer in $L^2$ as in the 3D case, and
 the term
\EQ{\label{drift-integral}
\int_\Om b \otimes u : \nabla( u \eta^4)
}
is not integrable. Hence we will mollify the drift $b$, so that we can derive an a priori bound. Since there is no theorem that extends a div-free vector field in $L^{2,\infty}(\Omega)$ to $L^{2,\infty}(\mathbb{R}^2)$ for a general bounded domain, we will consider the approximation problems in subdomains $\Om_n$ that converge to $\Om$.

The second new difficulty comes from the lower integrability of the pressure.
We will
bound the pressure using Wolf's local pressure projection with a uniform constant as in \cite{CT1}.
Denote $\Omega_R=\Omega\cap B_R(x_0)$ for some $x_0\in \overline \Om$. In \cite{CT1}, it suffices to bound $\pi -(\pi)_{\Omega_R}$ in $L^2(\Omega_R)$. However,  for dimension 2  we need to bound $\pi -(\pi)_{\Omega\cap B_R}$ in $L^q(\Omega_R)$, $q<2$. (We choose $q=4/3$ for simplicity.)
We have to choose $q<2$ because
\[
\pi {\ \lesssim \ } \nabla u + b \otimes u -\mathbb{G}
\]
and $b \otimes u$ is at best in $L^{2-}$ for $u \in W^{1,2}_\mathrm{loc}$ even if we assume $b \in L^2$. This would not be an issue if $b \in L^{r}$, $r>2$, and we can only assume $u \in W^{1,2}_\mathrm{loc}$ which is the assumption of the reversed H\"older inequality.

To obtain a uniform constant of Wolf's local pressure projection in $L^q(\Omega_R)$ for $q<2$ gives a significant higher restriction on the regularity of the domain. Instead of a Lipschitz domain with Lipschitz constant $L$ less than 1/2, we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.

The third difficulty is the approximation of a Lipschitz domain. In order to construct the solution we are using an approximation method, therefore the first logical idea would be to construct a sequence of $b_n\in L^{2,\infty}(\Om)$, $\mathop{\rm div}\nolimits b=0$, such that $b_n$ converge in a sense that allows us to prove that the limit will satisfy the equation, for example convergence almost everywhere is sufficient and $\|b_n\|_{L^{2,\infty}(\Om)}$ are bounded uniformly. For example in star-shaped domains it is trivial, but for arbitrary Lipschitz domains this question is not so clear. To deal with this issue, following Verchota \cite{Verchota_1,Verchota_2} and later Kwon \cite{Kwon}, Chernobai-Shilkin \cite{CS24}, we instead approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant and on sub-domains we can use standard mollification of $b$ and construct an approximation sequence as zero extension of solutions in these sub-domains. See Section \ref{sec6} for details.

The rest of the paper is organized as the following. In section 2 we have preliminary results concerning Lorentz spaces and the pressure, which will be used in the later sections. In section 3 we derive the existence of weak solutions in $W^{1,2}(\Omega)$. Section 4 contains Wolf's local pressure projection method that will be used to estimate the pressure. In section 5, assuming higher integrability of the drift, we prove uniform local estimate for the gradient which will imply higher integrability due to Gehring's lemma. Section 6 contains proof of the main Theorem \ref{main1}. In the Appendix we put a proof of geometrical lemma that was used in section 4.

We will still use Gehring's approach: “do local energy estimate in every ball and prove a reversed H\"older inequality in a general ball with a uniform constant.”

New issues/steps specific to proving Theorem~\ref{main1} in 2D:
\begin{itemize}
\item \textbf{Non-integrable drift term; mollify and approximate domains.} One “cannot use $\zeta = u\eta^4$ … as the test function … because $b\otimes u$ is no longer in $L^2$ … and the term $\int_\Om b \otimes u : \nabla( u \eta^4)$ is not integrable.” Thus “we will mollify the drift $b$, so that we can derive an a priori bound.” Since there is no general extension theorem to $L^{2,\infty}(\mathbb{R}^2)$, they “consider the approximation problems in subdomains $\Om_n$ that converge to $\Om$.”
\item \textbf{Pressure control via Wolf projection in $L^q$, $q<2$.} “The second new difficulty comes from the lower integrability of the pressure.” They “bound the pressure using Wolf's local pressure projection with a uniform constant.” In 2D they need “to bound $\pi-(\pi)_{\Omega\cap B_R}$ in $L^q(\Omega_R)$, $q<2$ (we choose $q=4/3$),” because “$\pi\lesssim \nabla u + b\otimes u -\mathbb{G}$” and “$b\otimes u$ is at best in $L^{2-}$ for $u\in W^{1,2}_{\mathrm{loc}}$.” Achieving a uniform constant for the projection in $L^q$, $q<2$ forces stronger boundary regularity: “we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.”
\item \textbf{Approximation of Lipschitz domains for constructing solutions.} To construct solutions they use an approximation method, but approximating $b$ directly in arbitrary Lipschitz domains is “not so clear.” Instead, “we approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant,” on which “we can use standard mollification of $b$,” and then “construct an approximation sequence as zero extension of solutions in these sub-domains.”
\end{itemize}

