\begin{prop}\label{pro ortho}
		If  Eq. \eqref{eq-k} is solvable, then $f$ satisfies

		\begin{eqnarray}\label{ortho-condition}
    \int_{\C_{\theta}}f(\xi)\<\xi, E_{\alpha}\>d\H^{n}=0,\quad \forall~ 1\leq \alpha\leq n.
\end{eqnarray}
	\end{prop}

\begin{thm}\label{main theorem}
  Let $\theta\in (0,\frac{\pi}{2}]$ and $1\leq k\leq n-1$.  Suppose $f\in C^{2}(\C_{\theta})$ is connected to $1$ in $\mathcal{L}_{-\frac{1}{k}}$.  Then there exists a $C^{3, \gamma}$ $(\gamma\in(0,1))$ strictly convex capillary hypersurface $\Sigma$ in $ \ov{{\RR}^{n+1}_{+}}$, such that $\widehat{\S}$ satisfies the prescribed  $k$-th capillary area measure problem.
  Moreover, $\S$ is unique up to a horizontal translation in $\ov{{\RR}^{n+1}_{+}}$.
\end{thm}

\begin{eqnarray}\label{measure}
    S_{k}^c(\widehat{\Sigma}, \beta)\coloneqq \binom{n}{k}^{-1}\int_{\beta}(\sin^{2}\theta+\cos\theta\<\xi, e\>)\sigma_{k}(r)d{\H}^{n},\quad \text{for}~0\leq k\leq n.
\end{eqnarray}

\begin{prop}\label{pro-2.6}
 The capillary  Christoffel-Minkowski problem 		is equivalent to finding a strictly convex solution to  Eq.  \eqref{eq-k}.\end{prop}

\begin{defn}\label{def1.1}
  We say that a function $f\in C^{2}(\C_{\theta})$ is (strictly) convex if the matrix $\n^{2}f+f\sigma$ is positive semi-definite (respectively, positive definite) on $\C_{\theta}$.
\end{defn}

In convex geometry, the prescribed $k$-th area measure problem is a well-known problem. It asks whether, for a given positive function $f$ on $\mathbb{S}^{n}$, one can find a closed, strongly convex hypersurface $\Sigma$ with $k$-th symmetric function of the principal curvature radii equal to $f$ as a function on its normals? When $k=n$, it corresponds to the famous Minkowski problem, which has been fully solved through the works of Minkowski \cite{Min}, Alexandrov \cite{Alex}, Lewy \cite{Lewy}, Nirenberg \cite{Nire}, Pogorelov \cite{Pog52, Pog}, Cheng-Yau \cite{CY76} and many others. Analytically, the Minkowski problem is equivalent to solving a Monge-Amp\`ere equation on $\mathbb{S}^{n}$. The study of the Minkowski problem has played a significant role in advancing research on fully nonlinear equations, especially on the Monge-Amp\`ere equation. We can also see a closely related work by Xia \cite{Xia} for an anisotropic Minkowski problem for closed hypersurfaces.
When $k=1$, it is known as the Christoffel problem, which is equivalent to finding convex solutions to the Laplace equation on $\mathbb{S}^{n}$.  Firey \cite{Firey, Firey1} and Berg \cite{Berg}  found necessary and sufficient conditions using the Green function of the Laplacian on $\mathbb{S}^{n}$. More recently, Li-Wan-Wang \cite{LWW} expressed the Christoffel problem as a Laplace equation on the entire space $\mathbb{R}^{n+1}$ and provided a new and simpler necessary and sufficient condition for the Christoffel problem.  Further details on Christoffel's problem can be found in \cite{Chri,  Hilbert, Hur, Suss}. For the intermediate cases, i.e., $2\leq k\leq n-1$, it is referred to as the Christoffel-Minkowski problem. Guan-Ma \cite{GM}  solved this problem under a continuous homotopy condition, which was removed later by  Sheng-Wang-Trudinger \cite{STW}. In addition to the elliptic method, Chou-Wang \cite{CW00} and Bryan-Ivaki-Scheuer \cite{BIS2023} provided a flow method to solve the above problems.
For a more comprehensive description of the prescribed area measure problem in convex geometry, see \cite{Guan-note} and \cite{Sch}.

