\begin{remark}\label{remdecv} Note that we have also the following regularity properties of $Y_\chi v$ and $Z_\chi v$:
 \begin{equation}\label{YZregbis}
\left\{\begin{array}{c}Y_\chi w\in \Psi^{0} w,\quad \divb Y_\chi w\in \Psi^{-\infty} w,\\  Z_\chi w\in \Psi^{-2}\divb w,\end{array}\right.
  \end{equation}
that are useful if we have regularity information on $\divb w$ or if we need regularity on $\divb Y_\chi w$. Typically when the regularity is different for $v$ and $\divb v$, as in \cite{Br}, both decompositions of $v$ and $w$ are needed to define the product $vw$.
In our case the Hodge decomposition \eqref{Hodge} is not on free divergence but on smooth divergence  fields, and the product formula \eqref{prod} can be extended to
\begin{equation}\label{prodbis}
v\cdot w =Y_\chi v\cdot Y_\chi w +\divb (Z_\chi v Y_\chi w)-(\divb Y_\chi w)\,Z_\chi v-(\divb Y_\chi v)\,Z_\chi w+\divb(Y_\chi v\,Z_\chi w)+\nabla Z_\chi v\cdot \nabla Z_\chi w.
\end{equation}
In this note we stick to the classical framework of same regularity for $v$ and $\divb v$ as well as for $w$ and $\curl w$, and we will need to decompose only $w$ and to use \eqref{YZreg}.
\end{remark}

\begin{equation}\label{Hodge}
 Z_\chi w := -\chi(D)(-\Delta)^{-1}\divb w \in \mathcal D ', \quad Y_\chi w:=w-\nabla Z_\chi w\in\mathcal  D '
 \end{equation}

\begin{theorem}\label{theo1}
The map
$$(v,w) \in \mathcal{M}_{0,\divb}\times L^\infty_{\curl}\mapsto (v\cdot w)_H \in \mathcal{D'},$$
is weakly continuous.
More precisely, let $(v_n,w_n)$ be  a bounded sequence in $\mathcal{M}_{0,\divb} \times L^\infty_{\curl}$ converging in the sense of distributions to $(v,w)$:
$$ v_n \overset{\mathcal{D}'}{\rightharpoonup} v, \qquad w_n \overset{\mathcal{D}'}{\rightharpoonup}  w,$$
so in particular $(v,w)\in  \mathcal{M}_{0,\divb}\times L^\infty_{\curl}$.
Then we have the product convergence:
$$ (v_n\cdot w_n)_H \overset{\mathcal{D}'}{\rightharpoonup}  (v\cdot  w)_H,$$
where the products are defined by Proposition~\ref{prop1}.
The result is valid also with $\mathcal{M}_{0,\divb} \times L^\infty_{\curl}$ replaced by:\par
\,\hspace{3mm}{\it{(i) $\mathcal M_{0,\divb} \times W_{\curl}^{-1+ \epsilon, \infty}$, for $\epsilon>0$,}} \par
\,\hspace{1mm}{\it{(ii)$^*$ $\mathcal M_{0,\divb} \times W_{\curl}^{-1+\frac dp+\epsilon, p}$, for $d<p< \infty$, $\epsilon>0$,}}\par
\,\hspace{1mm}\it{(iii)}  $W^{-1+\epsilon,\infty}_{\divb} \times \mathcal M_{0,\curl}$, for $\epsilon>0$, \par
\,\hspace{2mm}\it{(iv)} $W^{-\delta+\epsilon,p'}_{\divb} \times \mathcal M_{0,\curl}$, for $1< p\leq \frac d{d-1+\delta}$ and $\delta\in(0,1),\epsilon>0$, \par
\,\hspace{2mm}\it{(v)$^*$}  $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha+\epsilon, p}$, for $1< p< \infty $, $\epsilon>0$, $\alpha\in \mathbb R$, \par
\,\hspace{2mm}\it{(vi)}  $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha+\epsilon, p}$, for $p\in\{1,\infty\}$, $\epsilon>0$, $\alpha\in \mathbb R$.

\end{theorem}

Div-curl lemmas have been introduced by Murat and Tartar while establishing the compensated compactness theory in the end of the seventies (\cite{Mu1},\cite{Mu2},\cite{Ta1},\cite{Ta2}, see also \S 7,17 of \cite{Tartarbook}, and \cite{Mu3}).

The classical result states, for $1<p<\infty$ and $p'$ its conjugate, that given two sequences $(v_n)$ and $(w_n)$ uniformly bounded and weakly converging in $L^{p}$ and $L^{p'}$respectively, such that $(\divb\, v_n)$ is uniformly bounded in $L^p$ and $(\curl w_n)$ is uniformly bounded in $L^{p'}$, their product $(v_n\cdot w_n)$ converges in the sense of distributions\footnote{Another slightly different classical setting is the case, based mainly on the case $p=2$, where the sequences $(v_n)$ and $(w_n)$ are bounded and weakly converging in $L^{p}$ and $L^{p'}$respectively, and such that $(\divb\, v_n)$ and $(\curl w_n)$ are only compact in $W^{-1,p}$ and $W^{-1,p'}$ respectively.}. The classical div-curl lemma has several different proofs\footnote{In the case $p=2$ an approach based on default measures can be used. However these objects do not have an obvious extension to the $L^1/L^\infty$ regularity.}, see for instance \cite{Pr} for a review. Its classical proof uses Hodge's decomposition on divergence-free fields $w_n=y_n+\nabla z_n$, retrieving for instance uniform boundedness of $z_n$ in $W^{1,p'}$, which allows for compactness in $L^{p'}$. We note that here $1<p<\infty$, thus Hodge's decomposition on divergence-free fields is classical. We also note that in the conjugate-exponent spaces $L^p-L^{p'}$ there is no issue with defining the products $v_n\cdot w_n$.

A recent extension has been obtained by Briane, Casado-D\'iaz and Murat in \cite{Br} by considering for the regularity of $v_n$ and $w_n$, instead of $L^p-L^{p'}$, the spaces $L^p-L^q$ with $\frac 1p+\frac 1q=1+\frac 1d$ and $1<p,q<\infty$, and by considering for $\divb v_n$ and $\curl w_n$ some other appropriate regularity, see Theorem 2.3 in \cite{Br}. They also considered the spaces $\mathcal M-L^d$ and $L^d-\mathcal M$, where $\mathcal{M}$ is the set of Radon measures, see Theorems 3.1 and 4.1 respectively in \cite{Br}. In the first case $L^p-L^q$ the authors give a definition of the product $v_nw_n$ in the sense of distributions, based on Hodge's decompositions on divergence-free fields for both $v_n$ and $w_n$. In the remaining two cases, involving $\mathcal M$, firstly the product is defined by proving moreover an extension to measures of the representation of divergence free functions in $L^1$ of Br\'ezis and Van Schaftingen from \cite{BrVS}.
Then, once the product is well defined, for instance in the first case $L^p-L^q$, the uniform boundedness of $z_n$ in $W^{1,q}$ does not allow for compactness in $L^{p'}$, since $W^{1,q}$ is embedded in $L^{q^*}=L^{p'}$ but without compactness. The authors use the defect of compactness of the embedding, with the help of the celebrated concentration compactness principle of Lions \cite{Li}, to get a $\mathcal D'$-limit for $v_nz_n$, involving $v\cdot w$ and a combination of Dirac measures. \\

The main aim\footnote{The motivation to look at div-curl lemmas came to us, after having obtained in \cite{BBWFL1} results of $L^1$-regularity in linear situations, by having in mind nonlinear situations.} of this note is to give a short proof for the limit conjugate case $\mathcal M-L^\infty$. We shall use a non unique microlocal Hodge's decomposition that, contrary to the classical one, does not involve a divergence free field part. It will be our main ingredient, that will allow us to define the products $v_n\cdot w_n$ and $v\cdot w$, and modifying slightly the standard approach to obtain very easily a limit in the sense of distributions for $v_n\cdot w_n$. The method applies to much more spaces than $\mathcal M-L^\infty$, as stated in Proposition \ref{prop1} and Theorem \ref{theo1}.

Let $d\geq 2$, $1\leq  p \leq +\infty$, and $\mathcal F$ a function space among $L^p$, Sobolev spaces and $\mathcal M_0$, the set of finite Radon measures. We denote
\begin{equation}
\begin{aligned}
\mathcal{F}_{\curl}&:= \{ u \in \mathcal F( \mathbb{R}^d, \mathbb{R}^d), \curl u \in \mathcal F( \mathbb{R}^d, \mathbb{R}^{d^2})\},\\
\mathcal{F}_{\divb}&:= \{ u \in \mathcal F( \mathbb{R}^d, \mathbb{R}^d),\divb u \in \mathcal F( \mathbb{R}^d, \mathbb{R})\},\\
\end{aligned}
\end{equation}
and we endow these spaces with their natural norms. For sake of simplicity we do not specify anymore the natural domain and range of the functions. The spaces $\mathcal{F}_{\divb}$ and $\mathcal{F}_{\curl}$ that we shall consider in the sequel are continuously stable by multiplication by a $\mathcal C^\infty_0$ localization.

\begin{remark} Since the conclusions in Proposition \ref{prop1} and Theorem~\ref{theo1} below are in $\mathcal{D}'$, thus local, we can weaken the global hypothesis and assume only local conditions as for instance
$ \mathcal{M}_{\divb}, (L^\infty_{\curl})_{\text{loc}}$ instead of $\mathcal{M}_{0,\divb},L^\infty_{\curl}$. In particular we could place ourselves in the case of an open set $\Omega$ of $\mathbb R^d$ instead of $\mathbb R^d$ and consider local spaces in $\Omega$.
\end{remark}

We start with a result ensuring the existence of products in the sense of distributions between various spaces.

\begin{prop}\label{prop1} Let $ (v,w) \in \mathcal M_{0,\divb} \times L_{\curl}^\infty$ or belonging to one of  the following spaces:
\begin{enumerate}
\item $\mathcal M_{0,\divb} \times W_{\curl}^{-1+ \epsilon, \infty}$, for $\epsilon>0$,
\item $\mathcal M_{0,\divb} \times W_{\curl}^{-1+\frac dp, p}$, for $d<p< \infty$,
\item  $W^{-1+\epsilon,\infty}_{\divb} \times \mathcal M_{0,\curl}$, for $\epsilon>0$,
\item  $W^{-\delta+\epsilon,p'}_{\divb} \times \mathcal M_{0,\curl}$, for $1< p\leq \frac d{d-1+\delta}$\footnote{The condition $1<p\leq \frac d{d-1+\delta}$ is equivalent to $\frac{d}{1-\delta}\leq p'< \infty$, thus varying $\delta$ in $(0,1)$ we have the range $d<p'< \infty$}. and $\delta\in(0,1),\epsilon>0$,
\item  $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha, p}$, for $1< p< \infty $, $\alpha\in \mathbb R$,
\item  $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha+\epsilon, p}$, for $p\in\{1,\infty\}$, $\epsilon>0$, $\alpha\in \mathbb R$.
\end{enumerate}
Then $w$ admits a class of Hodge-type decompositions $w=y+\nabla z$ such that the following quantity
\begin{equation}\label{Hodgeprod}
(v\cdot w)_H:=v\cdot y -(\divb\,v)z+\divb\,(vz).
\end{equation}
is well-defined as a distribution, is independent of the choice of the decomposition, and coincides with the usual product if $(v,w) \in\mathcal C^\infty_0\times \mathcal C ^\infty_0$ or if $(v,w) \in L_{\divb}^{p'}\times L_{\curl}^p$ for $1\leq p\leq \infty$.