They indicate the proof is completed by these ingredients and is carried out later: “See Section~\ref{sec6} for details,” and “Section 6 contains proof of the main Theorem~\ref{main1}.”

We will still use Gehring's approach: “do local energy estimate in every ball and prove a reversed H\"older inequality in a general ball with a uniform constant.”

New issues/steps specific to establishing the main theorem in 2D:
\begin{itemize}
\item \textbf{Non-integrable drift term; mollify and approximate domains.} One “cannot use $\zeta = u\eta^4$ … as the test function … because $b\otimes u$ is no longer in $L^2$ … and the term $\int_\Om b \otimes u : \nabla( u \eta^4)$ is not integrable.” Thus “we will mollify the drift $b$, so that we can derive an a priori bound.” Since there is no general extension theorem to $L^{2,\infty}(\mathbb{R}^2)$, they “consider the approximation problems in subdomains $\Om_n$ that converge to $\Om$.”
\item \textbf{Pressure control via Wolf projection in $L^q$, $q<2$.} “The second new difficulty comes from the lower integrability of the pressure.” They “bound the pressure using Wolf's local pressure projection with a uniform constant.” In 2D they need “to bound $\pi-(\pi)_{\Omega\cap B_R}$ in $L^q(\Omega_R)$, $q<2$ (we choose $q=4/3$),” because “$\pi\lesssim \nabla u + b\otimes u -\mathbb{G}$” and “$b\otimes u$ is at best in $L^{2-}$ for $u\in W^{1,2}_{\mathrm{loc}}$.” Achieving a uniform constant for the projection in $L^q$, $q<2$ forces stronger boundary regularity: “we have to assume $L$ is sufficiently small and do boundary stretching in the proof for the 2D case.”
\item \textbf{Approximation of Lipschitz domains for constructing solutions.} To construct solutions they use an approximation method, but approximating $b$ directly in arbitrary Lipschitz domains is “not so clear.” Instead, “we approximate the domain by Lipschtiz sub-domains with controlled Lipschitz constant,” on which “we can use standard mollification of $b$,” and then “construct an approximation sequence as zero extension of solutions in these sub-domains.”
\end{itemize}

They indicate the proof is completed by these ingredients and is carried out later: “See Section~\ref{sec6} for details,” and “Section 6 contains proof of the main theorem.”

\label{main1}
Let $\Om$ be a bounded Lipschitz domain in $\mathbb{R}^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\in L^{2,\infty}(\Omega)^2$ and $\mathop{\rm div}\nolimits b=0$.
Then there exists $p_0(\|b\|_{L^{2,\infty}})>2$ such that for any $p_0'<q<p_0$ and $\mathbb{G}\in L^q(\Om)^{2\times 2}$, there exists a  weak solution pair $u\in W^{1,q}_{0,\si}(\Omega)$ and $\pi\in L^q_0(\Omega)$ of \label{drifteq}
-\Delta u +b\cdot\nabla u+\nabla \pi=\div \GG,\quad \div u=0, \quad u|_{\pd \Om}=0,
}
with given drift $b$ in the critical space, $b\in L^{2,\oo}(\Omega)^2$, $\div b=0$, and  force $\G such that
\begin{theorem}\label{main1}
Let $\Om$ be a bounded Lipschitz domain in $\R^2$ with Lipschitz constant $L < L_0$, where $0<L_0<1$ is a sufficiently small constant. Assume $b\in L^{2,\infty}(\Omega)^2$ and $\div b=0$.
Then there exists $p_0(\|b\|_{L^{2,\infty}})>2$ such that for any $p_0'<q<p_0$ and $\GG\in L^q(\Om)^{2\times 2}$, there exists a  weak solution pair $u\in W^{1,q}_{0,\si}(\Omega)$ and $\pi\in L^q_0(\Omega)$ of \eqref{drifteq} such that
\EQ{\label{main1-eq}
\|u\|_{ W^{1,q}(\Omega)}+\|\pi\|_{L^q(\Omega)}\le C(\Omega,\|b\|_{L^{2,\infty}})\|\GG\|_{L^q(\Omega)}.
}
For $p_0'<q<p_0$, the above pair $(u,\pi)$ is the unique weak solution pair  in $W^{1,q}_0\times L^q_0(\Om)$ without assuming \eqref{main1-eq}.
\end{theorem}