\subsection{Motivation and set up}
In recent years, there has been growing interest in the study of capillary hypersurfaces, including free boundary hypersurfaces, along with their geometry and topology, see, for instance, \cite{FL14, FS16, JWXZ, MWW-GCF, SWX, WW20, WWX22, WX2019, WeX21} and references therein.
Capillary hypersurfaces originated from physics through the work of Thomas Young. Later,
Young, Liouville, and Gauss formulated it as a variational problem using the so-called wetting energy.  For capillary hypersurfaces, we refer to the elegant book by Finn \cite{Finn}, and also a book by Maggi  \cite{Maggi}. In this paper, with a slight abuse of terminology,
a smooth compact embedded hypersurface in $\overline{{\mathbb{R}}_+^{n+1}}$  whose boundary lies on $\partial\overline{{\mathbb{R}}_+^{n+1}}$ is called {\it capillary hypersurface} if
it intersects $\partial\overline{{{\mathbb{R}}}_+^{n+1}}$ at a constant contact angle $\theta\in (0,\pi)$.
   Let $\nu$ be the unit outward normal of $\Sigma$. The contact angle $\theta $ is defined by $$ \cos (\pi -\theta)=\langle \nu, e \rangle,\quad \rm{along~\partial\Sigma},$$
	where $e\coloneqq -E_{n+1}$, $E_{n+1}$ is the $(n+1)$-th unit coordinate vector in $\overline{\mathbb{R}^{n+1}_+}$ and thus $e$ is the unit outward normal of $\partial \overline{\mathbb{R}^{n+1}_+}$.
This means that a capillary hypersurface in this paper is not necessarily a hypersurface of constant mean curvature.
If $\Sigma$ is strictly convex, then it is clear that the image of the Gauss map, namely,  $\nu$, lies on the spherical cap
\begin{eqnarray*}
		\mathbb{S}^n_\theta \coloneqq  \left\{ x\in \mathbb{S}^n \,|\,  \langle x, E_{n+1} \rangle \geq \cos \theta \right\}.
\end{eqnarray*}
Instead of using the usual Gauss map $\nu$, it is more convenient to use the following map
$$\tilde \nu\coloneqq  T \circ \nu: \Sigma \to \C_\theta,$$
where $\C_\theta$ is a spherical cap, defined by
\begin{eqnarray*}
		\C_{\theta }  \coloneqq  \left\{\xi\in \overline{\mathbb{R}^{n+1}_+} \mid |\xi-\cos\theta e|= 1 \right\},
\end{eqnarray*}
	and $T:\mathbb{S}^n_\theta\to\C_\theta$ is defined by $ T(y)\coloneqq y+\cos\theta e$,  which is a translation in the vertical direction.
 We refer to $\tilde \nu$  as the {\it capillary Gauss map} of $\Sigma$.  It turns out that $\tilde \nu: \Sigma\to \C_\theta$ is a diffeomorphism. Consequently,  one can reparametrize $\Sigma$ using its inverse on $\C_\theta$, see, e.g., \cite[Section 2]{MWWX}.

In \cite{MWW-AIM}, the authors formulated a capillary Minkowski problem, which concerns the existence of a strictly convex capillary hypersurface $\Sigma\subset \overline{\mathbb{R}^{n+1}_{+}}$ with a prescribed Gauss-Kronecker curvature on $\C_{\theta}$. We have established the following result in \cite[Theorem 1.1]{MWW-AIM}.

\

\noindent{\bf Theorem A}.
{\it Let $\theta\in (0,\frac{\pi}{2}]$ and $f\in C^2(\C_\theta)$ be a positive function  satisfying  \eqref{ortho-condition1} below.  Then there exists a $C^{3,\gamma}$ $(\gamma\in(0,1))$ strictly convex capillary hypersurface $\Sigma\subset \overline{\mathbb{R}^{n+1}_+}$ such that its Gauss-Kronecker curvature $K$ satisfying $$K(\tilde \nu^{-1}(\xi))=
		{f^{-1}(\xi)},$$ for all $\xi\in \C_\theta$. Moreover, $\Sigma$ is unique up to a horizontal translation in $\overline{\mathbb{R}^{n+1}_+}$.}

  \

The capillary Minkowski problem can also be viewed as a prescribed capillary area measure problem, finding a strictly convex capillary hypersurface such that the induced capillary area measure equals a prescribed Borel measure on $\C_{\theta}$. See \cite{MWW-AIM}. The area measures are an important subject in the study of convex bodies, and they are the local versions of quermassintegrals in the Brunn-Minkowski theory.  For more details about the classical Christoffel-Minkowski problem, see \cite[Chapter~8]{Sch}.

The primary objective of this article is to extend our work presented in \cite{MWW-AIM} and study the capillary Christoffel-Minkowski problem or the prescribed  $k$-th capillary area measure problem for the convex body with a capillary boundary in $\overline{\mathbb{R}^{n+1}_{+}}$.
 Firstly, we introduce a novel concept of local parallel sets for the capillary convex bodies. We say $\widehat{\Sigma}$ is a \textit{capillary convex body} (a compact convex set with non-empty interior) in $\mathbb{R}^{n+1}$ if it is a bounded closed region in $\overline{\mathbb{R}^{n+1}_+}$ enclosed by a strictly convex capillary hypersurface $\Sigma$ and  $\partial \overline{\mathbb{R}^{n+1}_{+}}$. The class of capillary convex bodies in $\overline{{\mathbb{R}}^{n+1}_+}$ is denoted by $\K_{\theta}$.   Given a Borel set $\beta\subset \C_{\theta}$ and a capillary convex body $\widehat{\Sigma}\subset \K_{\theta}$, we consider the following set :
\begin{eqnarray*}
    B_{s}(\widehat{\Sigma}, \beta)\coloneqq  \left\{Y\in \overline{{\mathbb{R}}^{n+1}_{+}} \mid  Y=X+t\tilde{\nu}(X),~0<t<s,~\text{for}~X\in \Sigma ~ \text{and} ~\tilde{\nu}(X)\in \beta \right\}.
\end{eqnarray*}
When $\beta=\C_{\theta}$, the set
\begin{eqnarray*}
    A_{s}(\widehat{\Sigma}, \C_{\theta}) \coloneqq \left\{Y\in \overline{{\mathbb{R}}^{n+1}_{+}} \mid  Y=X+s\tilde{\nu}(X), ~X\in \Sigma~\text{and}~s>0 \right\}
\end{eqnarray*}
is also a strictly convex capillary hypersurface in $\overline{{\mathbb{R}}^{n+1}_{+}}$ (see, for instance, \cite[Remark 2.17]{MWWX} or \cite[Proposition~2.3]{JWXZ}).