Moreover, in all space $\mathcal{F}_{\divb}\times\mathcal{\tilde F}_{\curl}$ of the cases (ii)-(iv)-(v), thus not involving $L^\infty$ for which $C^\infty_0$ is not dense, the classical product map
$$ (v,w) \in\mathcal C^\infty_0\times \mathcal C ^\infty_0 \mapsto v \cdot w  \in \mathcal C ^\infty_0 $$
admits $(vw)_H$ as a unique continuous extension from  $\mathcal{F}_{\divb}\times\mathcal{\tilde F}_{\curl}$ to $\mathcal{D}' $.
\end{prop}

\begin{remark}

In the remaining spaces $\mathcal{F}_{\divb}\times\mathcal{\tilde F}_{\curl}$ of the cases (i)-(iii)-(vi), involving $L^\infty$, the product $(v\cdot w)_H$ is still the natural one. More precisely, if $(v_n,w_n)$
is any sequence bounded in $\mathcal{F}_{\divb}\times\mathcal{\tilde F}_{\curl}$  converging in the sense of distributions to $(v,w)$, such that the product $v_n\cdot w_n$ is well-defined in the classical sense, for instance a mollified approximation of $(v,w)$, then we will obtain by Theorem \ref{theo1} that
$$(v\cdot w)_H\overset{\mathcal D'}{=}\lim_{n \rightarrow \infty}v_n\cdot  w_n,$$
and consequently, the product $(v\cdot w)_H$ is in this cases the unique extension from $\mathcal F_{\divb}\cap\mathcal C^0\times\mathcal {\tilde F}_{\curl}\cap\mathcal C^0$ to $\mathcal D'$ which enjoys this weak continuity property.
\end{remark}

The proof of Proposition \ref{prop1} is based on the Hodge decomposition defined in \eqref{Hodge} and on the action of pseudo-differential operators on $L^p$ and $\mathcal M_0$. The definition, notations and properties of pseudo-differential operators that will be of use in this note are given in Appendix \ref{app}.

We note that the space from {\em{(v)}} with $\alpha=0$, i.e. $W_{\divb}^{p}\times W_{\curl}^{-1, p'}$ for $1< p<\infty$, extends the classical $L^p-L^{p'}$ framework. Moreover, as a subcase we obtain from it the space $L^p_{\divb}\times L^q_{\curl}$ with  $\frac 1p+\frac 1q=1+\frac 1d$, since\footnote{We recall some Sobolev embeddings that will be used in the sequel: $W^{k,\alpha}\subset L^p$ for $k\in(0,1), k<\frac d\alpha, \alpha\in [1,\infty), p\in[\alpha,\frac{d\alpha}{d-k\alpha}]$,
$W^{k,\alpha}\subset \mathcal C^{0,\frac{k\alpha-d}{\alpha}}$ for $k\in (0,1), k>\frac d\alpha, \alpha\in [1,\infty]$, and $W^{\delta,1}\subset L^p$ for $p\in [1,\frac d{d-\delta}]$.}
 in this case $L^q=L^{(p^*)'}\subset W^{-1,p'}$.
We also note that from {\em{(ii)}}, since $L^d\subset W^{-1+\frac{d}{d^+},d^+}$, we obtain as a subcase the space $\mathcal M_{0,\divb}\times L^d_{\curl}$, and similarly we obtain the space $L^d_{\divb}\times \mathcal M_{0,\curl}$ as a subcase of {\em{(iv)}}. These last three cases corresponds to the ones in \cite{Br} modulo the fact that here we have the same regularity for $v_n$ and $\divb v_n$ and for $w_n$ and $\curl w_n$. To extend to different appropriate regularities for $\divb v_n$ and $\curl w_n$ as in \cite{Br} the proof could be extended in the spirit of Remark \ref{remdecv}.

Now we can state the div-curl lemma.

A recent extension has been obtained by Briane, Casado-D\'iaz and Murat in \cite{Br} by considering for the regularity of $v_n$ and $w_n$, instead of $L^p-L^{p'}$, the spaces $L^p-L^q$ with $\frac 1p+\frac 1q=1+\frac 1d$ and $1<p,q<\infty$, and by considering for $\divb v_n$ and $\curl w_n$ some other appropriate regularity, see Theorem 2.3 in \cite{Br}. They also considered the spaces $\mathcal M-L^d$ and $L^d-\mathcal M$, where $\mathcal{M}$ is the set of Radon measures, see Theorems 3.1 and 4.1 respectively in \cite{Br}. In the first case $L^p-L^q$ the authors give a definition of the product $v_nw_n$ in the sense of distributions, based on Hodge's decompositions on divergence-free fields for both $v_n$ and $w_n$. In the remaining two cases, involving $\mathcal M$, firstly the product is defined by proving moreover an extension to measures of the representation of divergence free functions in $L^1$ of Br\'ezis and Van Schaftingen from \cite{BrVS}.
Then, once the product is well defined, for instance in the first case $L^p-L^q$, the uniform boundedness of $z_n$ in $W^{1,q}$ does not allow for compactness in $L^{p'}$, since $W^{1,q}$ is embedded in $L^{q^*}=L^{p'}$ but without compactness. The authors use the defect of compactness of the embedding, with the help of the celebrated concentration compactness principle of Lions \cite{Li}, to get a $\mathcal D'$-limit for $v_nz_n$, involving $v\cdot w$ and a combination of Dirac measures. \\

\begin{prop}\label{prop1} Let $ (v,w) \in \mathcal M_{0,\divb} \times L_{\curl}^\infty$ or belonging to one of the following spaces:
\begin{enumerate}
\item $\mathcal M_{0,\divb} \times W_{\curl}^{-1+ \epsilon, \infty}$, for $\epsilon>0$,
\item $\mathcal M_{0,\divb} \times W_{\curl}^{-1+\frac dp, p}$, for $d<p< \infty$,
\item $W^{-1+\epsilon,\infty}_{\divb} \times \mathcal M_{0,\curl}$, for $\epsilon>0$,
\item $W^{-\delta+\epsilon,p'}_{\divb} \times \mathcal M_{0,\curl}$, for $1< p\leq \frac d{d-1+\delta}$\footnote{The condition $1<p\leq \frac d{d-1+\delta}$ is equivalent to $\frac{d}{1-\delta}\leq p'< \infty$, thus varying $\delta$ in $(0,1)$ we have the range $d<p'< \infty$}. and $\delta\in(0,1),\epsilon>0$,
\item $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha, p}$, for $1< p< \infty $, $\alpha\in \mathbb R$,
\item $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha+\epsilon, p}$, for $p\in\{1,\infty\}$, $\epsilon>0$, $\alpha\in \mathbb R$.
\end{enumerate}
Then $w$ admits a class of Hodge-type decompositions $w=y+\nabla z$ such that the following quantity
\begin{equation}\label{Hodgeprod}
(v\cdot w)_H:=v\cdot y -(\divb\,v)z+\divb\,(vz).
\end{equation}
is well-defined as a distribution, is independent of the choice of the decomposition, and coincides with the usual product if $(v,w) \in\mathcal C^\infty_0\times \mathcal C ^\infty_0$ or if $(v,w) \in L_{\divb}^{p'}\times L_{\curl}^p$ for $1\leq p\leq \infty$.

In the remaining spaces $\mathcal{F}_{\divb}\times\mathcal{\tilde F}_{\curl}$ of the cases (i)-(iii)-(vi), involving $L^\infty$, the product $(v\cdot w)_H$ is still the natural one. More precisely, if $(v_n,w_n)$
is any sequence bounded in $\mathcal{F}_{\divb}\times\mathcal{\tilde F}_{\curl}$ converging in the sense of distributions to $(v,w)$, such that the product $v_n\cdot w_n$ is well-defined in the classical sense, for instance a mollified approximation of $(v,w)$, then we will obtain by Theorem \ref{theo1} that
$$(v\cdot w)_H\overset{\mathcal D'}{=}\lim_{n \rightarrow \infty}v_n\cdot w_n,$$
and consequently, the product $(v\cdot w)_H$ is in this cases the unique extension from $\mathcal F_{\divb}\cap\mathcal C^0\times\mathcal {\tilde F}_{\curl}\cap\mathcal C^0$ to $\mathcal D'$ which enjoys this weak continuity property.
\end{remark}

We note that the space from {\em{(v)}} with $\alpha=0$, i.e. $W_{\divb}^{p}\times W_{\curl}^{-1, p'}$ for $1< p<\infty$, extends the classical $L^p-L^{p'}$ framework. Moreover, as a subcase we obtain from it the space $L^p_{\divb}\times L^q_{\curl}$ with $\frac 1p+\frac 1q=1+\frac 1d$, since\footnote{We recall some Sobolev embeddings that will be used in the sequel: $W^{k,\alpha}\subset L^p$ for $k\in(0,1), k<\frac d\alpha, \alpha\in [1,\infty), p\in[\alpha,\frac{d\alpha}{d-k\alpha}]$,
$W^{k,\alpha}\subset \mathcal C^{0,\frac{k\alpha-d}{\alpha}}$ for $k\in (0,1), k>\frac d\alpha, \alpha\in [1,\infty]$, and $W^{\delta,1}\subset L^p$ for $p\in [1,\frac d{d-\delta}]$.}
 in this case $L^q=L^{(p^*)'}\subset W^{-1,p'}$.
We also note that from {\em{(ii)}}, since $L^d\subset W^{-1+\frac{d}{d^+},d^+}$, we obtain as a subcase the space $\mathcal M_{0,\divb}\times L^d_{\curl}$, and similarly we obtain the space $L^d_{\divb}\times \mathcal M_{0,\curl}$ as a subcase of {\em{(iv)}}. These last three cases corresponds to the ones in \cite{Br} modulo the fact that here we have the same regularity for $v_n$ and $\divb v_n$ and for $w_n$ and $\curl w_n$. To extend to different appropriate regularities for $\divb v_n$ and $\curl w_n$ as in \cite{Br} the proof could be extended in the spirit of Remark \ref{remdecv}.

Now we can state the div-curl lemma.

We note that the space from {\em{(v)}} with $\alpha=0$, i.e. $W_{\divb}^{p}\times W_{\curl}^{-1, p'}$ for $1< p<\infty$, extends the classical $L^p-L^{p'}$ framework. Moreover, as a subcase we obtain from it the space $L^p_{\divb}\times L^q_{\curl}$ with $\frac 1p+\frac 1q=1+\frac 1d$, since\footnote{We recall some Sobolev embeddings that will be used in the sequel: $W^{k,\alpha}\subset L^p$ for $k\in(0,1), k<\frac d\alpha, \alpha\in [1,\infty), p\in[\alpha,\frac{d\alpha}{d-k\alpha}]$,
$W^{k,\alpha}\subset \mathcal C^{0,\frac{k\alpha-d}{\alpha}}$ for $k\in (0,1), k>\frac d\alpha, \alpha\in [1,\infty]$, and $W^{\delta,1}\subset L^p$ for $p\in [1,\frac d{d-\delta}]$.}
 in this case $L^q=L^{(p^*)'}\subset W^{-1,p'}$.
We also note that from {\em{(ii)}}, since $L^d\subset W^{-1+\frac{d}{d^+},d^+}$, we obtain as a subcase the space $\mathcal M_{0,\divb}\times L^d_{\curl}$, and similarly we obtain the space $L^d_{\divb}\times \mathcal M_{0,\curl}$ as a subcase of {\em{(iv)}}. These last three cases corresponds to the ones in \cite{Br} modulo the fact that here we have the same regularity for $v_n$ and $\divb v_n$ and for $w_n$ and $\curl w_n$. To extend to different appropriate regularities for $\divb v_n$ and $\curl w_n$ as in \cite{Br} the proof could be extended in the spirit of Remark \ref{remdecv}.

The proof of Theorem \ref{theo1} is based on Proposition \ref{prop1} and on the fact that the smooth localization $u\in\ W^{\epsilon,p}\to \chi u \in L^p$ is compact for $\epsilon>0$ and $1\leq p\leq \infty$. Thus a bit of regularity is lost with respect to Proposition \ref{prop1}. This explains why Theorem \ref{theo1} is valid in all the functional settings listed in Proposition \ref{prop1} with slight modifications for the spaces in {\em{(ii)}} and {\em{(v)}}, i.e. for instance $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha, p}$, for $1< p<\infty $, $\alpha\in \mathbb R$, is replaced by $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha+\epsilon, p}$.