For $p_0'<q<p_0$, the above pair $(u,\pi)$ is the unique weak solution pair  in $W^{1,q}_0\times L^q_0(\Om)$ without assuming the preceding estimate.

Question:
Let Ω ⊂ ℝ² be a bounded Lipschitz domain whose Lipschitz constant L satisfies L < L0 for some sufficiently small 0 < L0 < 1. Assume b ∈ L^{2,\infty}(Ω)^2 and div b = 0. For q > 1, set W^{1,q}_{0,σ}(Ω) = {u ∈ W^{1,q}_0(Ω)^2 : div u = 0} and L^q_0(Ω) = {\pi ∈ L^q(Ω) : \int_Ω \pi\,dx = 0}. A weak solution pair (u,\pi) of
-Δu + b\cdot∇u + ∇\pi = div \mathbb{G}, \quad div\,u = 0 \text{ in } Ω, \quad u|_{\partial Ω}=0,
is one with u ∈ W^{1,q}_{0,σ}(Ω), \pi ∈ L^q_0(Ω), and
\int_Ω (∇u - b\otimes u):∇\zeta - \pi\,div\,\zeta = -\int_Ω \mathbb{G}:∇\zeta \quad \text{for all } \zeta \in C_c^\infty(Ω)^2.
If p0' denotes the Hölder conjugate p0/(p0-1), which statement holds for this problem?

There exists p0 = p0(\|b\|_{L^{2,\infty}}) > 2 such that for every q with p0' < q < p0 and every \mathbb{G} ∈ L^q(Ω)^{2\times 2}, there exists a unique weak solution pair (u,\pi) ∈ W^{1,q}_{0,σ}(Ω) \times L^q_0(Ω) of
-Δu + b\cdot∇u + ∇\pi = div \mathbb{G}, \quad div\,u = 0 \text{ in } Ω, \quad u|_{\partial Ω}=0,
and this pair satisfies
\|u\|_{W^{1,q}(Ω)} + \|\pi\|_{L^q(Ω)} \le C(Ω,\|b\|_{L^{2,\infty}})\,\|\mathbb{G}\|_{L^q(Ω)}.
Equivalently, for each such q, the solution pair is unique in W^{1,q}_{0,σ}(Ω) \times L^q_0(Ω) even without assuming the estimate a priori.