Direct calculation yields the following local Steiner-type formula (see also \cite[Remark~2.17]{MWWX} for a global version),
\begin{eqnarray*}
    \text{Vol}(B_{s}(\widehat{\Sigma}, \beta))&=&\sum\limits_{k=0}^{n}\frac{s^{n+1-k}}{n+1-k}\int_{\Sigma \cap \tilde{\nu}^{-1}(\beta)}(1+\cos\theta \<\nu, e\>)\sigma_{n-k}(\kappa)d\mu\notag\\
    &=&\frac{1}{n+1}\sum\limits_{k=0}^{n}s^{n+1-k}\binom{n+1}{k}S_{k}^c(\widehat{\Sigma}, \beta),
\end{eqnarray*}
where $S_k^c(\widehat{\Sigma},\beta)$ is given by
\begin{eqnarray}\label{measure}
    S_{k}^c(\widehat{\Sigma}, \beta)\coloneqq \binom{n}{k}^{-1}\int_{\beta}(\sin^{2}\theta+\cos\theta\<\xi, e\>)\sigma_{k}(r)d{\mathcal{H}}^{n},\quad \text{for}~0\leq k\leq n.
\end{eqnarray}
Here, $\kappa$ and $r$ are the set of the principal curvatures and principal curvature radii of $\Sigma$ respectively, $\sigma_{k}$ is the $k$-th elementary symmetric polynomial function, $d \mathcal{H}^n$ and $d\mu$ are the $n$-dimensional Hausdorff measure on $\C_\theta$
 and  $\Sigma$ respectively. The superscript $c$ indicates the presence of the \textit{capillary} effect in our notation. Moreover, $S_k^c (\widehat{\Sigma}, \C_{\theta})$ $(0\leq k\leq n)$ coincide with the quermassintegrals of convex capillary hypersurface introduced in  \cite[Eq. (1.8)]{WWX22}, see also \cite[Lemma~2.14]{MWWX}.

 For each capillary convex body $\widehat{\Sigma}$, by \eqref{measure} we introduce  a \textit{$k$-th capillary area measure} on $\C_{\theta}$, which is given  by \begin{eqnarray}\notag
     dS_k^c \coloneqq  \ell\sigma_{k}(\nabla^{2}h+h\sigma)d\mathcal{H}^{n}, ~~~0\leq k\leq n,
 \end{eqnarray}
 where $$\ell(\xi) \coloneqq \sin^{2}\theta+\cos\theta\<\xi, e\>,$$ and $h$ is the support function of $\Sigma$. In particular, when $\theta=\frac \pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\widehat{\Sigma}$, such that the  $k$-th capillary area measure is prescribed on $\C_{\theta}$. We formulate the problem as follows.

 \

\noindent	\textbf{Capillary Christoffel-Minkowski problem:} \textit{Given a positive, $C^{2}$ function $f$ on $\C_{\theta}$, when does  there exist a capillary convex body $\widehat{\Sigma}$ in $\overline{\mathbb{R}^{n+1}_+}$ such that
\begin{eqnarray*}
    dS_k^c=\ell fd\mathcal{H}^{n}~?
\end{eqnarray*}}

\

When $k=n$, it reduces to the $n$-th capillary area measure and corresponds to the capillary Minkowski problem, which was solved by the authors in \cite{MWW-AIM}.
For the intermediate case  $2\leq k\leq n-1$, we refer to this problem as the \textit{capillary Christoffel-Minkowski problem}. From Proposition~\ref{pro-2.6} below, the capillary Christoffel-Minkowski problem is equivalent to finding  convex solutions to the following Hessian type equation with Robin boundary value condition,  \begin{eqnarray}
\label{eq-k}
	\begin{array}{rcll}\vspace{2mm}
		\sigma_{k}(\nabla^2 h+h\sigma)&=&f,& \quad  \hbox{ in }  \C_{\theta},\\
		\n_\mu h&=&\cot\theta h, &\quad \hbox{ on }  \partial \C_\theta,
	\end{array}
 \end{eqnarray}
where   $h$ is an unknown function, $\nabla h$ and $\nabla^2h$ are the gradient and the Hessian of $h$ on $\C_\theta$ with respect to the standard spherical metric $\sigma$ on $\C_{\theta}$ respectively  and $\mu$ is the unit outward normal of $\partial\C_\theta\subset \C_{\theta}$.
If Eq. \eqref{eq-k} is solvable, the function $f$  satisfies
\begin{eqnarray}\label{ortho-condition1}
    \int_{\C_{\theta}}f(\xi)\<\xi, E_{\alpha}\>d\mathcal{H}^{n}=0,\quad \forall~ 1\leq \alpha\leq n,
\end{eqnarray} (cf. Proposition \ref{pro ortho} for more details)
where  $\{E_i\}_{i=1}^{n+1}$ is the standard coordinate unit vector of $\overline{\mathbb{R}^{n+1}_+}$. For the capillary Minkowski problem, the necessary condition \eqref{ortho-condition1} is also sufficient, while the situation becomes subtle for the general case $k\neq n$. This phenomenon also occurs in the closed case, see for instance \cite{GM, GMZ, STW}. Before presenting the main result,  we introduce some definitions.
\begin{defn}\label{def1.1}
  We say that a function $f\in C^{2}(\C_{\theta})$ is (strictly) convex if the matrix $\nabla^{2}f+f\sigma$ is positive semi-definite (respectively, positive definite) on $\C_{\theta}$.
\end{defn}
\begin{defn}
  For $s\in \mathbb{R}$, let $\mathcal{L}_{s}$ be the set of positive $C^{2}$ functions $f$ on $\C_{\theta}$ such that $f$ satisfies \eqref{ortho-condition1} and $f^{s}$ is convex on $\C_{\theta}$, in the sense of Definition \ref{def1.1}. Moreover, we say $f$ is connected to $g$ in $\mathcal{L}_{s}$ if there exists a continuous path $q(t)\in \mathcal{L}_{s}$, $0\leq t\leq 1$, such that $q(0)=g$, $q(1)=f$, and $\n_{\mu}q(t)\geq 0$ on $\partial\C_{\theta}$ for any $t\in[0, 1]$.
\end{defn}

It is worth noting that, compared to the definition of homotopy given in \cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \eqref{eq-k}.