Now we are ready to define the product in the statement of Proposition \ref{prop1}. We use the Hodge decomposition \eqref{Hodgedec} of $w$ to define, whenever each product term in the following is well defined in the sense of distributions, the quantity:
\begin{equation}\label{prod}
(v\cdot w)_\chi =v\cdot Y_\chi w -(\divb\,v)Z_\chi w+\divb\,(v\,Z_\chi w).
\end{equation}
We notice that for $ (v,w) \in \mathcal C^\infty_0\times \mathcal C ^\infty_0$ or if $(v,w) \in L_{\divb}^{p'}\times L_{\curl}^p$ for $1\leq p\leq \infty$, this quantity identifies with the usual product:
$$v\cdot w=(v\cdot w)_\chi.$$

To prove the last extension result in Proposition \ref{prop1}, say for $\mathcal{M}_{0,\divb}\times W^{-1+\frac dp+\epsilon,p}_{\curl}$ with $d<p<\infty$ (the other cases follow similarly), we use the same arguments before to get that the map
$$ (v,w) \in \mathcal C^\infty_0\times \mathcal C^\infty_0\mapsto v w \in \mathcal{D}'$$
is continuous for the $\mathcal{M}_{0,\divb}\times W^{-1+\frac dp+\epsilon,p}_{\curl}$ topology.
Indeed, as before, we get from \eqref{YZreg} that along with
$$
Y_\chi w, Z_\chi w\in \Psi^{-1}W^{-1+\frac dp+\epsilon,p}\subset W^{\frac dp+\epsilon,p}\subset \mathcal C^0,
$$
we have also the continuity for the $W^{-1+\frac dp+\epsilon,p}_{\curl}$ topology of the maps
$$
w\in C^\infty_0 \mapsto Y_\chi w \in \mathcal C^0,\quad w\in C^\infty_0 \mapsto Z_\chi w \in \mathcal C^0.
$$
As a consequence we deduce that the maps
\begin{equation}\label{maps}
 (v,w) \in \mathcal C^\infty_0\times \mathcal C^\infty_0 \mapsto \begin{cases}v\cdot Y_\chi w \in \mathcal{D}',\\ (\divb v) Z_\chi w \in \mathcal{D}',\\
v\,Z_\chi w \in \mathcal{D}',
\end{cases}
\end{equation}
are continuous for the $\mathcal{M}_{0,\divb}\times W^{-1+\frac dp+\epsilon,p}_{\curl}$ topology, thus so is the map
$$ (w\cdot v) \in \mathcal C^\infty_0 \times \mathcal C^\infty_0 \mapsto w\cdot v =v\cdot Y_\chi w-(\divb\, v)Z_\chi w+\divb\,(v\,Z_\chi w) \in \mathcal{D}'.$$
Consequently the product map has a unique extension from $\mathcal{M}_{0,\divb}\times W^{-1+\frac dp+\epsilon,p}_{\curl}$ to $\mathcal D'$. This extension defines the product on $ \mathcal{M}_{0,\divb}\times W^{-1+\frac dp+\epsilon,p}_{\curl}$ as a distribution, and it does not depend on the choice of the cut-off function $\chi$ because it coincides, whatever this choice, with the usual product on the dense subspace $\mathcal C^\infty_0\times \mathcal C^\infty _0$.\\

\appendix
\section{}\label{app}
In this appendix we recall the definition of pseudodifferential operators and describe their action on $L^p$ spaces and on $\mathcal M_0$. We refer to~\cite[Chapter XVIII]{Ho} for a general presentation or~\cite[\S VI.6]{Stein} for a presentation closer to our needs of the pseudodifferential calculus (see also \cite{AlGe}).
Let us first recall the definitions of the class of symbol of order $\delta\in\mathbb R$:
$$
 S^\delta( \mathbb{R}^d) =\{a\in C^\infty( \mathbb{R}^{d}); \\
 \forall \alpha, \beta \in \mathbb{N}^d, \sup_{x,\xi \in \mathbb{R}^d} |\partial_x^\alpha \partial_\xi^\beta a(x, \xi)| (1+ |\xi|)^{-\delta+|\gamma|}=: \|a\|_{\delta, \alpha, \beta}<+\infty\}.
$$
To any symbol $a \in S^\delta_{\text{cl}}(\mathbb{R}^d)$ we can associate an operator on the temperate distributions set $\mathcal{S}'( \mathbb{R}^d)$ by the formula
$$ a(x,D_x) u (x)= \text{Op} (a) u(x)= \frac{ 1 }{(2\pi)^d} \int e^{i(x-y) \cdot \xi} a(x, \xi) u(y) dy d\xi.$$
The set of such operators, that are called pseudodifferential operators of order $\delta$, is denoted by $\Psi^\delta$. The set $\Psi^{-\infty}$ is defined as $\cap_{\delta<0}\Psi^\delta$. In the following proposition we gather the results needed in this note regarding to the action of pseudodifferential operators.

\begin{proof}
The results (i)-(ii) are classical: for the first one see for instance \cite[Section VI.5.2]{Stein}, and the second follows by the dual estimate of the Lemma in~\cite[Section VI.5.3.1]{Stein}. The result (iii) is less classical and we give here a complete (simple) proof.
We use a dyadic partition of unity
$$ 1 =\sum_{j\geq 0} \phi_j (\xi) , \, \phi_0 \in C^\infty _ 0 (\mathbb{R}^d), \,\forall j \geq 1, \phi_j (\xi) = \phi( 2^{-j} \xi), \,\phi \in C^\infty_0 ( \{ \frac 1 2 < \| \xi\| <2 \} ),$$
 to decompose
 $$ A = \sum_{j\geq 0} A\, \phi_j (D).$$
 Each operator of the sum has the kernel
$$
K_j(x,y):= \frac 1 {(2\pi)^d} \int e^{i(x-y) \cdot \xi } a(x,\xi) \phi_j(\xi) d\xi.$$
In view of the decay of $a$ and of the localization of $\phi_j$ we have
$$|K_j(x,y)|\leq C 2^{j(d+\delta)}\|a\|_{\delta,0,0}.$$
Also, integrating by parts $N$ times using the identity
$$ L(e^{i(x-y) \cdot \xi} )= - e^{i(x-y) \cdot \xi}, \qquad L= \frac {i (x-y) \cdot \nabla_\xi} {\| x-y\|^2} ,$$
we get for $N>1$ if $\|x-y\|\neq 0$, that
$$
|K_j(x,y)|=\Big| \frac {1} {(2\pi)^d} \int e^{i(x-y) \cdot \xi } L^N (a(x,\xi) \phi_j(\xi) ) d\xi\Big| \leq C 2^{j(d+\delta)}\frac{\|a\|_{\delta,0,N}}{(2^j\|x-y\|)^N}.
$$
Therefore by taking $N=d+1$ we obtain
$$\int|K_j(x,y)|dx=\int_{2^j\|x-y\|<1}|K_j(x,y)|dx+\int_{2^j\|x-y\|>1}|K_j(x,y)|dx
\leq C 2^{j\delta}(\|a\|_{\delta,0,0}+\|a\|_{\delta,0,N}),
$$
and similarly
$$\int|K_j(x,y)|dy\leq C(a)2^{j\delta}.$$
For $\mu\in\mathcal M_0$ we have, since $\delta<0$,
$$\|A\mu\|_{L^1}\leq \sum_{j\geq 0}\Big\|\int K_j(x,y)d\mu(y)\Big\|\leq C(a) \|\mu\|\sum_{j\geq 0}2^{j\delta}\leq C(a)\|\mu\|.$$
In conclusion $\Psi^\delta$ acts continuously from $\mathcal M_0(\mathbb R^n)$ to $L^1(\mathbb R^n)$ for any $\delta<0$. Combining with (ii) we obtain that $\Psi^\delta$ acts continuously from $\mathcal M_0(\mathbb R^n)$ to $W^{-\delta-\epsilon,1}(\mathbb R^n)$ for any $\delta<0$ and $\epsilon>0$.
\end{proof}

The proof of Theorem \ref{theo1} is based on Proposition \ref{prop1} and on  the fact that the smooth localization $u\in\ W^{\epsilon,p}\to \chi u \in L^p$ is compact for $\epsilon>0$ and $1\leq p\leq \infty$. Thus a bit of regularity is lost with respect to Proposition \ref{prop1}. This explains why Theorem \ref{theo1} is valid in all the functional settings listed in Proposition \ref{prop1} with slight modifications for the spaces in {\em{(ii)}} and {\em{(v)}}, i.e. for instance $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha, p}$, for $1< p<\infty $, $\alpha\in \mathbb R$, is replaced by $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha+\epsilon, p}$.

Theorem \ref{theo1} is not considering the spaces $W_{\divb}^{-\alpha,p}\times W_{\curl}^{-1+\alpha, p'}$, for $1< p<\infty$, and in particular the subset $L^p_{\divb}\times L^q_{\curl}$ with $\frac 1p+\frac 1q=1+\frac 1d, 1<p<\infty$. This is normal, since in this case concentration effects in terms of Dirac measures can appear in the limit of the product, as in Example 2.10 in \cite{Br}, and more generally in Theorem 2.1 in \cite{Br} that describes the limit of $v_nw_n$ in a more involved than simply $vw$. To get such a result here one could continue the analysis in the spirit of \cite{Br} by exploiting the defect of compactness of the embedding $W^{1,q}\subset L^{p'}$.  \\

{\bf{Aknowledgements: }} The authors thank Anne-Laure Dalibard, Gilles Francfort, Antoine Gloria and Fran\c{c}ois Murat for enriching discussions on the div-curl lemma.
Both authors acknowledge the funding from the European Research Council (ERC) under the
European Union's Horizon 2020
research and innovation programme (Grant agreement 101097172 -- GEOEDP). The first author was also partially supported by the French ANR project BOURGEONS.

The proof of Theorem~\ref{theo1} is based on Proposition~\ref{prop1} and on the fact that the smooth localization $u\in W^{\epsilon,p}\mapsto \chi u\in L^p$ is compact for $\epsilon>0$ and $1\le p\le \infty$. As a consequence, “a bit of regularity is lost with respect to Proposition~\ref{prop1}$,” which explains why Theorem~\ref{theo1} holds in the functional settings of Proposition~\ref{prop1} but with “slight modifications” in cases \emph{(ii)} and \emph{(v)}, e.g. replacing $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha,p}$ by $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha+\epsilon,p}$.

The proof of Theorem~\ref{theo1} is based on the following proposition.

\begin{prop}\label{prop1} Let $ (v,w) \in \mathcal M_{0,\divb} \times L_{\curl}^\infty$ or belonging to one of  the following spaces:
\begin{enumerate}
\item $\mathcal M_{0,\divb} \times W_{\curl}^{-1+ \epsilon, \infty}$, for $\epsilon>0$,
\item $\mathcal M_{0,\divb} \times W_{\curl}^{-1+\frac dp, p}$, for $d<p< \infty$,
\item  $W^{-1+\epsilon,\infty}_{\divb} \times \mathcal M_{0,\curl}$, for $\epsilon>0$,
\item  $W^{-\delta+\epsilon,p'}_{\divb} \times \mathcal M_{0,\curl}$, for $1< p\leq \frac d{d-1+\delta}$\footnote{The condition $1<p\leq \frac d{d-1+\delta}$ is equivalent to $\frac{d}{1-\delta}\leq p'< \infty$, thus varying $\delta$ in $(0,1)$ we have the range $d<p'< \infty$}. and $\delta\in(0,1),\epsilon>0$,
\item  $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha, p}$, for $1< p< \infty $, $\alpha\in \mathbb R$,
\item  $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha+\epsilon, p}$, for $p\in\{1,\infty\}$, $\epsilon>0$, $\alpha\in \mathbb R$.
\end{enumerate}
Then $w$ admits a class of Hodge-type decompositions $w=y+\nabla z$ such that the following quantity
\begin{equation}\label{Hodgeprod}
(v\cdot w)_H:=v\cdot y -(\divb\,v)z+\divb\,(vz).
\end{equation}
is well-defined as a distribution, is independent of the choice of the decomposition, and coincides with the usual product if $(v,w) \in\mathcal C^\infty_0\times \mathcal C ^\infty_0$ or if $(v,w) \in L_{\divb}^{p'}\times L_{\curl}^p$ for $1\leq p\leq \infty$.