Choices:
B.
There exists p0 = p0(\|b\|_{L^{2,\infty}}) > 2 such that for every q with 1 < q < p0 and every \mathbb{G} \in L^q(\Omega)^{2\times 2}, there exists a unique weak solution pair (u,\pi) \in W^{1,q}_{0,\sigma}(\Omega) \times L^q_0(\Omega) of
-\Delta u + b\cdot\nabla u + \nabla\pi = \operatorname{div} \mathbb{G}, \quad \operatorname{div} u = 0 \text{ in } \Omega, \quad u|_{\partial\Omega}=0,
and this pair satisfies
\|u\|_{W^{1,q}(\Omega)} + \|\pi\|_{L^q(\Omega)} \le C(\Omega,\|b\|_{L^{2,\infty}})\,\|\mathbb{G}\|_{L^q(\Omega)}.
C.
There exists p0 = p0(\|b\|_{L^{2,\infty}}) > 2 such that for every q with p0' < q < p0 and every \mathbb{G} \in L^q(\Omega)^{2\times 2}, there exists a weak solution pair (u,\pi) \in W^{1,q}_{0,\sigma}(\Omega) \times L^q_0(\Omega) of
-\Delta u + b\cdot\nabla u + \nabla\pi = \operatorname{div} \mathbb{G}, \quad \operatorname{div} u = 0 \text{ in } \Omega, \quad u|_{\partial\Omega}=0,
such that
\|u\|_{W^{1,q}(\Omega)} + \|\pi\|_{L^q(\Omega)} \le C(\Omega,\|b\|_{L^{2,\infty}})\,\|\mathbb{G}\|_{L^q(\Omega)}.
D.
There exists p0 = p0(\|b\|_{L^{2,\infty}}) > 2 such that for every q with p0' < q < p0 and every \mathbb{G} \in L^q(\Omega)^{2\times 2}, there exists a unique weak solution pair (u,\pi) \in W^{1,q}_{0,\sigma}(\Omega) \times L^q_0(\Omega) of
-\Delta u + b\cdot\nabla u + \nabla\pi = \operatorname{div} \mathbb{G}, \quad \operatorname{div} u = 0 \text{ in } \Omega, \quad u|_{\partial\Omega}=0,
and this pair satisfies
\|u\|_{W^{1,q}(\Omega)} + \|\pi\|_{L^q(\Omega)} \le C(\Omega)\,\|\mathbb{G}\|_{L^q(\Omega)},
with a constant independent of b.
E.
If \Omega \subset \mathbb{R}^2 is a bounded Lipschitz domain and b \in L^{2,\infty}(\Omega)^2 is divergence free, then for every q with p0' < q < p0 and every \mathbb{G} \in L^q(\Omega)^{2\times 2}, there exists a unique weak solution pair (u,\pi) \in W^{1,q}_{0,\sigma}(\Omega) \times L^q_0(\Omega) of
-\Delta u + b\cdot\nabla u + \nabla\pi = \operatorname{div} \mathbb{G}, \quad \operatorname{div} u = 0 \text{ in } \Omega, \quad u|_{\partial\Omega}=0,
and this pair satisfies
\|u\|_{W^{1,q}(\Omega)} + \|\pi\|_{L^q(\Omega)} \le C(\Omega,\|b\|_{L^{2,\infty}})\,\|\mathbb{G}\|_{L^q(\Omega)}.

dependence of constants on \|b\|_{L^{2,\infty}}

small Lipschitz constant requirement for Wolf projection/boundary stretching

The stem does not explicitly reveal the correct option. It gives the PDE setting and definition of weak solution, but the key conclusion—exact q-range, uniqueness strength, and parameter dependence—must be inferred from the choices.

The item is very close to a theorem-recall question: the correct choice is essentially the precise theorem statement. Still, the alternatives vary in subtle ways (endpoint inclusion, uniqueness, dependence on b, q-range), so it is not a pure verbatim restatement.

Some reasoning is required to identify the strongest valid statement among nearby variants, especially regarding open vs. closed ranges and dependence of p0. However, the task mainly tests recognition of the exact theorem rather than substantial derivation or generative mathematical reasoning.

The distractors are plausible and mathematically meaningful: they alter lower endpoints, endpoint inclusion, uniqueness, and norm/parameter dependence in ways that match common failure modes when recalling regularity theorems.

A solid theorem-discrimination MCQ with strong distractors and no direct answer leakage, but it leans heavily toward precise theorem recall rather than genuinely generative reasoning.


Question:
Let Ω ⊂ ℝ² be a bounded Lipschitz domain whose Lipschitz constant L satisfies L < L0 for some sufficiently small 0 < L0 < 1. Assume b ∈ L^{2,\infty}(Ω)^2 and div b = 0. For q > 1, set W^{1,q}_{0,σ}(Ω) = {u ∈ W^{1,q}_0(Ω)^2 : div u = 0} and L^q_0(Ω) = {\pi ∈ L^q(Ω) : \int_Ω \pi\,dx = 0}. A weak solution pair (u,\pi) of
-Δu + b\cdot∇u + ∇\pi = div \mathbb{G}, \quad div\,u = 0 \text{ in } Ω, \quad u|_{\partial Ω}=0,
is one with u ∈ W^{1,q}_{0,σ}(Ω), \pi ∈ L^q_0(Ω), and
\int_Ω (∇u - b\otimes u):∇\zeta - \pi\,div\,\zeta = -\int_Ω \mathbb{G}:∇\zeta \quad \text{for all } \zeta \in C_c^\infty(Ω)^2.
If p0' denotes the Hölder conjugate p0/(p0-1), which statement holds for this problem?