\subsection {Main result}
The main theorem of this paper is as follows.

\noindent{\bf Theorem A}.
{\it Let $\theta\in (0,\frac{\pi}{2}]$ and $f\in C^2(\C_\theta)$ be a positive function satisfying \eqref{ortho-condition1} below. Then there exists a $C^{3,\gamma}$ $(\gamma\in(0,1))$ strictly convex capillary hypersurface $\Sigma\subset \overline{\mathbb{R}^{n+1}_+}$ such that its Gauss-Kronecker curvature $K$ satisfying $$K(\tilde \nu^{-1}(\xi))=
 {f^{-1}(\xi)},$$ for all $\xi\in \C_\theta$. Moreover, $\Sigma$ is unique up to a horizontal translation in $\overline{\mathbb{R}^{n+1}_+}$.}

For each capillary convex body $\widehat{\Sigma}$, by \eqref{measure} we introduce a \textit{$k$-th capillary area measure} on $\C_{\theta}$, which is given by \begin{eqnarray}\notag
 dS_k^c \coloneqq \ell\sigma_{k}(\nabla^{2}h+h\sigma)d\mathcal{H}^{n}, ~~~0\leq k\leq n,
 \end{eqnarray}
 where $$\ell(\xi) \coloneqq \sin^{2}\theta+\cos\theta\<\xi, e\>,$$ and $h$ is the support function of $\Sigma$. In particular, when $\theta=\frac \pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\widehat{\Sigma}$, such that the $k$-th capillary area measure is prescribed on $\C_{\theta}$. We formulate the problem as follows.

When $k=n$, it reduces to the $n$-th capillary area measure and corresponds to the capillary Minkowski problem, which was solved by the authors in \cite{MWW-AIM}.
For the intermediate case $2\leq k\leq n-1$, we refer to this problem as the \textit{capillary Christoffel-Minkowski problem}. From Proposition~\ref{pro-2.6} below, the capillary Christoffel-Minkowski problem is equivalent to finding convex solutions to the following Hessian type equation with Robin boundary value condition, \begin{eqnarray}
\label{eq-k}
 \begin{array}{rcll}\vspace{2mm}
 \sigma_{k}(\nabla^2 h+h\sigma)&=&f,& \quad \hbox{ in } \C_{\theta},\\
 \n_\mu h&=&\cot\theta h, &\quad \hbox{ on } \partial \C_\theta,
 \end{array}
 \end{eqnarray}
where $h$ is an unknown function, $\nabla h$ and $\nabla^2h$ are the gradient and the Hessian of $h$ on $\C_\theta$ with respect to the standard spherical metric $\sigma$ on $\C_{\theta}$ respectively and $\mu$ is the unit outward normal of $\partial\C_\theta\subset \C_{\theta}$.
If Eq. \eqref{eq-k} is solvable, the function $f$ satisfies
\begin{eqnarray}\label{ortho-condition1}
 \int_{\C_{\theta}}f(\xi)\<\xi, E_{\alpha}\>d\mathcal{H}^{n}=0,\quad \forall~ 1\leq \alpha\leq n,
\end{eqnarray} (cf. Proposition \ref{pro ortho} for more details)
where $\{E_i\}_{i=1}^{n+1}$ is the standard coordinate unit vector of $\overline{\mathbb{R}^{n+1}_+}$. For the capillary Minkowski problem, the necessary condition \eqref{ortho-condition1} is also sufficient, while the situation becomes subtle for the general case $k\neq n$. This phenomenon also occurs in the closed case, see for instance \cite{GM, GMZ, STW}. Before presenting the main result, we introduce some definitions.
\begin{defn}\label{def1.1}
 We say that a function $f\in C^{2}(\C_{\theta})$ is (strictly) convex if the matrix $\nabla^{2}f+f\sigma$ is positive semi-definite (respectively, positive definite) on $\C_{\theta}$.
\end{defn}
\begin{defn}
 For $s\in \mathbb{R}$, let $\mathcal{L}_{s}$ be the set of positive $C^{2}$ functions $f$ on $\C_{\theta}$ such that $f$ satisfies \eqref{ortho-condition1} and $f^{s}$ is convex on $\C_{\theta}$, in the sense of Definition \ref{def1.1}. Moreover, we say $f$ is connected to $g$ in $\mathcal{L}_{s}$ if there exists a continuous path $q(t)\in \mathcal{L}_{s}$, $0\leq t\leq 1$, such that $q(0)=g$, $q(1)=f$, and $\n_{\mu}q(t)\geq 0$ on $\partial\C_{\theta}$ for any $t\in[0, 1]$.
\end{defn}

It is worth noting that, compared to the definition of homotopy given in \cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \eqref{eq-k}.

\subsection {Main result}
The main theorem of this paper is as follows.