Moreover, in all space $\mathcal{F}_{\divb}\times\mathcal{\tilde F}_{\curl}$ of the cases (ii)-(iv)-(v), thus not involving $L^\infty$ for which $C^\infty_0$ is not dense, the classical product map
$$ (v,w) \in\mathcal C^\infty_0\times \mathcal C ^\infty_0 \mapsto v \cdot w  \in \mathcal C ^\infty_0 $$
admits $(vw)_H$ as a unique continuous extension from  $\mathcal{F}_{\divb}\times\mathcal{\tilde F}_{\curl}$ to $\mathcal{D}' $.
\end{prop}

It also uses the fact that the smooth localization $u\in W^{\epsilon,p}\mapsto \chi u\in L^p$ is compact for $\epsilon>0$ and $1\le p\le \infty$. As a consequence, “a bit of regularity is lost with respect to the proposition above,” which explains why, in establishing the main theorem, the result holds in the functional settings of the proposition above but with “slight modifications” in cases \emph{(ii)} and \emph{(v)}, e.g. replacing $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha,p}$ by $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha+\epsilon,p}$.,

\label{theo1}
The map
$$(v,w) \in \mathcal{M}_{0,\divb}\times L^\infty_{\curl}\mapsto (v\cdot w)_H \in \mathcal{D'},$$
is weakly continuous.
More precisely, let $(v_n,w_n)$ be  a bounded sequence in $\mathcal{M}_{0,\divb} \times L^\infty_{\curl}$ converging in the sense of distributions to $(v,w)$:
$$ v_n \overset{\mathcal{D}'}{\rightharpoonup} v, \qquad w_n \overset{\mathcal{D}'}{\rightharpoonup}  w,$$
so in particular $(v,w)\in  \mathcal{M}_{0,\divb}\times L^\infty_{\curl}$.
Then we have the product convergence:
$$ (v_n\cdot w_n)_H \overset{\mathcal{D}'}{\rightharpoonup}  (v\cdot  w)_H,$$
where the products are defined as follows.

\begin{prop}\label{prop1} Let $ (v,w) \in \mathcal M_{0,\divb} \times L_{\curl}^\infty$ or belonging to one of  the following spaces:
\begin{enumerate}
\item $\mathcal M_{0,\divb} \times W_{\curl}^{-1+ \epsilon, \infty}$, for $\epsilon>0$,
\item $\mathcal M_{0,\divb} \times W_{\curl}^{-1+\frac dp, p}$, for $d<p< \infty$,
\item  $W^{-1+\epsilon,\infty}_{\divb} \times \mathcal M_{0,\curl}$, for $\epsilon>0$,
\item  $W^{-\delta+\epsilon,p'}_{\divb} \times \mathcal M_{0,\curl}$, for $1< p\leq \frac d{d-1+\delta}$\footnote{The condition $1<p\leq \frac d{d-1+\delta}$ is equivalent to $\frac{d}{1-\delta}\leq p'< \infty$, thus varying $\delta$ in $(0,1)$ we have the range $d<p'< \infty$}. and $\delta\in(0,1),\epsilon>0$,
\item  $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha, p}$, for $1< p< \infty $, $\alpha\in \mathbb R$,
\item  $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha+\epsilon, p}$, for $p\in\{1,\infty\}$, $\epsilon>0$, $\alpha\in \mathbb R$.
\end{enumerate}
Then $w$ admits a class of Hodge-type decompositions $w=y+\nabla z$ such that the following quantity
\begin{equation}\label{Hodgeprod}
(v\cdot w)_H:=v\cdot y -(\divb\,v)z+\divb\,(vz).
\end{equation}
is well-defined as a distribution, is independent of the choice of the decomposition, and coincides with the usual product if $(v,w) \in\mathcal C^\infty_0\times \mathcal C ^\infty_0$ or if $(v,w) \in L_{\divb}^{p'}\times L_{\curl}^p$ for $1\leq p\leq \infty$.

Moreover, in all space $\mathcal{F}_{\divb}\times\mathcal{\tilde F}_{\curl}$ of the cases (ii)-(iv)-(v), thus not involving $L^\infty$ for which $C^\infty_0$ is not dense, the classical product map
$$ (v,w) \in\mathcal C^\infty_0\times \mathcal C ^\infty_0 \mapsto v \cdot w  \in \mathcal C ^\infty_0 $$
admits $(vw)_H$ as a unique continuous extension from  $\mathcal{F}_{\divb}\times\mathcal{\tilde F}_{\curl}$ to $\mathcal{D}' $.
\end{prop}

The result is valid also with $\mathcal{M}_{0,\divb} \times L^\infty_{\curl}$ replaced by:\par
\,\hspace{3mm}{\it{(i) $\mathcal M_{0,\divb} \times W_{\curl}^{-1+ \epsilon, \infty}$, for $\epsilon>0$,}} \par
\,\hspace{1mm}{\it{(ii)$^*$ $\mathcal M_{0,\divb} \times W_{\curl}^{-1+\frac dp+\epsilon, p}$, for $d<p< \infty$, $\epsilon>0$,}}\par
\,\hspace{1mm}\it{(iii)}  $W^{-1+\epsilon,\infty}_{\divb} \times \mathcal M_{0,\curl}$, for $\epsilon>0$, \par
\,\hspace{2mm}\it{(iv)} $W^{-\delta+\epsilon,p'}_{\divb} \times \mathcal M_{0,\curl}$, for $1< p\leq \frac d{d-1+\delta}$ and $\delta\in(0,1),\epsilon>0$, \par
\,\hspace{2mm}\it{(v)$^*$}  $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha+\epsilon, p}$, for $1< p< \infty $, $\epsilon>0$, $\alpha\in \mathbb R$, \par
\,\hspace{2mm}\it{(vi)}  $W_{\divb}^{-\alpha,p'}\times W_{\curl}^{-1+\alpha+\epsilon, p}$, for $p\in\{1,\infty\}$, $\epsilon>0$, $\alpha\in \mathbb R$.,

Question:
Let \(\mathcal M_0\) denote the space of finite Radon measures on \(\mathbb R^d\). For a distribution space \(X\), write \(X_{\divb}:=\{v: v\in X,\ \divb v\in X\}\) and \(X_{\curl}:=\{w: w\in X,\ \curl w\in X\}\). For any admissible pair \((v,w)\) in one of the classes below, choose a Hodge-type decomposition \(w=y+\nabla z\) and define
\[
(v\cdot w)_H:=v\cdot y-(\divb v)z+\divb(vz),
\]
which is a well-defined distribution independent of the chosen decomposition. Let \(p'\) be the H\"older-conjugate exponent of \(p\) (with \(1'=\infty\) and \(\infty'=1\)). Suppose \((v_n,w_n)\) is a bounded sequence converging in \(\mathcal D'(\mathbb R^d)\) to \((v,w)\),
\[
v_n\rightharpoonup v,\qquad w_n\rightharpoonup w,
\]
and assume that all \((v_n,w_n)\) belong to one fixed class among
\[
\mathcal M_{0,\divb}\times L^\infty_{\curl},
\]
or
\[
\text{(i) }\mathcal M_{0,\divb}\times W^{-1+\epsilon,\infty}_{\curl}\quad (\epsilon>0),
\]
\[
\text{(ii) }\mathcal M_{0,\divb}\times W^{-1+\frac dp+\epsilon,p}_{\curl}\quad (d<p<\infty,\ \epsilon>0),
\]
\[
\text{(iii) }W^{-1+\epsilon,\infty}_{\divb}\times \mathcal M_{0,\curl}\quad (\epsilon>0),
\]
\[
\text{(iv) }W^{-\delta+\epsilon,p'}_{\divb}\times \mathcal M_{0,\curl}\quad \Big(1<p\le \frac d{d-1+\delta},\ \delta\in(0,1),\ \epsilon>0\Big),
\]
\[
\text{(v) }W^{-\alpha,p'}_{\divb}\times W^{-1+\alpha+\epsilon,p}_{\curl}\quad (1<p<\infty,\ \epsilon>0,\ \alpha\in\mathbb R),
\]
or
\[
\text{(vi) }W^{-\alpha,p'}_{\divb}\times W^{-1+\alpha+\epsilon,p}_{\curl}\quad (p\in\{1,\infty\},\ \epsilon>0,\ \alpha\in\mathbb R).
\]
Which statement holds for every such bounded distribution-convergent sequence?

The Hodge products converge weakly in distributions:
\[
(v_n\cdot w_n)_H\rightharpoonup (v\cdot w)_H\qquad\text{in }\mathcal D'(\mathbb R^d).
\]
Equivalently, on each listed class, the map \((v,w)\mapsto (v\cdot w)_H\) is weakly continuous from that class into \(\mathcal D'\).

Choices:
B.
The Hodge products converge weakly in distributions:
\[
(v_n\cdot w_n)_H\rightharpoonup (v\cdot w)_H\qquad\text{in }\mathcal D'(\mathbb R^d),
\]
but this remains valid already for the larger endpoint classes obtained by deleting the \(\epsilon>0\) losses in \textup{(ii)} and \textup{(v)}, i.e. for
\[
\mathcal M_{0,\divb}\times W^{-1+\frac dp,p}_{\curl}\quad(d<p<\infty)
\]
and
\[
W^{-\alpha,p'}_{\divb}\times W^{-1+\alpha,p}_{\curl}\quad(1<p<\infty,\ \alpha\in\mathbb R).
\]
Equivalently, the map \((v,w)\mapsto (v\cdot w)_H\) is weakly continuous on those endpoint classes as well.
C.
For every such bounded distribution-convergent sequence, after passing to a subsequence one has
\[
(v_{n_k}\cdot w_{n_k})_H\rightharpoonup (v\cdot w)_H\qquad\text{in }\mathcal D'(\mathbb R^d).
\]
In particular, every bounded sequence in any one fixed listed class has at least one distributionally convergent subsequence of Hodge products with limit equal to the Hodge product of the distributional limits.
D.
For every such bounded distribution-convergent sequence, the conclusion improves from weak distributional convergence to strong convergence in the local Sobolev topology: for every compactly supported \(\chi\in C_c^\infty(\mathbb R^d)\),
\[
\chi (v_n\cdot w_n)_H\to \chi (v\cdot w)_H
\]
strongly in some fixed negative Sobolev space depending only on the chosen class. Equivalently, the map \((v,w)\mapsto (v\cdot w)_H\) is compact from each listed class into \(\mathcal D'\) after localization.
E.
For every such bounded distribution-convergent sequence and for every choice of Hodge-type decompositions \(w_n=y_n+\nabla z_n\) and \(w=y+\nabla z\), one has termwise weak convergence in \(\mathcal D'(\mathbb R^d)\):
\[
v_n\cdot y_n\rightharpoonup v\cdot y,
\qquad
(\divb v_n)z_n\rightharpoonup (\divb v)z,
\qquad
\divb(v_n z_n)\rightharpoonup \divb(vz),
\]
and hence
\[
(v_n\cdot w_n)_H\rightharpoonup (v\cdot w)_H.
\]
Thus weak continuity holds separately for each term appearing in the definition of \((v\cdot w)_H\).

The stem does not explicitly reveal the marked answer. It states hypotheses and asks for the valid conclusion, without giving away the distinctive clauses that separate A from the other options.

This is largely a theorem-recall/restatement item: the stem lists assumptions in detail and the choices restate candidate conclusions of the same theorem with small perturbations. It is not purely tautological, since the options differ in subtle claims, but it is close to direct theorem matching.

There is some reasoning pressure because the options differ by subtle regularity losses, uniqueness-extension clauses, and decomposition-independence claims. However, the task mainly rewards precise theorem recall/comparison rather than generating a conclusion from first principles. Also, choice C appears to be a weaker true statement, which reduces the force of the reasoning task.

Several distractors are mathematically plausible and target realistic failure modes: overextending continuity, removing epsilon losses, or strengthening decomposition claims. But the distractor set is weakened because C is apparently also true (just weaker than A), so the item is not cleanly single-correct under the wording 'Which conclusion ... is valid?'

Technically sophisticated and mostly free of answer leakage, but it is heavily theorem-restatement based and suffers from ambiguity because at least one distractor appears genuinely true as a weaker conclusion. This limits its quality as a single-best-answer MCQ testing generative reasoning.


Erd\H{o}s and Selfridge famously proved that the product of two or more positive consecutive integers is never a perfect power \cite{ErdSel1975}. That is, they showed the equation
\begin{align}\label{eq:ErdSelf eqn}
n(n+1)(n+2)\cdots(n+k-1) = y^\ell
\end{align}
has no solutions in positive integers $n,y,k,\ell$ with $k,\ell \geq 2$.

There are many ways to generalize the Diophantine equation \eqref{eq:ErdSelf eqn}. One generalization, which has been studied a great deal, is as follows: determine whether there are solutions to
\begin{align}\label{eq:generalized ErdSelf eqn}
n(n+d)(n+2d) \cdots (n+(k-1)d) = y^\ell
\end{align}
in positive integers $n,d,y,k,\ell$ with $\text{gcd}(n,d)=1$ and $k,\ell \geq 2$. Hence, rather than taking products of consecutive integers as in \eqref{eq:ErdSelf eqn}, one takes products of consecutive terms in an arithmetic progression. (See \cite[pp. 22--26]{Sho2006} for a survey of some older results on \eqref{eq:generalized ErdSelf eqn} and its variants. See \cite{Ben2018,BenSik2016,BenSik2020,DLS2018,DLSS2023,GHP2009,HK2011,HTT2009}, say, and the associated references for an introduction to more recent work on this and related problems.) It is widely believed that \eqref{eq:generalized ErdSelf eqn} has no solutions once $k$ is sufficiently large (see, e.g., \cite[p. 356]{BenSik2020} and \cite{ErdProb672}). In fact, it ought to be the case that one can remove some terms $n+jd$ from the left-hand side of \eqref{eq:generalized ErdSelf eqn}, and the resulting equation will still be unsolvable (for $k$ sufficiently large).