Choices:
B.
There exists p0 = p0(\|b\|_{L^{2,\infty}}) > 2 such that for every q with 1 < q < p0 and every \mathbb{G} \in L^q(\Omega)^{2\times 2}, there exists a unique weak solution pair (u,\pi) \in W^{1,q}_{0,\sigma}(\Omega) \times L^q_0(\Omega) of
-\Delta u + b\cdot\nabla u + \nabla\pi = \operatorname{div} \mathbb{G}, \quad \operatorname{div} u = 0 \text{ in } \Omega, \quad u|_{\partial\Omega}=0,
and this pair satisfies
\|u\|_{W^{1,q}(\Omega)} + \|\pi\|_{L^q(\Omega)} \le C(\Omega,\|b\|_{L^{2,\infty}})\,\|\mathbb{G}\|_{L^q(\Omega)}.
C.
There exists p0 = p0(\|b\|_{L^{2,\infty}}) > 2 such that for every q with p0' < q < p0 and every \mathbb{G} \in L^q(\Omega)^{2\times 2}, there exists a weak solution pair (u,\pi) \in W^{1,q}_{0,\sigma}(\Omega) \times L^q_0(\Omega) of
-\Delta u + b\cdot\nabla u + \nabla\pi = \operatorname{div} \mathbb{G}, \quad \operatorname{div} u = 0 \text{ in } \Omega, \quad u|_{\partial\Omega}=0,
such that
\|u\|_{W^{1,q}(\Omega)} + \|\pi\|_{L^q(\Omega)} \le C(\Omega,\|b\|_{L^{2,\infty}})\,\|\mathbb{G}\|_{L^q(\Omega)}.
D.
There exists p0 = p0(\|b\|_{L^{2,\infty}}) > 2 such that for every q with p0' < q < p0 and every \mathbb{G} \in L^q(\Omega)^{2\times 2}, there exists a unique weak solution pair (u,\pi) \in W^{1,q}_{0,\sigma}(\Omega) \times L^q_0(\Omega) of
-\Delta u + b\cdot\nabla u + \nabla\pi = \operatorname{div} \mathbb{G}, \quad \operatorname{div} u = 0 \text{ in } \Omega, \quad u|_{\partial\Omega}=0,
and this pair satisfies
\|u\|_{W^{1,q}(\Omega)} + \|\pi\|_{L^q(\Omega)} \le C(\Omega)\,\|\mathbb{G}\|_{L^q(\Omega)},
with a constant independent of b.
E.
If \Omega \subset \mathbb{R}^2 is a bounded Lipschitz domain and b \in L^{2,\infty}(\Omega)^2 is divergence free, then for every q with p0' < q < p0 and every \mathbb{G} \in L^q(\Omega)^{2\times 2}, there exists a unique weak solution pair (u,\pi) \in W^{1,q}_{0,\sigma}(\Omega) \times L^q_0(\Omega) of
-\Delta u + b\cdot\nabla u + \nabla\pi = \operatorname{div} \mathbb{G}, \quad \operatorname{div} u = 0 \text{ in } \Omega, \quad u|_{\partial\Omega}=0,
and this pair satisfies
\|u\|_{W^{1,q}(\Omega)} + \|\pi\|_{L^q(\Omega)} \le C(\Omega,\|b\|_{L^{2,\infty}})\,\|\mathbb{G}\|_{L^q(\Omega)}.

The stem does not explicitly reveal the correct option. It gives the PDE setup and definitions, but the decisive theorem-level conclusion (exact q-range, uniqueness, and estimate dependence) is left to the choices.

The item is essentially a theorem-identification question: it asks which theorem statement is correct for the given problem. While the options differ in meaningful ways, the task is still close to selecting the precise restatement of a known result rather than deriving a new conclusion from the setup.

Some reasoning is needed to distinguish subtle variants: the sharp range p0' < q < p0, the uniqueness clause, dependence on ||b||_{L^{2,\infty}}, and the small-Lipschitz assumption. However, the problem mainly tests recall/recognition of the exact theorem statement rather than substantial generative mathematical reasoning.

The distractors are plausible and mathematically targeted. They reflect common failure modes: using the wrong q-range, dropping uniqueness, claiming b-independent constants, or omitting the small Lipschitz requirement. They are distinct and close enough to the correct theorem to be effective.

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