For each capillary convex body $\widehat{\S}$, by \eqref{measure} we introduce a \textit{$k$-th capillary area measure} on $\C_{\theta}$, which is given by \begin{eqnarray}\notag
 dS_k^c \coloneqq \ell\sigma_{k}(\n^{2}h+h\sigma)d\mathcal{H}^{n}, ~~~0\leq k\leq n,
 \end{eqnarray}
 where $$\ell(\xi) \coloneqq \sin^{2}\theta+\cos\theta\<\xi, e\>,$$ and $h$ is the support function of $\S$. In particular, when $\theta=\frac \pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\widehat{\S}$, such that the $k$-th capillary area measure is prescribed on $\C_{\theta}$. We formulate the problem as follows.

It is worth noting that, compared to the definition of homotopy given in \cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \eqref{eq-k}.

\

In this section, we establish the a priori estimates for the strictly convex solution of Eq. \eqref{eq-k}.
For any solution $h$ to Eq. \eqref{eq-k}, it is easy to verify that
\begin{eqnarray*}
 h(\xi)+\<\xi,E_\alpha\> ,\quad \text{ for all } 1\leq \a\leq n,
\end{eqnarray*} is still a solution. Up to a horizontal translation, we assume $h$ satisfies the following orthogonal condition:
\begin{eqnarray}\label{orth cond}
 \int_{\C_\theta} h(\xi) \<\xi,E_\alpha\> d\H^{n}=0, \quad \text{ for all } 1\leq \a\leq n.
\end{eqnarray}
Under the condition \eqref{orth cond}, we are able to establish the following a priori estimates.
\begin{thm}\label{priori estimate}
Let $\theta\in (0, \frac{\pi}{2})$. Suppose that $h$ is a strictly convex solution of Eq. \eqref{eq-k} and satisfies the condition \eqref{orth cond}. For any integer $m\geq 2$ and $\gamma\in(0,1)$, there exists a constant $C>0$ depending on $n, k,$ $\min\limits_{\C_{\theta}} f$ and $\|f\|_{C^{m}(\C_\theta)}$, such that
 \begin{eqnarray}\label{sch est}
 \|h\|_{C^{m+1,\gamma}(\C_\theta)}\leq C.
 \end{eqnarray}
\end{thm}