Some recent work in this direction is due to Bennett \cite{Ben2023}, who studied equations of the form
\begin{align}\label{eq:Bennett eqn}
\prod_{\substack{j=0 \\ j \neq i}}^{k-1} (n+jd) = y^\ell,
\end{align}
where $5 \leq k \leq 8, i \in \{0,1,\ldots,k-1\}$, and $\ell \geq 3$ is a prime. Thus, these are equations where a single term $n+id$ is removed from the product. A variety of methods are used in \cite{Ben2023} to study this equation, such as the modularity of elliptic curves and a clever analysis of the resulting congruences (for $\ell \geq 7$), rational points on elliptic curves of rank zero (for $\ell=3$), and Chabauty-type methods (for $\ell=5$, relying on work of Hajdu and Kov\'{a}cs \cite{HK2011}). It is claimed in \cite[Theorem 1.2]{Ben2023} that all solutions to \eqref{eq:Bennett eqn} with $5 \leq k \leq 8$ arise when $k=5$ and $\ell=3$. However, the analysis in \cite{Ben2023} is incomplete in the case $\ell=3$, as some solutions to \eqref{eq:Bennett eqn} are missing.

The main result of this paper is the following theorem.

\begin{abstract}
Let $5 \leq k \leq 11$ and $0\leq i \leq k-1$ be integers. We determine all solutions to the equation
\begin{align*}
n(n+d)(n+2d)\cdots(n+(i-1)d)(n+(i+1)d) \cdots (n+(k-1)d) = y^3
\end{align*}
in integers $n,d,y$ with $ny \neq 0$, $d\geq 1$, and $\textup{gcd}(n,d) = 1$. Our method relies on the theory of elliptic curves, including elliptic curve Chabauty over a number field. As an application, we answer a question of Das, Laishram, Saradha, and Sharma concerning rational points on a certain superelliptic curve.
\end{abstract}

Some recent work in this direction is due to Bennett \cite{Ben2023}, who studied equations of the form
\begin{align}\label{eq:Bennett eqn}
\prod_{\substack{j=0 \\ j \neq i}}^{k-1} (n+jd) = y^\ell,
\end{align}
where $5 \leq k \leq 8, i \in \{0,1,\ldots,k-1\}$, and $\ell \geq 3$ is a prime. Thus, these are equations where a single term $n+id$ is removed from the product. A variety of methods are used in \cite{Ben2023} to study this equation, such as the modularity of elliptic curves and a clever analysis of the resulting congruences (for $\ell \geq 7$), rational points on elliptic curves of rank zero (for $\ell=3$), and Chabauty-type methods (for $\ell=5$, relying on work of Hajdu and Kov\'{a}cs \cite{HK2011}). It is claimed in \cite[Theorem 1.2]{Ben2023} that all solutions to \eqref{eq:Bennett eqn} with $5 \leq k \leq 8$ arise when $k=5$ and $\ell=3$. However, the analysis in \cite{Ben2023} is incomplete in the case $\ell=3$, as some solutions to \eqref{eq:Bennett eqn} are missing.

The main result of this paper is the following theorem.

It would be interesting to extend Theorem \ref{thm:main thm} to larger values of $k$, and to allow for more terms to be missing from the products.

\begin{lemma}\label{lem:factorization of terms}
Let $5 \leq k \leq 11$ and $0 \leq i \leq k-1$ be integers. Assume there are nonzero integers $y,n,d$ with $d\geq 1$ and $\textup{gcd}(n,d)=1$ such that \eqref{eq:main eqn of study} holds. Then for $0 \leq j \leq k-1, j \neq i$, we may uniquely write $n+jd=a_jx_j^3$, where:
\begin{itemize}
\item $a_j$ is a positive integer,
\item $a_j$ is cube-free, and divisible only by primes $\leq k-1$,
\item $\textup{gcd}(a_j,a_\ell) \mid (\ell-j)$ if $0 \leq j < \ell \leq k-1$ and $j,\ell \neq i$,
\item the integers $x_j$ are nonzero, and if $k \leq 8$ they are pairwise coprime,
\item $\prod_{\substack{j=0 \\ j \neq i}}^{k-1}a_j$ is a perfect cube.
\end{itemize}
\end{lemma}
\begin{proof}
Let $0 \leq j < \ell \leq k-1$ with $j,\ell \neq i$, and let $g = \text{gcd}(n+jd,n+\ell d)$. Then $g$ divides $(n+\ell d)-(n+jd) = (\ell-j)d$, and since $\text{gcd}(n,d)=1$ we find $g$ is coprime to $d$. Hence $g \mid (\ell - j)$. It follows that the greatest common divisor of $n+jd,n+\ell d$ is at most $k-1$. Therefore, if $p\geq k$ is a prime and $p \mid y$, then $p$ divides a unique term $n+jd$.

Our general strategy is to determine the coefficient vectors that can correspond to solutions to \eqref{eq:main eqn of study}. Since the integers $a_j$ are cube-free and their largest prime factors are bounded, there are only a finite number of potential coefficient vectors to consider for given values of $k$ and $i$. Given a potential coefficient vector $\textbf{v} = (a_0,\ldots,a_{i-1},a_{i+1},\ldots,a_{k-1})$, we may try to show $\textbf{v}$ cannot correspond to solutions of \eqref{eq:main eqn of study} by reducing to various cubic equations. More particularly, given distinct integers $r,s,t$ with $0 \leq r,s,t \leq k-1$ and $r,s,t \neq i$, we have
\begin{equation}\label{eq:ternary cubic eqn}
\begin{split}
0 &= (s-t)(n+rd)+(t-r)(n+sd)+(r-s)(n+td) \\
&= (s-t)a_rx_r^3 + (t-r)a_sx_s^3 + (r-s)a_tx_t^3.
\end{split}
\end{equation}
If the cubic equation $(s-t)a_rx_r^3 + (t-r)a_sx_s^3 + (r-s)a_tx_t^3 = 0$ has no solutions in nonzero integers $x_r,x_s,x_t$, then $\textbf{v}$ cannot be a coefficient vector corresponding to any solutions to \eqref{eq:main eqn of study}.

\begin{lemma}\label{lem:HTT}
Suppose we have a solution to the equation
\begin{align*}
\prod_{j=0}^{k-1}(n+jd) = y^3
\end{align*}
in integers $k,n,d,y$ with $4 \leq k \leq 38, d \geq 1, ny \neq 0$, and $\textup{gcd}(n,d)=1$. Then $(k,n,d) = (4,-9,5)$ or $(4,-6,5)$.
\end{lemma}

Hence, we may assume $1 \leq i \leq k-2$. Suppose we have a solution to \eqref{eq:main eqn of study}. If we set $m=n+(k-1)d$, then
\begin{align*}
y^3 &= (-1)^{k-1}\prod_{\substack{j=0 \\ j \neq i}}^{k-1}(-m+(k-1-j)d) = (-1)^{k-1} \prod_{\substack{\ell = 0 \\ \ell \neq k-1-i}}^{k-1} (-m+\ell d).
\end{align*}
If we absorb the sign $(-1)^{k-1}$ into $y^3$, then this is an equation of the form \eqref{eq:main eqn of study} with $i$ replaced by $k-1-i$. In fact, this yields a bijective involution between different solutions to \eqref{eq:main eqn of study} given by
\begin{align}\label{eq:flipping bijection}
(k,i,n,d) \leftrightarrow (k,k-1-i,-n-(k-1)d,d).
\end{align}
It therefore suffices to consider $1 \leq i \leq \frac{k-1}{2}$ in what follows.

\begin{align}\label{eq:main eqn of study}
\prod_{\substack{j=0 \\ j \neq i}}^{k-1}(n+jd) = y^3
\end{align}

\begin{theorem}\label{thm:main thm}
Let $5 \leq k \leq 11$ and $0\leq i \leq k-1$ be integers. The only solutions to the equation
\begin{align}\label{eq:main eqn of study}
\prod_{\substack{j=0 \\ j \neq i}}^{k-1}(n+jd) = y^3
\end{align}
in integers $n,d,y$ with $ny \neq 0$, $d \geq 1$, and $\textup{gcd}(n,d) = 1$ are given by
\begin{equation}\label{eq:solns for main thm}
\begin{split}
(k,i,n,d) = \, &(5,0,-14,5),(5,0,-11,5),(5,1,-8,3),(5,3,-4,3), \\
&(5,4,-9,5),(5,4,-6,5), (7,3,-10,7),(7,3,-32,7).
\end{split}
\end{equation}
\end{theorem}

It would be interesting to extend Theorem \ref{thm:main thm} to larger values of $k$, and to allow for more terms to be missing from the products.

Theorem \ref{thm:main thm} allows us to answer a question raised by Das, Laishram, Saradha, and Sharma \cite[p. 1710]{DLSS2023} concerning rational points on a certain superelliptic curve.

\begin{corollary}\label{cor:rat pts}
All rational points $(x,y) \in \mathbb{Q}^2$ on the curve
\begin{align}\label{eq:dlss curve}
y^3=(x+1)(x+2)(x+3)(x+5)(x+6)(x+7)
\end{align}
satisfy $y=0$, or $(x,y) = \left(-\frac{17}{7},\frac{120}{49} \right)$, or $(x,y) = \left(-\frac{39}{7},\frac{120}{49} \right)$.
\end{corollary}

The points on \eqref{eq:dlss curve} with $y=0$ are the ``trivial'' rational points
\begin{align*}
(-1,0),(-2,0),(-3,0),(-5,0),(-6,0),(-7,0).
\end{align*}
The authors of \cite{DLSS2023} noted the existence of the ``nontrivial'' rational point $\left(-\frac{17}{7},\frac{120}{49} \right)$, but did not provably determine all of the rational points on \eqref{eq:dlss curve}.

\subsection{Outline of the paper}

In Section \ref{sec:prelim}, we show how Corollary \ref{cor:rat pts} follows from Theorem \ref{thm:main thm}. We then prove an elementary but important lemma (Lemma \ref{lem:factorization of terms}) detailing the structure of the terms $n+jd$ in any solution to \eqref{eq:main eqn of study}. This gives rise to the notion of a \emph{coefficient vector}  (Definition \ref{defn: coeff vector}). A large part of our work is devoted to determining the coefficient vectors that can correspond to solutions to \eqref{eq:main eqn of study}. It will transpire that solutions to \eqref{eq:main eqn of study} give rise to many different ternary cubic equations (see \eqref{eq:ternary cubic eqn}), and these cubic equations can be shown to be unsolvable in many cases by relating them to elliptic curves of rank zero (Lemma \ref{lem:selmer}). We end the section with a general description of our strategy for determining potential coefficient vectors, which relies extensively on computer calculation. (All the computer code used in this paper, written in Magma \cite{Magma} and Sage \cite{Sage}, is available at our ``Cubes from products with one term missing'' GitHub repository \cite{Pratt_cubes_2025}.)

In Section \ref{sec:proving most of main thm}, we use elementary arguments and the results of Section \ref{sec:prelim} to solve \eqref{eq:main eqn of study} in all cases of $k$ and $i$ except $k=7,i=3$. There are two coefficient vectors in this case that correspond to solutions to \eqref{eq:main eqn of study} (see \eqref{eq:solns for main thm}). All the associated ternary cubic equations correspond to elliptic curves of positive rank, so the results of Section \ref{sec:prelim} are inapplicable.

In the last section of the paper, Section \ref{sec:finishing proof of main thm}, we finish the analysis in the case $k=7,i=3$ and give the proof of Theorem \ref{thm:main thm}, using a result on the simultaneous solutions to a pair of ternary cubic equations (Proposition \ref{prop:only solns to pair of cubic eqns}). Computations in the pure cubic field $K = \mathbb{Q}(\sqrt[3]{5})$ show that any solution to the pair of cubic equations gives rise to a point on an elliptic curve $E$ over $K$. All the points on $E$ arising in this way can be determined with the elliptic curve Chabauty method (see Lemma \ref{lem:EC over K}).

In Section \ref{sec:prelim}, we show how Corollary \ref{cor:rat pts} follows from Theorem \ref{thm:main thm}. We then prove an elementary but important lemma (Lemma \ref{lem:factorization of terms}) detailing the structure of the terms $n+jd$ in any solution to \eqref{eq:main eqn of study}. This gives rise to the notion of a \emph{coefficient vector} (Definition \ref{defn: coeff vector}). A large part of our work is devoted to determining the coefficient vectors that can correspond to solutions to \eqref{eq:main eqn of study}. Solutions to \eqref{eq:main eqn of study} give rise to many different ternary cubic equations (see \eqref{eq:ternary cubic eqn}), and these cubic equations can be shown to be unsolvable in many cases by relating them to elliptic curves of rank zero (Lemma \ref{lem:selmer}).

In Section \ref{sec:proving most of main thm}, we use elementary arguments and the results of Section \ref{sec:prelim} to solve \eqref{eq:main eqn of study} in all cases of $k$ and $i$ except $k=7,i=3$; in this remaining case, “all the associated ternary cubic equations correspond to elliptic curves of positive rank, so the results of Section \ref{sec:prelim} are inapplicable.”

In Section \ref{sec:finishing proof of main thm}, we finish the analysis for $k=7,i=3$ and thereby prove Theorem \ref{thm:main thm}, “using a result on the simultaneous solutions to a pair of ternary cubic equations” (Proposition \ref{prop:only solns to pair of cubic eqns}). Computations in the pure cubic field $K=\mathbb{Q}(\sqrt[3]{5})$ show that any solution to the pair of cubic equations gives rise to a point on an elliptic curve $E$ over $K$, and “all the points on $E$ arising in this way can be determined with the elliptic curve Chabauty method” (Lemma \ref{lem:EC over K}).

Next we show how \begin{corollary}\label{cor:rat pts}
All rational points $(x,y) \in \mathbb{Q}^2$ on the curve
\begin{align}\label{eq:dlss curve}
y^3=(x+1)(x+2)(x+3)(x+5)(x+6)(x+7)
\end{align}
satisfy $y=0$, or $(x,y) = \left(-\frac{17}{7},\frac{120}{49} \right)$, or $(x,y) = \left(-\frac{39}{7},\frac{120}{49} \right)$.
\end{corollary}
follows from the main theorem. We then prove an elementary but important lemma detailing the structure of the terms $n+jd$ in any solution to
\begin{align}\label{eq:main eqn of study}
\prod_{\substack{j=0 \\ j \neq i}}^{k-1}(n+jd) = y^3
\end{align}
namely:
\begin{lemma}\label{lem:factorization of terms}
Let $5 \leq k \leq 11$ and $0 \leq i \leq k-1$ be integers. Assume there are nonzero integers $y,n,d$ with $d\geq 1$ and $\textup{gcd}(n,d)=1$ such that \eqref{eq:main eqn of study} holds. Then for $0 \leq j \leq k-1, j \neq i$, we may uniquely write $n+jd=a_jx_j^3$, where:
\begin{itemize}
\item $a_j$ is a positive integer,
\item $a_j$ is cube-free, and divisible only by primes $\leq k-1$,
\item $\textup{gcd}(a_j,a_\ell) \mid (\ell-j)$ if $0 \leq j < \ell \leq k-1$ and $j,\ell \neq i$,
\item the integers $x_j$ are nonzero, and if $k \leq 8$ they are pairwise coprime,
\item $\prod_{\substack{j=0 \\ j \neq i}}^{k-1}a_j$ is a perfect cube.
\end{itemize}
\end{lemma}
This gives rise to the notion of a \emph{coefficient vector}:
\begin{definition}\label{defn: coeff vector}
Assume there are nonzero integers $y,n,d$ with $d\geq 1$ and $\textup{gcd}(n,d)=1$ such that \eqref{eq:main eqn of study} holds. Let $n+jd=a_jx_j^3$ be the factorization as described in Lemma \ref{lem:factorization of terms}. We call $(a_0,a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_{k-1})$ the \emph{coefficient vector} corresponding to $y,n,d$.
\end{definition}
A large part of our work is devoted to determining the coefficient vectors that can correspond to solutions to the equation above. Solutions to that equation give rise to many different ternary cubic equations, for instance
\begin{equation}\label{eq:ternary cubic eqn}
\begin{split}
0 &= (s-t)(n+rd)+(t-r)(n+sd)+(r-s)(n+td) \\
&= (s-t)a_rx_r^3 + (t-r)a_sx_s^3 + (r-s)a_tx_t^3.
\end{split}
\end{equation}
and these cubic equations can be shown to be unsolvable in many cases by relating them to elliptic curves of rank zero. Specifically, we use the following lemma:
\begin{lemma}\label{lem:selmer}
Let
\begin{align*}
\mathcal{L} = \{&1, 3, 4, 5, 10, 14, 18, 21, 25, 36, 45, 60, 100, 147, 150, \\
                &175, 196, 225, 245, 252, 300, 315, 350, 882, 980, \\
                &1050, 1470, 1575, 1764, 2940, 7350, 14700\}.
\end{align*}
Let $a,b,c$ be nonzero integers, and let $D$ denote the cube-free part of $abc$. If $D \in \mathcal{L}$, then the equation $ax^3+by^3+cz^3=0$ has no solutions in nonzero integers $x,y,z$.
\end{lemma}

Next, we use elementary arguments and the results established earlier to solve
\begin{align}\label{eq:main eqn of study}
\prod_{\substack{j=0 \\ j \neq i}}^{k-1}(n+jd) = y^3
\end{align}
in all cases of $k$ and $i$ except $k=7,i=3$; in this remaining case, “all the associated ternary cubic equations correspond to elliptic curves of positive rank, so the results established earlier are inapplicable.”

Finally, we finish the analysis for $k=7,i=3$ and thereby prove the main theorem, using the following result on the simultaneous solutions to a pair of ternary cubic equations:
\begin{proposition}\label{prop:only solns to pair of cubic eqns}
The only solutions $(x,y,z,w) \in \mathbb{Z}^4$ to the pair of equations
\begin{align*}
x^3+y^3&=9z^3 \\
5x^3-y^3&=3w^3
\end{align*}
with $xyzw \neq 0$ and $x,y$ coprime are $(x,y,z,w) = \pm (1,2,1,-1)$.
\end{proposition}
Computations in the pure cubic field $K=\mathbb{Q}(\sqrt[3]{5})$ show that any solution to the pair of cubic equations gives rise to a point on an elliptic curve $E$ over $K$, and “all the points on $E$ arising in this way can be determined with the elliptic curve Chabauty method,” using the following lemma:
\begin{lemma}\label{lem:EC over K}
Let $\alpha = \sqrt[3]{5}$, and set $K = \mathbb{Q}(\alpha)$. Write $\mathcal{O}_K = \mathbb{Z}[\alpha]$ for the ring of integers of $K$. The only solutions $(x,y,u) \in \mathbb{Z} \times \mathbb{Z}\times \mathcal{O}_K$ to
\begin{align}\label{eq:cubic over K}
(\alpha x - y)(x^2-xy+y^2) = 3(2-\alpha) u^3
\end{align}
with $xy \neq 0$ and $x,y$ coprime are $(x,y,u) = \pm (1,2,-1)$.
\end{lemma}

\label{thm:main thm}
Let $5 \leq k \leq 11$ and $0\leq i \leq k-1$ be integers. The only solutions to the equation
\begin{align}\label{eq:main eqn of study}
\prod_{\substack{j=0 \\ j \neq i}}^{k-1}(n+jd) = y^3
\end{align}
in integers $n,d,y$ with $ny \neq 0$, $d \geq 1$, and $\textup{gcd}(n,d) = 1$ are given by
\begin{equation}\label{eq:solns for main thm}
\begin{split}
(k,i,n,d) = \, &(5,0,-14,5),(5,0,-11,5),(5,1,-8,3),(5,3,-4,3), \\
&(5,4,-9,5),(5,4,-6,5), (7,3,-10,7),(7,3,-32,7).
\end{split}
\end{equation}.

Classification or Bijection
Nonexistence

Question:
Let integers $k,i,n,d,y$ satisfy $5\le k\le 11$, $0\le i\le k-1$, $d\ge 1$, $ny\ne 0$, and $\gcd(n,d)=1$. Consider the equation obtained by deleting the $i$th term from a length-$k$ arithmetic progression product,
\[
\prod_{\substack{j=0 \\ j\ne i}}^{k-1}(n+jd)=y^3.
\]
Which statement correctly classifies all integer solutions?

The only possible values of $(k,i,n,d)$ are
\[
(5,0,-14,5),\ (5,0,-11,5),\ (5,1,-8,3),\ (5,3,-4,3),\ (5,4,-9,5),\ (5,4,-6,5),\ (7,3,-10,7),\ (7,3,-32,7),
\]
and there are no other integer solutions with $5\le k\le 11$, $0\le i\le k-1$, $d\ge 1$, $ny\ne 0$, and $\gcd(n,d)=1$.

Choices:
B.
The only possible values of $(k,i,n,d)$ are
\[
(5,0,-14,5),\ (5,0,-11,5),\ (5,1,-8,3),\ (5,3,-4,3),\ (5,4,-9,5),\ (5,4,-6,5),\ (7,3,-10,7),\ (7,3,-32,7),
\]
together with their images under the involution
\[
(k,i,n,d)\mapsto (k,k-1-i,-n-(k-1)d,d),
\]
and there are no other integer solutions with $5\le k\le 11$, $0\le i\le k-1$, $d\ge 1$, $ny\ne 0$, and $\gcd(n,d)=1$.
C.
If
\[
\prod_{\substack{j=0 \\ j\ne i}}^{k-1}(n+jd)=y^3
\]
has an integer solution with $5\le k\le 11$, $0\le i\le k-1$, $d\ge 1$, $ny\ne 0$, and $\gcd(n,d)=1$, then necessarily $k\in\{5,7\}$; in particular, there are no solutions for $k\in\{6,8,9,10,11\}$.
D.
The only possible values of $(k,i,n,d)$ are
\[
(5,0,-14,5),\ (5,0,-11,5),\ (5,1,-8,3),\ (5,3,-4,3),\ (5,4,-9,5),\ (5,4,-6,5),\ (7,3,-10,7),\ (7,3,-32,7),
\]
and, moreover, every integer solution with $5\le k\le 11$, $0\le i\le k-1$, $d\ge 1$, $ny\ne 0$, and $\gcd(n,d)=1$ must satisfy $1\le i\le \frac{k-1}{2}$.
E.
The only possible values of $(k,i,n,d)$ are
\[
(5,0,-14,5),\ (5,0,-11,5),\ (5,1,-8,3),\ (5,3,-4,3),\ (5,4,-9,5),\ (5,4,-6,5),\ (7,3,-10,7),\ (7,3,-32,7),
\]
and for each such quadruple there is exactly one corresponding integer $y$ satisfying
\[
\prod_{\substack{j=0 \\ j\ne i}}^{k-1}(n+jd)=y^3.
\]
There are no other integer solutions with $5\le k\le 11$, $0\le i\le k-1$, $d\ge 1$, $ny\ne 0$, and $\gcd(n,d)=1$.

The stem does not explicitly or implicitly reveal the correct tuple list. It asks for an exact classification, and no wording in the question itself singles out choice A over the alternatives.

The item is close to a theorem-classification recall question: it asks for the exact exceptional tuples and 'no others.' However, it is not a pure verbatim restatement, since the options vary in completeness, logical strength, and misuse of symmetry.

Some reasoning is needed to distinguish an exact classification from a weaker true statement, an incomplete list, and overgenerated cases. Still, the task mainly rewards recognition/recall of the classification rather than substantial fresh mathematical derivation.

The distractors are strong and mathematically meaningful: one omits exceptional cases, one is a weaker true statement, one overextends a special case, and one misuses an involution. These reflect plausible failure modes and are clearly distinct.

A solid MCQ with no stem-based answer leakage and high-quality distractors, but it is somewhat theorem-restatement-like and only moderately tests generative reasoning.


\label{thmPoincare}
    Let $(M,g)$ be a compact Riemannian manifold of dimension $n$, and let
    $p,q \in (1,\infty)$ satisfy
    \[
        p \geq \frac{n}{n-1}
        \quad \text{and} \quad
        q > \frac{n}{2}.
    \]
    Let $r \in (1,\infty]$, and assume that, if $p < n$, then
    \[
        \frac{1}{r} \geq \frac{1}{p} - \frac{1}{n}.
    \]
    Then, for all functions $f \in W^{1,p}(M,\mathbb{R})$ and all non-negative densities
    $\omega \in L^q(M)$ normalized by $\mathbb{E}[\omega] = 1$, the weighted average
    $\bE_\omega[f]$ is well-defined and there exists a constant $C>0$, depending
    only on $(M,g,p,q,r)$, such that
    \[
        \left\| f - \bE_\omega[f] \right\|_{L^r}
        \leq
        C\,\|\omega\|_{L^q}^{\frac{n}{(n-1)p}}\,\|df\|_{L^p}.
    \]

Inequalities of Poincaré and Sobolev type are among the most fundamental tools
in the analysis of partial differential equations. They provide quantitative
control of the oscillation of a function in terms of its derivatives and lie at
the heart of \emph{a priori} estimates, compactness arguments, and regularity
theory for elliptic and parabolic equations.