\begin{lem}\label{c0 est}
Let $\theta\in(0, \pi)$ and $h$ be a strictly convex solution of Eq. \eqref{eq-k} that satisfies \eqref{orth cond}.
Then there exists a positive constant $C$ depending on $n, \min\limits_{\C_{\theta}}f$, and $\max\limits_{\C_\theta} f$, such that
\begin{eqnarray*}
 0<\min_{\C_\theta} h\leq \max_{\C_\theta} h\leq C.
\end{eqnarray*}
\end{lem}
\begin{proof}
First of all, due to \eqref{orth cond}, by \cite[Lemma 3.1]{MWW-AIM}, we have $h>0$ and the origin lies in the interior of $\widehat{\partial\Sigma}$.
Let $R_{0}>0$ be the smallest number of $R$ such that
 \begin{eqnarray}\label{enclosed}
 \widehat \S\subset \widehat{R\C_\theta},
 \end{eqnarray}where $R\C_\theta\coloneqq \{\xi \in \ov{\RR^{n+1}_+}: |\xi-\cos\theta R e|=R\}$. It is clear that there exists $X_0\in \Sigma \cap R_0\C_{\theta } $. Denote $\widehat{X}_0\coloneqq \frac{X_0 }{R_0}\in \C_\theta$.
For any $\xi \in \C_\theta$, we have
 \begin{eqnarray}\label{lower bdd of h}
 h(\xi)&=&\max_{Y\in \S}\<\xi-\cos\theta e, Y\>\notag \\
 &\geq& \max \left\{0, \<\xi-\cos\theta e,X_0\> \right\}\notag \\
 &=& \max \left\{0, R_0\<\xi-\cos\theta e,\widehat X_0\> \right\}.
 ~~~\end{eqnarray}
Multiplying $f$ on both sides of \eqref{lower bdd of h} and integrating over $ \C_\theta$, we see
\begin{eqnarray}\label{R1}
 R_0\leq \left(\int_{\C_\theta}h(\xi)\cdot f(\xi)d \H^{n}\right) \left(\int_{\C_\theta}\max \left\{0, \<\xi-\cos\theta e,\widehat X_0\> \right \}\cdot f(\xi) d \H^{n}\right) ^{-1}.
\end{eqnarray}
Note that
\begin{eqnarray}\label{R2}
 \int_{\C_{\theta}}h(\xi)\cdot f(\xi) d \H^{n}=
\int_{\tilde{\nu}(\Sigma)}h(\xi)\cdot f(\xi)d\H^{n}=\int_{\S}\<X, \nu\>\sigma_{n-k}(\kappa)d\mu.
\end{eqnarray}
Together with the Alexandrov-Fenchel inequalities for convex capillary hypersurfaces \cite[Theorem~1.2]{MWWX} for $\theta\in(0, \pi)$ and the Minkowski formulas for capillary hypersurfaces \cite[Proposition~2.6]{WWX22}, we obtain
\begin{eqnarray}
 C_{n, k}\left(\int_{\S}\<X, \nu\>\sigma_{n-k}(\kappa)d\mu\right)^{\frac{1}{k+1}}&\leq& \left(\int_{\Sigma}\<X, \nu\>\sigma_{n-k+1}(\kappa)d\mu\right)^{\frac{1}{k}}\notag \\
 &=&\left(\frac{k}{n-k+1}\int_{\C_{\theta}}f(\xi)d\H^{n}\right)^{\frac{1}{k}}.\label{R3}
\end{eqnarray}
Together with \eqref{R1}, \eqref{R2} and \eqref{R3}, we conclude that
\begin{eqnarray}
 R_{0}
 &\leq& C_{n,k}\left(\int_{\C_{\theta}}f(\xi)d\H^{n}\right)^{\frac{k+1}{k}}\cdot\left(\inf\limits_{\eta\in \C_{\theta}}\int_{\C_\theta}\max\{0, \<\xi-\cos\theta e, \eta\>\}\cdot f(\xi) d \H^{n}\right) ^{-1}.\quad ~~~\label{upper bound of R0}
\end{eqnarray}
Using \eqref{enclosed} and \eqref{upper bound of R0}, we see
\begin{eqnarray*}
 h(\xi) =\<\xi-\cos\theta e , \tilde{\nu}^{-1}(\xi)\>\leq |\tilde{\nu}^{-1}(\xi)| <2R_0\leq C(n, \min_{\C_\theta} f,\max_{\C_\theta} f).
\end{eqnarray*}
 We complete the proof of Lemma \ref{c0 est}.
\end{proof}
\subsection{\texorpdfstring{$C^1$ }{}estimate}
 In this subsection, we get the gradient estimate for the strictly convex solution $h$ of Eq. \eqref{eq-k}. This estimate is a direct consequence of the convexity and the $C^{0}$ estimate.
\begin{lem}\label{C1 est}
 Let $\theta\in (0, \pi)$, and let $h$ be a strictly convex solution of Eq. \eqref{eq-k}. Then \begin{eqnarray}\label{gradient ps}
 \max\limits_{\C_{\theta}} |\n h|\leq C,
 \end{eqnarray}
 where the constant $C$ only depends on $\|h\|_{C^{0}(\C_{\theta})}$.
\end{lem}
\begin{proof}
 Consider the function
 \begin{eqnarray*}
 P=|\n h|^{2}+h^{2}.
 \end{eqnarray*}
Suppose that $P$ attains its maximum value at some point, say $\xi_{0}\in \C_{\theta}$. If $\xi_{0}\in \C_{\theta}\setminus \partial\C_{\theta}$, by the maximal condition, we have
\begin{eqnarray*}
 0=\n_{e_i}P=2h_{k}h_{ki}+2hh_{i},\quad \text{for}~1\leq i\leq n.
\end{eqnarray*}
Since $h$ is a strictly convex function, we have $\n h(\xi_{0})=0$. Combining with Lemma \ref{c0 est}, we conclude that \eqref{gradient ps} holds.

\section{Convexity}\label{sec-convex}
The primary goal of this section is to obtain a strictly convex solution of Eq. \eqref{eq-k}. Based on the celebrated result of Guan-Ma \cite{GM} and with suitable assumptions on the function $f$, we show that the convex solution of Eq. \eqref{eq-k} is strictly convex. To begin with, we establish the following key lemma.
\begin{lem}\label{lem-def}
 Let $\theta\in (0, \frac{\pi}{2})$ and $\tilde{F}(W) \coloneqq \sigma_{k}(W)$. Suppose that $h\in C^{2}(\C_{\theta})$ is a solution of Eq. \eqref{eq-k}, and assume that $W$ is positive semi-definite on $\C_\theta$. Suppose there exists a positive constant $C_{0}$ such that for a fixed integer $l $ $(k\leq l\leq n-1)$, $\sigma_{l}(W)\geq C_{0}$ for all $\xi\in \C_{\theta}$. If $f\in \mathcal{L}_{-\frac{1}{k}}$ and $\n_{\mu}f\geq 0$ on $\partial \C_{\theta}$, denote $$\phi\coloneqq \sigma_{l+1}(W(\xi)),$$ then there exist constants $C_{1}, C_{2}$ depending only on $n, k, C_{0}, \|h\|_{C^{3}},$ and $\|f\|_{C^{2}}$, such that $\phi$ satisfies
 \begin{eqnarray}
 & \tilde{F}^{ij}\phi_{ij}\leq C_{1}|\n \phi|+C_{2}\phi, & \quad \rm{in} ~~ \C_{\theta} ,\label{Diff-ine}\\
 &\n_{\mu} \phi\geq 0, & \quad {\rm{on}}~\partial \C_{\theta}.\label{boundary ine}