In its simplest form, the Poincaré inequality asserts that, on a compact
Riemannian manifold $(M,g)$, the deviation of a function from its mean value is
controlled by the $L^p$-norm of its gradient: for any $p \in [1,\infty)$, there
exists a constant $C = C(M,g,p)$ such that, for any function $f \in
 W^{1,p}(M,\mathbb{R})$,
\[
 \left\| f - \mathbb{E}[f] \right\|_{L^p}
 \leq C \|df\|_{L^p},
\]
where $\mathbb{E}[f]$ denotes the average value of $f$,
\[
 \mathbb{E}[f]
 = \frac{1}{\operatorname{vol}(M,g)} \int_M f\,d\mu^g.
\]
We refer the reader to~\cite[Section~5.8]{Evans} for a proof of this inequality
on bounded domains of $\mathbb{R}^n$ and to~\cite[Lemma~3.8]{Hebey} for compact
manifolds. More refined versions, based on the Sobolev inequality, allow one to
estimate this deviation in stronger Lebesgue norms and are indispensable in the
study of nonlinear elliptic equations.

In many applications, however, the reference average is not the uniform mean
with respect to the Riemannian volume measure. Instead, one is led to consider
averages defined using a density $\omega$,
\[
 \bE_\omega[f] = \int_M f\,\omega\,d\mu^g,
\]
where $\omega$ is a non-negative function normalized by $\int_M \omega\,d\mu^g
 = 1$. Such weighted averages arise naturally in problems involving
normalization constraints, conservation laws, or coupled systems of equations,
where the density $\omega$ may depend on other unknowns of the system.

For a fixed density $\omega$ and $p \in (1,\infty)$, it is not difficult to
prove a Poincaré-type inequality controlling $f - \bE_\omega[f]$ by
$\|df\|_{L^p}$. A standard compactness argument analogous
to~\cite[Lemma~3.8]{Hebey} shows that there exists a constant $C =
 C(M,g,p,\omega)$ such that
\[
 \|f - \bE_\omega[f]\|_{L^p}
 \leq C \|df\|_{L^p}
 \qquad \text{for all } f \in W^{1,p}(M,\mathbb{R}).
\]
However, this argument provides no information on how the constant depends on
$\omega$, and in particular does not allow one to obtain uniform estimates when
the density varies in a family of weights.

Understanding this dependence is a subtle issue. To our knowledge, the only
case in which such an estimate is explicitly addressed in the literature is
when $\omega$ is the characteristic function of a measurable subset of a
bounded convex domain, see~\cite[Lemma 7.16]{GilbargTrudinger}, a situation
related to nonlinear capacity theory (see, for instance,~\cite{MazyaSobolev}).
Another source of interest in Poincaré-type inequalities involving densities
comes from the theory of Bakry--Émery curvature-dimension conditions (see,
e.g.,~\cite{LiWang}); in that framework, however, the density defines the
underlying measure itself and modifies the geometric structure of the space, so
that the resulting inequalities belong to a fundamentally different setting.

The aim of this paper is to establish a Poincaré--Sobolev type inequality in
which the deviation of a function from a weighted average is controlled by the
unweighted Sobolev norm of its gradient, with an explicit quantitative
dependence of the constant on the density. More precisely, we prove the
following result.

To the best of our knowledge, such a quantitative estimate has not previously
been established, even in the Euclidean setting.

We now pass to polar coordinates for $y-x$. Note that $\omega$ can be extended
 to zero outside of $U$. We also extend $df$ by zero outside of $U$ without
 changing the notation. This allows us not to care about restricting the domain
 of integration. Let also $d = \diam(U)$. We have
 \[
 |f(x) - \bE_\omega[f]|^t
 \leq \int_{r=0}^d \int_{\bS^{n-1}} r^{n+t-2} \omega(x + r \theta) \int_0^r |df_{x + s \theta}|^t ds d\theta dr,
 \]
 where $d\theta$ is the surface measure on the unit sphere $\bS^{n-1} \subset
 \bR^n$. Hence,
 \[
 |f(x) - \bE_\omega[f]|^t
 \leq \int_{0 \leq s \leq r \leq d} \int_{\bS^{n-1}} r^{n+t-2} \omega(x + r \theta) |df_{x + s \theta}|^t d\theta dr ds.
 \]
 Integrating against $x \in U$, we get
 \begin{align*}
 \int_U |f(x) - \bE_\omega[f]|^t dx
 & \leq \int_U \int_{0 \leq s \leq r \leq d} \int_{\bS^{n-1}} r^{n+t-2} \omega(x + r \theta) |df_{x + s \theta}|^t d\theta dr ds dx \\
 & \leq \int_U \int_{\bS^{n-1}} \int_{s=0}^d \left(\int_{s \leq r \leq d} r^{n+t-2} \omega(x + r \theta) dr \right) |df_{x + s \theta}|^t d\theta ds dx \\
 & \leq d^{n+t-2} \int_U \int_{\bS^{n-1}} \int_{s=0}^d \left(\int_{s \leq r \leq d} \omega(x + r \theta) dr \right) |df_{x + s \theta}|^t d\theta ds dx.
 \end{align*}
 We change the order of integration and perform the change of variable $z = x + s \theta$ to get:
 \begin{align}
 \int_U |f(x) - \bE_\omega[f]|^t dx
 & \leq d^{n+t-2} \int_{\bS^{n-1}} \int_{s=0}^d \int_{x \in U} \left(\int_{s \leq r \leq d} \omega(x + r \theta) dr \right) |df_{x + s \theta}|^t dx d\theta ds\nonumber \\
 & = d^{n+t-2} \int_{\bS^{n-1}} \int_{s=0}^d \int_{z \in U} \left(\int_{s \leq r \leq d} \omega(z + (r-s) \theta) dr \right) |df_{z}|^t dz d\theta ds\nonumber \\
 & = d^{n+t-2} \int_{z \in U} \left(\int_{\bS^{n-1}} \int_{s=0}^d \int_{s \leq r \leq d} \omega(z + (r-s) \theta) dr ds d\theta\right) |df_{z}|^t dz\nonumber \\
 & \leq d^{n+t-1} \int_{z \in U} \left(\int_{\bS^{n-1}} \int_{\rho=0}^d \omega(z + \rho \theta) d\rho d\theta\right) |df_{z}|^t dz,\label{eqPoincare3}
 \end{align}
 where we also performed the change of variable $\rho = r-s$ to obtain the last line. Now remark that the inner integral can be rewritten as follows:
 \begin{align*}
 \omegatil(z)
 & = \int_{\bS^{n-1}} \int_{t=0}^d \omega(z + \rho \theta) d\rho d\theta \\
 & = \int_{\bS^{n-1}} \int_{t=0}^d \omega(z + \rho \theta) \rho^{-n-1} \rho^{n-1} d\rho d\theta \\
 & = \int_{\bR^n} \omega(z - x) \frac{\chi_{|x| \leq d}}{|x|^{n-1}} dx,
 \end{align*}
 where we have set $x = \rho \theta$. Note that the function $x \mapsto \frac{\chi_{|x| \leq d}}{|x|^{n-1}}$ belongs to $L^r(\bR^n, \bR)$ for any $r < \frac{n}{n-1}$\footnote{This function is actually a truncated Riesz potential for which we could use the Hardy-Littlewood-Sobolev inequality~\cite[Section 7.8]{GilbargTrudinger}. This is not needed here.} so we use Young's inequality together with the fact that $q > \frac{n}{2}$ to get
 \[
 \|\omegatil\|_{L^n} \leq c \|\omega\|_{L^q}
 \]
 for some constant $c = c(n, q, d)$. We finally conclude
 from~\eqref{eqPoincare3} that
 \[
 \int_U |f(x) - \bE_\omega[f]|^t dx \leq c d^{n+p-2} d^{n+t-1} \int_{z \in U} \omegatil(z) |df_{z}|^t dz \leq \|\omegatil\|_{L^n} \left(\int_U |df|^p dx\right)^{\frac{n-1}{n}}.
 \]
\end{proof}

The next step is to extend the previous result to a broader range for $t$ by
means of the Sobolev inequality:
\begin{lemma}\label{lmPoincare2}
 Let $U$ be an open non-empty bounded convex subset of $\bR^n$ with smooth boundary. Let $p, q \in (1, \infty)$ be such that $p \geq \frac{n}{n-1}$ and $q > \frac{n}{2}$. Let $r \in [1, \infty]$ be such that
 \[
 \frac{1}{r} \geq \frac{1}{p} - \frac{1}{n} \quad\text{if } p < n.
 \]
 There exists a constant $c' = c'(U, p, q, r)$ such that, for any $\omega \in
 L^q(U, \bR)$, $\omega \geq 0$ a.e., $\bE[\omega] = 1$ and any function $f \in
 W^{1, p}(U, \bR)$,
 \[
 \|f - \bE_\omega[f]\|_{L^r} \leq c' \|\omega\|_{L^q}^{\frac{n}{(n-1)p}} \|df\|_{L^p}.
 \]
\end{lemma}

\begin{proof}
 Let $t = \frac{n-1}{n} p$ be as in Lemma~\ref{lmPoincare1}. The conclusion of this lemma can be written as follows:
 \begin{equation}\label{eqPoincare4}
 \|f - \bE_\omega[f]\|_{L^t} \leq (c \|\omega\|_{L^q})^{\frac{n}{(n-1)p}} \|df\|_{L^p}.
 \end{equation}
 As $U$ has finite volume, the conclusion of the lemma is immediate if $r \leq t$ as there is a continuous inclusion $L^t(U, \bR) \hookrightarrow L^r(U, \bR)$. If, instead, $r > t$, we let $p'$ be defined as follows:
 \[
 p' =
 \left\lbrace
 \begin{aligned}
 \frac{np}{n-p} & \quad\text{if } p < n, \\
 \infty & \quad\text{if } p > n, \\
 \text{arbitrary} > p & \quad\text{if } p = n.
 \end{aligned}
 \right.
 \]
 Let $\ftil = f - \bE_\omega[f]$. The Sobolev embedding theorem implies that
 there exists a constant $s > 0$ such that
 \begin{equation}\label{eqPoincare5}
 \|\ftil\|_{L^{p'}} \leq s \left(\|d\ftil\|_{L^p} + \|\ftil\|_{L^p}\right).
 \end{equation}
 As $t < p < p'$, we have
 \[
 \|\ftil\|_{L^p} \leq \|\ftil\|_{L^t}^\theta \|\ftil\|_{L^{p'}}^{1-\theta}\quad\text{with } \theta \in (0, 1) \text{ such that } \frac{1}{p} = \frac{\theta}{t} + \frac{1-\theta}{p'}.
 \]
 By the $\epsilon$-Young inequality~\cite[Equation $(7.6)$]{GilbargTrudinger},
 we deduce that, for any $\epsilon > 0$,
 \[
 \|\ftil\|_{L^p} \leq \theta \epsilon^{-\frac{1-\theta}{\theta}} \|\ftil\|_{L^t} + (1-\theta) \epsilon \|\ftil\|_{L^{p'}}.
 \]
 From~\eqref{eqPoincare4}, we deduce that
 \[
 \|\ftil\|_{L^p} \leq \theta \epsilon^{-\frac{1-\theta}{\theta}} (c \|\omega\|_{L^q})^{\frac{n}{(n-1)p}} \|df\|_{L^p} + (1-\theta) \epsilon \|\ftil\|_{L^{p'}}.
 \]
 Hence,~\eqref{eqPoincare5} implies
 \[
 \|\ftil\|_{L^{p'}} \leq s \left(\|d\ftil\|_{L^p} + \theta \epsilon^{-\frac{1-\theta}{\theta}} (c \|\omega\|_{L^q})^{\frac{n}{(n-1)p}} \|df\|_{L^p} + (1-\theta) \epsilon \|\ftil\|_{L^{p'}}\right),
 \]
 which we rewite as follows:
 \[
 \left(\frac{1}{s} - (1-\theta) \epsilon\right) \|\ftil\|_{L^{p'}} \leq \left(1 + \theta \epsilon^{-\frac{1-\theta}{\theta}} (c \|\omega\|_{L^q})^{\frac{n}{(n-1)p}}\right) \|d\ftil\|_{L^p}.
 \]
 Choosing $\epsilon$ such that $(1-\theta) \epsilon = \frac{1}{2s}$, we conclude
 that
 \[
 \|\ftil\|_{L^{p'}} \leq 2s \left(1 + \theta \epsilon^{-\frac{1-\theta}{\theta}} (c \|\omega\|_{L^q})^{\frac{n}{(n-1)p}}\right) \|d\ftil\|_{L^p}.
 \]
 As $1 = \bE[\omega] = \|\omega\|_{L^1} \leq \vol(U)^{1 - \frac{1}{q}}
 \|\omega\|_{L^q}$, we can replace the constant $1$ in the previous inequality
 by
 \[
 1 = 1^{\frac{n}{(n-1)p}} \leq \vol(U)^{\left(1 - \frac{1}{q}\right)\frac{n}{(n-1)p}} \|\omega\|_{L^q}^{\frac{n}{(n-1)p}}.
 \]
 Thus, we conclude that there is a constant $c'$ depending only on $U$ and $p$
 but not on $f$ such that
 \[
 \|\ftil\|_{L^{p'}} \leq c' \|\omega\|_{L^q}^{\frac{n}{(n-1)p}} \|d\ftil\|_{L^p},
 \]
 i.e.\ such that
 \[
 \|f - \bE_\omega[f]\|_{L^{p'}} \leq c' \|\omega\|_{L^q}^{\frac{n}{(n-1)p}} \|df\|_{L^p}.
 \]
 Using once again that, if $r \leq p'$, we have a continuous embedding
 $L^{p'}(U, \bR) \hookrightarrow L^r(U, \bR)$, we obtain the conclusion of the
 lemma.
\end{proof}

Let $f \in W^{1,p}(M)$ and $\omega \in L^q(M)$ be non-negative with
 $\mathbb{E}[\omega] = 1$. We define $\tilde{f} := f \circ \Psi$, and wish to
 construct a weight $\tilde{\omega}$ on $B$ such that
 $\mathbb{E}[\tilde{\omega}] = 1$. For this, we use the coarea formula (see
 e.g.~\cite[Chapter 5.2]{KrantzParks}):
 \begin{equation}\label{eqCoarea}
 \int_B h(x) \, dx = \int_M \left(\sum_{x \in \Psi^{-1}(y)} \frac{h(x)}{|\det d\Psi_x|} \right) d\mu^g(y),
 \end{equation}
 which holds for any integrable function $h : B \to \bR$.

To the best of our knowledge, such a quantitative estimate has not previously
been established, even in the Euclidean setting.

The outline of the paper is as follows. In Section~\ref{secRn}, we address the
case of bounded convex open subsets of $\mathbb{R}^n$. In Section~\ref{secDiffeo}, we
construct a local diffeomorphism $\Psi: B \to M$ from the open unit ball $B$ in
$\mathbb{R}^n$ onto the manifold $M$. Finally, in Section~\ref{secProof}, we use the
coarea formula to lift the function and $\omega$ from $M$ to $B$ to prove our
main result.

\subsection*{Acknowledgments}

I am grateful to Laurent Véron for useful discussion about the paper.

To prove the main result (Theorem~\ref{thmPoincare}), the paper proceeds as follows: first it treats the case of bounded convex open subsets of $\mathbb{R}^n$ (Section~\ref{secRn}); then it "construct[s] a local diffeomorphism $\Psi: B \to M$ from the open unit ball $B$ in $\mathbb{R}^n$ onto the manifold $M$" (Section~\ref{secDiffeo}); finally, in Section~\ref{secProof}, it "use[s] the coarea formula to lift the function and $\omega$ from $M$ to $B$" in order to prove the main result.

To prove the main theorem, the paper proceeds as follows: first it treats the case of bounded convex open subsets of $\mathbb{R}^n$ (Section~\ref{secRn}); then it "construct[s] a local diffeomorphism $\Psi: B \to M$ from the open unit ball $B$ in $\mathbb{R}^n$ onto the manifold $M$" (Section~\ref{secDiffeo}); finally, it "use[s] the coarea formula to lift the function and $\omega$ from $M$ to $B$" in order to prove the main theorem.

Inequality or Bound
Existential–Universal

Question:
Let $(M,g)$ be a compact Riemannian manifold of dimension $n$. Let $p,q\in(1,\infty)$ satisfy
\[
p\ge \frac{n}{n-1}
\qquad\text{and}\qquad
q>\frac{n}{2}.
\]
Let $r\in(1,\infty]$, and assume that if $p<n$, then
\[
\frac{1}{r}\ge \frac{1}{p}-\frac{1}{n}.
\]
For a nonnegative function $\omega\in L^q(M)$ with
\[
\int_M \omega\,d\mu^g = 1,
\]
define the weighted average of $f$ by
\[
\mathbb E_\omega[f] := \int_M f\,\omega\,d\mu^g.
\]
Here $f\in W^{1,p}(M,\mathbb R)$, $df$ denotes its differential, and all $L^r$ and $L^q$ norms are taken with respect to the Riemannian volume measure $d\mu^g$. Which quantitative estimate holds for every such $f$ and $\omega$?

The quantity $\mathbb E_\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\in W^{1,p}(M,\mathbb R)$ and all nonnegative $\omega\in L^q(M)$ with $\int_M \omega\,d\mu^g=1$,
\[
\|f-\mathbb E_\omega[f]\|_{L^r}
\le C\,\|\omega\|_{L^q}^{\frac{n}{(n-1)p}}\,\|df\|_{L^p}.
\]

Choices:
B.
The quantity $\mathbb E_\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\in W^{1,p}(M,\mathbb R)$ and all nonnegative $\omega\in L^q(M)$ with $\int_M \omega\,d\mu^g=1$,
\[
\|f-\mathbb E_\omega[f]\|_{L^r}
\le C\,\|\omega\|_{L^q}^{\frac{n}{(n-1)p}}\,\|df\|_{L^p},
\]
provided that, if $p<n$, then
\[
\frac{1}{r}> \frac{1}{p}-\frac{1}{n}.
\]
C.
The quantity $\mathbb E_\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\in W^{1,p}(M,\mathbb R)$ and all nonnegative $\omega\in L^q(M)$ with $\int_M \omega\,d\mu^g=1$,
\[
\|f-\mathbb E_\omega[f]\|_{L^r}
\le C\,\|\omega\|_{L^q}^{\frac{n}{(n-1)p}}\bigl(\|df\|_{L^p}+\|f\|_{L^p}\bigr).
\]
D.
The quantity $\mathbb E_\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\in W^{1,p}(M,\mathbb R)$ and all nonnegative $\omega\in L^q(M)$ with $\int_M \omega\,d\mu^g=1$,
\[
\|f-\mathbb E_\omega[f]\|_{L^r}
\le C\,\|\omega\|_{L^q}^{\frac{n}{(n-1)p}}\,\|df\|_{L^p},
\]
where the constant $C$ may depend on $\omega$ but is independent of $f$.
E.
The quantity $\mathbb E_\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\in W^{1,p}(M,\mathbb R)$ and all nonnegative $\omega\in L^q(M)$ with $\int_M \omega\,d\mu^g=1$,
\[
\|f-\mathbb E_\omega[f]\|_{L^r}
\le C\,\|\omega\|_{L^q}^{\frac{n}{(n-1)q}}\,\|df\|_{L^p}.
\]

pure gradient control by adding an extra $\|f\|_{L^p}$ term

uniformity of the constant with respect to the density after lifting via coarea

power of $\|\omega\|_{L^q}$ coming from the Euclidean estimate

The stem gives hypotheses and definitions but does not reveal the specific inequality, exponent of \|\omega\|_{L^q}, or the precise uniformity claim. The correct answer is not stated or strongly signaled.

The item is largely a theorem-identification question: it asks for the correct uniform conclusion under the stated assumptions. The options do introduce meaningful competing formulations, but the task remains close to recalling the exact statement rather than applying it in a new setting.

Some reasoning is needed to distinguish the sharp weight exponent, the role of uniform dependence on \omega, and the admissible target space L^r versus L^\infty. However, the question mostly tests recognition/recall of the precise result rather than substantial generative mathematical reasoning.

The distractors are plausible and mathematically targeted: they vary the \|\omega\|_{L^q} exponent, weaken uniformity by allowing dependence on \omega, or overstate the target norm. These reflect common failure modes and are clearly distinct.

A solid theorem-recall MCQ with no answer leakage and strong, plausible distractors, but it only moderately avoids tautology and does not strongly test generative reasoning.


Question:
Let $(M,g)$ be a compact Riemannian manifold of dimension $n$. Let $p,q\in(1,\infty)$ satisfy
\[
p\ge \frac{n}{n-1}
\qquad\text{and}\qquad
q>\frac{n}{2}.
\]
Let $r\in(1,\infty]$, and assume that if $p<n$, then
\[
\frac{1}{r}\ge \frac{1}{p}-\frac{1}{n}.
\]
For a nonnegative function $\omega\in L^q(M)$ with
\[
\int_M \omega\,d\mu^g = 1,
\]
define the weighted average of $f$ by
\[
\mathbb E_\omega[f] := \int_M f\,\omega\,d\mu^g.
\]
Here $f\in W^{1,p}(M,\mathbb R)$, $df$ denotes its differential, and all $L^r$ and $L^q$ norms are taken with respect to the Riemannian volume measure $d\mu^g$. Which quantitative estimate holds for every such $f$ and $\omega$?

Choices:
B.
The quantity $\mathbb E_\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\in W^{1,p}(M,\mathbb R)$ and all nonnegative $\omega\in L^q(M)$ with $\int_M \omega\,d\mu^g=1$,
\[
\|f-\mathbb E_\omega[f]\|_{L^r}
\le C\,\|\omega\|_{L^q}^{\frac{n}{(n-1)p}}\,\|df\|_{L^p},
\]
provided that, if $p<n$, then
\[
\frac{1}{r}> \frac{1}{p}-\frac{1}{n}.
\]
C.
The quantity $\mathbb E_\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\in W^{1,p}(M,\mathbb R)$ and all nonnegative $\omega\in L^q(M)$ with $\int_M \omega\,d\mu^g=1$,
\[
\|f-\mathbb E_\omega[f]\|_{L^r}
\le C\,\|\omega\|_{L^q}^{\frac{n}{(n-1)p}}\bigl(\|df\|_{L^p}+\|f\|_{L^p}\bigr).
\]
D.
The quantity $\mathbb E_\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\in W^{1,p}(M,\mathbb R)$ and all nonnegative $\omega\in L^q(M)$ with $\int_M \omega\,d\mu^g=1$,
\[
\|f-\mathbb E_\omega[f]\|_{L^r}
\le C\,\|\omega\|_{L^q}^{\frac{n}{(n-1)p}}\,\|df\|_{L^p},
\]
where the constant $C$ may depend on $\omega$ but is independent of $f$.
E.
The quantity $\mathbb E_\omega[f]$ is well-defined, and there exists a constant $C>0$, depending only on $(M,g,p,q,r)$, such that for all $f\in W^{1,p}(M,\mathbb R)$ and all nonnegative $\omega\in L^q(M)$ with $\int_M \omega\,d\mu^g=1$,
\[
\|f-\mathbb E_\omega[f]\|_{L^r}
\le C\,\|\omega\|_{L^q}^{\frac{n}{(n-1)q}}\,\|df\|_{L^p}.
\]

The stem does not explicitly reveal the correct option or reproduce the target estimate itself. It specifies hypotheses and asks for the valid conclusion, so there is no direct answer leakage.

This is essentially a theorem-recall item: the stem lists the precise assumptions and the correct choice is the theorem's conclusion almost verbatim. The task is mainly to identify the exact stated estimate rather than derive a new consequence.

Some reasoning is needed to distinguish endpoint admissibility, the exponent of \(\|\omega\|_{L^q}\), uniformity of the constant, and the weaker true variant with an added \(\|f\|_{L^p}\) term. However, the item primarily tests recognition of the exact theorem statement rather than substantial generative reasoning.

The distractors are mathematically plausible and target realistic failure modes: excluding the critical Sobolev endpoint, weakening the estimate by adding an extra norm term, allowing non-uniform dependence on \(\omega\), and altering the exponent incorrectly. They are distinct and well aligned with common confusions.

A technically well-constructed theorem-recognition MCQ with strong distractors, but it is close to a direct restatement of the result and only moderately tests genuine reasoning.