 \end{eqnarray}
 \end{lem}
 \begin{proof}
 The differential inequality \eqref{Diff-ine} follows from \cite[Lemma~4.1]{GM} and $f\in \mathcal{L}_{-\frac{1}{k}}$. We just need to prove the boundary condition \eqref{boundary ine}.

\

We conclude the introduction by outlining the strategy for solving  Eq. \eqref{eq-k}. If $\theta=\frac{\pi}{2}$, this problem can be reduced to the closed case using a reflection argument. Hence, we focus on the case $\theta<\frac \pi 2$ in proving Theorem \ref{main theorem}. We will employ the method of continuity.  We need to establish a  priori estimates for solutions to Eq. \eqref{eq-k} and show that the convexity is preserved. To establish the $C^{0}$ estimate, we modify the approach in  \cite[Lemma~3]{CY76}, \cite[Lemma 3.1]{MWW-AIM} and combine it with a general form of capillary Alexandrov-Fenchel inequalities for capillary hypersurfaces presented in \cite[Theorem~1.2]{MWWX} for $\theta\in(0, \pi)$.  Hence our $C^0$-estimate for \eqref{eq-k} holds for all contact angle $\theta\in(0, \pi)$, see Lemma \ref{c0 est}. The $C^1$ estimate follows from the strict convexity of $h$ and the $C^0$ estimate. In contrast to the closed case considered in \cite[Proposition 3.2]{GM}, our global $C^2$ estimate depends on the gradient estimates, due to the specific choice of test functions used to derive the boundary $C^2$ estimate. The choice of the test functions is motivated by the works of \cite{LTU, MQ} and \cite{MWW-AIM}.
For the preservation of the convexity, we apply the celebrated result of Guan-Ma \cite{GM} on the Constant Rank Theorem.
It is worth noting that the assumption $\theta\in (0, \frac{\pi}{2})$ is crucial not only for deriving the $C^2$-estimate, but also for showing the preservation of the convexity.
A similar assumption was employed in our previous works \cite{MWW-Weingarten} and \cite{MWW-Lp}, concerning the problems of convex capillary surfaces with prescribed $k$-th Weingarten curvature and prescribed capillary $L_p$-surface area measures for convex capillary bodies, respectively.  Nevertheless, we expect that the above results still hold for $\theta > \frac{\pi}{2}$.

\

\textbf{The rest of the article is structured as follows.}
	In Section \ref{sec2}, we list some properties of elementary symmetric functions and introduce some notations and basic facts about capillary hypersurfaces. Moreover, we prove the necessary condition \eqref{ortho-condition1} for the capillary Christoffel-Minkowski problem.
	In Section \ref{sec3}, we establish a priori estimates for the solution to Eq. \eqref{eq-k}. In Section \ref{sec-convex}, under certain assumptions on $f$, we investigate the convexity of the solution to Eq. \eqref{eq-k}.
 In the final section, by using the method of continuity, we prove Theorem \ref{main theorem}.
	\vspace{.2cm}

To prove Theorem~\ref{main theorem}, they first note that if $\theta=\frac{\pi}{2}$ the problem “can be reduced to the closed case using a reflection argument,” so they “focus on the case $\theta<\frac \pi 2$.” The proof then “employ[s] the method of continuity,” which requires establishing “a priori estimates for solutions to Eq.~\eqref{eq-k} and show[ing] that the convexity is preserved.”

For the estimates: the $C^{0}$ estimate is obtained by “modify[ing] the approach in \cite[Lemma~3]{CY76}, \cite[Lemma 3.1]{MWW-AIM} and combin[ing] it with a general form of capillary Alexandrov-Fenchel inequalities… in \cite[Theorem~1.2]{MWWX},” yielding a $C^0$ bound “for all contact angle $\theta\in(0,\pi)$.” The “$C^1$ estimate follows from the strict convexity of $h$ and the $C^0$ estimate.” The “global $C^2$ estimate depends on the gradient estimates,” due to the “specific choice of test functions used to derive the boundary $C^2$ estimate,” motivated by \cite{LTU, MQ} and \cite{MWW-AIM}.

For convexity preservation, they “apply the celebrated result of Guan-Ma \cite{GM} on the Constant Rank Theorem,” and emphasize that the assumption $\theta\in(0,\frac{\pi}{2})$ is “crucial not only for deriving the $C^2$-estimate, but also for showing the preservation of the convexity.” Finally, “by using the method of continuity,” they conclude the existence/uniqueness statement of Theorem~\ref{main theorem}.

To prove the main theorem, they first note that if $\theta=\frac{\pi}{2}$ the problem “can be reduced to the closed case using a reflection argument,” so they “focus on the case $\theta<\frac \pi 2$.” The proof then “employ[s] the method of continuity,” which requires establishing “a priori estimates for solutions to
\begin{eqnarray}
\label{eq-k}
	\begin{array}{rcll}\vspace{2mm}
		\sigma_{k}(\n^2 h+h\sigma)&=&f,& \quad  \hbox{ in }  \C_{\theta},\\
		\n_\mu h&=&\cot\theta h, &\quad \hbox{ on }  \p \C_\theta,
	\end{array}
 \end{eqnarray}
 and show[ing] that the convexity is preserved.

For the estimates: the $C^{0}$ estimate is obtained by “modify[ing] the approach in \cite[Lemma~3]{CY76}, \cite[Lemma 3.1]{MWW-AIM} and combin[ing] it with a general form of capillary Alexandrov-Fenchel inequalities… in \cite[Theorem~1.2]{MWWX},” yielding a $C^0$ bound “for all contact angle $\theta\in(0,\pi)$.” The “$C^1$ estimate follows from the strict convexity of $h$ and the $C^0$ estimate.” The “global $C^2$ estimate depends on the gradient estimates,” due to the “specific choice of test functions used to derive the boundary $C^2$ estimate,” motivated by \cite{LTU, MQ} and \cite{MWW-AIM}.

For convexity preservation, they “apply the celebrated result of Guan-Ma \cite{GM} on the Constant Rank Theorem,” and emphasize that the assumption $\theta\in(0,\frac{\pi}{2})$ is “crucial not only for deriving the $C^2$-estimate, but also for showing the preservation of the convexity.” Finally, “by using the method of continuity,” they conclude the existence/uniqueness statement of the main theorem.

\label{main theorem}
  Let $\theta\in (0,\frac{\pi}{2}]$ and $1\leq k\leq n-1$.  Suppose $f\in C^{2}(\C_{\theta})$ is connected to $1$ in $\mathcal{L}_{-\frac{1}{k}}$.  Then there exists a $C^{3, \gamma}$ $(\gamma\in(0,1))$ strictly convex capillary hypersurface $\Sigma$ in $ \overline{{\mathbb{R}}^{n+1}_{+}}$, such that $\widehat{\Sigma}$ satisfies the prescribed  $k$-th capillary area measure problem.
  Moreover, $\Sigma$ is unique up to a horizontal translation in $\overline{{\mathbb{R}}^{n+1}_{+}}$.,

Question:
Let \(\theta\in(0,\pi/2]\), \(1\le k\le n-1\), \(e=E_{n+1}\), and let \(\mathcal C_\theta\subset \mathbb S^n\) be the spherical cap \(\{\xi\in\mathbb S^n:\langle \xi,e\rangle>-\cos\theta\}\). For \(s\in\mathbb R\), let \(\mathcal L_s\) be the set of positive \(C^2\)-functions \(g\) on \(\mathcal C_\theta\) such that
\[
\int_{\mathcal C_\theta} g(\xi)\langle \xi,E_\alpha\rangle\,d\mathcal H^n=0\quad\text{for }\alpha=1,\dots,n,
\]
and \(g^s\) is convex in the sense that \(\nabla^2(g^s)+g^s\sigma\) is positive semidefinite on \(\mathcal C_\theta\). Say that \(f\) is connected to \(1\) in \(\mathcal L_{-1/k}\) if there is a continuous path \(q(t)\in \mathcal L_{-1/k}\), \(0\le t\le1\), with \(q(0)=1\), \(q(1)=f\), and \(\partial_\mu q(t)\ge0\) on \(\partial\mathcal C_\theta\) for all \(t\). For a capillary convex body \(\widehat\Sigma\), its \(k\)-th capillary area measure on \(\mathcal C_\theta\) is
\[
dS_k^c=\ell(\xi)\,\sigma_k(\nabla^2h+h\sigma)\,d\mathcal H^n,
\qquad \ell(\xi)=\sin^2\theta+\cos\theta\langle \xi,e\rangle,
\]
where \(h\) is the support function of \(\Sigma\). If \(f\in C^2(\mathcal C_\theta)\) is connected to \(1\) in \(\mathcal L_{-1/k}\), which statement holds?

There exists a \(C^{3,\gamma}\) hypersurface \(\Sigma\subset\overline{\mathbb R^{n+1}_+}\), for some \(\gamma\in(0,1)\), which is strictly convex and capillary, such that its associated capillary convex body \(\widehat\Sigma\) has prescribed \(k\)-th capillary area measure
\[
dS_k^c=\ell(\xi)f(\xi)\,d\mathcal H^n\quad\text{on }\mathcal C_\theta.
\]
Moreover, any two such hypersurfaces differ only by a horizontal translation in \(\overline{\mathbb R^{n+1}_+}\).

Choices:
B.
There exists a \(C^{3,\gamma}\) hypersurface \(\Sigma\subset\overline{\mathbb R^{n+1}_+}\), for some \(\gamma\in(0,1)\), which is strictly convex and capillary, such that its associated capillary convex body \(\widehat\Sigma\) has prescribed \(k\)-th capillary area measure
\[
dS_k^c=\ell(\xi)f(\xi)\,d\mathcal H^n\quad\text{on }\mathcal C_\theta.
\]
Moreover, such a hypersurface is unique in \(\overline{\mathbb R^{n+1}_+}\) without allowing any horizontal translation.
C.
There exists a \(C^{3,\gamma}\) hypersurface \(\Sigma\subset\overline{\mathbb R^{n+1}_+}\), for some \(\gamma\in(0,1)\), which is strictly convex and capillary, such that its associated capillary convex body \(\widehat\Sigma\) has prescribed \(k\)-th capillary area measure
\[
dS_k^c=\ell(\xi)f(\xi)\,d\mathcal H^n\quad\text{on }\mathcal C_\theta.
\]
D.
For every \(\theta\in(0,\pi)\), if \(f\in C^2(\mathcal C_\theta)\) is connected to \(1\) in \(\mathcal L_{-1/k}\), then there exists a \(C^{3,\gamma}\) hypersurface \(\Sigma\subset\overline{\mathbb R^{n+1}_+}\), for some \(\gamma\in(0,1)\), which is strictly convex and capillary, such that its associated capillary convex body \(\widehat\Sigma\) has prescribed \(k\)-th capillary area measure
\[
dS_k^c=\ell(\xi)f(\xi)\,d\mathcal H^n\quad\text{on }\mathcal C_\theta,
\]
and any two such hypersurfaces differ only by a horizontal translation in \(\overline{\mathbb R^{n+1}_+}\).
E.
If \(f\in C^2(\mathcal C_\theta)\) satisfies the orthogonality condition
\[
\int_{\mathcal C_\theta} f(\xi)\langle \xi,E_\alpha\rangle\,d\mathcal H^n=0\quad\text{for }\alpha=1,\dots,n,
\]
then there exists a \(C^{3,\gamma}\) hypersurface \(\Sigma\subset\overline{\mathbb R^{n+1}_+}\), for some \(\gamma\in(0,1)\), which is strictly convex and capillary, such that its associated capillary convex body \(\widehat\Sigma\) has prescribed \(k\)-th capillary area measure
\[
dS_k^c=\ell(\xi)f(\xi)\,d\mathcal H^n\quad\text{on }\mathcal C_\theta.
\]
Moreover, any two such hypersurfaces differ only by a horizontal translation in \(\overline{\mathbb R^{n+1}_+}\).

homotopy/convexity-preserving connectedness in \(\mathcal L_{-1/k}\)


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 $\gcd(n,d)=1$.


\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