# Mean value formulas on surfaces in Grushin spaces

Valentina Franceschi∗ ,  Roberto Monti∗ and  Alessandro Socionovo†
∗Dipartimento di Matematica Tullio Levi Civita, Università di Padova, Italy
<EMAIL_ADDRESS><EMAIL_ADDRESS>†Laboratoire Jacques-Louis
Lions, CNRS, Inria, Sorbonne Université, Université de Paris, France
<EMAIL_ADDRESS>
###### Abstract.

We prove (sub)mean value formulas at the point $0\in\Sigma$ for (sub)harmonic
functions on a hypersurface $\Sigma\subset\mathbb{R}^{n+1}$ where the
differentiable structure and the surface measure depend on the ambient Grushin
structure.

###### 2020 Mathematics Subject Classification:

Main: 35B05. Secondary: 35H20, 49Q05.

## Acknowledgments

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 101034255.

##  1\. Introduction

For $n\in\mathbb{N}$ and $\alpha>0$, we consider the vector fields on
$\mathbb{R}^{n+1}$

$\begin{array}[]{l}\displaystyle X_{i}=\frac{\partial}{\partial{x_{i}}},\quad
i=1,\ldots,n,\quad\displaystyle X_{n+1}=|x|^{\alpha}\frac{\partial}{\partial
y}.\end{array}$ (1.1)

Here, a generic point in $\mathbb{R}^{n+1}$ is denoted by
$\xi=(x,y)=(x_{1},\ldots,x_{n},y)\in\mathbb{R}^{n+1}$. We also consider the
second order partial differential operator on $\mathbb{R}^{n+1}$ given by

$\mathscr{L}\varphi=\sum_{i=1}^{n+1}X_{i}^{2}\varphi=\Delta_{x}\varphi+|x|^{2\alpha}\partial^{2}_{y}\varphi,$
(1.2)

where $|x|=(x_{1}^{2}+\ldots+x_{n}^{2})^{1/2}$. The operator $\mathscr{L}$ in
(1.2) is known as Baouendi-Grushin operator, see [9] for a historical account
and see also [7].

When $\alpha$ is an even integer, this operator is hypoelliptic and admits a
fundamental solution with pole at any point $\xi_{0}\in\mathbb{R}^{n+1}$ (see
[1] for an explicit representation). When $\xi_{0}=0$, an explicit formula for
this fundamental solution is in fact known for any $\alpha>0$ (see [8]) and,
up to a normalization constant, it is the function
$\Gamma(\xi)=\varrho(\xi)^{1-n-\alpha}$, $\xi\neq 0$, where
$\varrho:\mathbb{R}^{n+1}\to\mathbb{R}$ is the gauge function

$\varrho(\xi)=\big{(}|x|^{2(\alpha+1)}+(\alpha+1)^{2}y^{2}\big{)}^{\frac{1}{2(\alpha+1)}}.$
(1.3)

Let $\Sigma\subset\mathbb{R}^{n+1}$ be a hypersurface of class $C^{2}$ with
$0\in\Sigma$. We declare the vector-fields (1.1) orthonormal and we project
them onto the tangent space to $\Sigma$, getting tangential operators
$\delta_{1},\ldots,\delta_{n+1}$. We fix on $\Sigma$ the hypersurface measure
$\sigma$ associated with (1.1) according to the theory of sub-elliptic
perimeters and then we define the adjoint operators
$\delta_{1}^{*},\ldots,\delta_{n+1}^{*}$ with respect to $\sigma$. The natural
restriction of $\mathscr{L}$ to $\Sigma$ is the differential operator

$\mathscr{L}_{\Sigma}=-\sum_{i=1}^{n+1}\delta_{i}^{*}\delta_{i}.$

In this paper, we investigate the validity of mean value formulas (sub-mean
value formulas) at the point $0\in\Sigma$ for functions $f\in C^{2}(\Sigma)$
satisfying $\mathscr{L}_{\Sigma}f=0$ ($\mathscr{L}_{\Sigma}f\geq 0$,
respectively).

The operator $\mathscr{L}$ is an example of “sub-Laplacian” or “sum of squares
of vector fields” satisfying the Hörmander condition [10]. When
$\mathscr{L}=\sum_{i=1}^{m}X_{i}^{2}$ is such an operator in
$\mathbb{R}^{n+1}$, the validity of mean-value formulas for
$\mathscr{L}$-harmonic functions is established in [4, 3, 5]. Denoting by
$\Gamma(\cdot,\xi_{0})$ the fundamental solution for $\mathscr{L}$ with pole
at $\xi_{0}$, if a function $f$ satisfies $\mathscr{L}f=0$ then for any $r>0$
and $\xi_{0}\in\mathbb{R}^{n+1}$

$f(\xi_{0})=\frac{1}{r}\int_{\Omega_{r}(\xi_{0})}f(\xi)K(\xi,\xi_{0})\;d\xi,$
(1.4)

where
$\Omega_{r}(\xi_{0})=\\{\xi\in\mathbb{R}^{n+1}:\Gamma(\xi,\xi_{0})>1/r\\}$ and
$K(\xi,\xi_{0})={|X\Gamma(\xi,\xi_{0})|^{2}}/{\Gamma(\xi,\xi_{0})^{2}}$, with
$|X\Gamma(\cdot,\xi_{0})|^{2}=\sum_{i=1}^{m}(X_{i}\Gamma(\cdot,\xi_{0}))^{2}$.
The appearance of the kernel $K$ is due to the different symmetry of Carnot-
Carathéodory balls associated with the vector-fields building up $\mathscr{L}$
and level sets of $\Gamma(\cdot,\xi_{0})$.

In the Riemannian case, the validity of mean-value formulas on metric balls
for harmonic functions leads to the notion of “harmonic manifold”. Starting
probably with [11], there exists a huge literature on the problem of
characterizing harmonic manifolds and it is not possible to give a full
account, here. In fact, our hypersurface $\Sigma$ embedded in
$\mathbb{R}^{n+1}$ with the Grushin structure is not a Riemannian manifold but
rather a weighted Riemannian manifold that becomes singular at the point
$0\in\Sigma$, see Remark 2.1.

In the Grushin space, the harmonicity at $0\in\Sigma$ is governed by the
following structural function $q_{\Sigma}:\Sigma\setminus\\{0\\}\to\mathbb{R}$

$q_{\Sigma}(\xi)=\langle X\varrho,\nu\rangle\Big{[}(n+3\alpha)\langle
X\log\varrho,\nu\rangle-2\alpha\langle\nabla_{x}\log|x|,\bar{\nu}\rangle+nH_{\Sigma}\Big{]}.$
(1.5)

Above, $X\varrho=(X_{1}\varrho,\ldots,X_{n+1}\varrho)$ is the $X$-gradient of
the gauge function $\varrho$, $\nu=(\bar{\nu},\nu_{n+1})$ is the
$\alpha$-normal to $\Sigma$, $\langle\cdot,\cdot\rangle$ are standard scalar
products in $\mathbb{R}^{n+1}$ and $\mathbb{R}^{n}$, and $H_{\Sigma}$ is the
mean curvature of $\Sigma$ associated with the Grushin structure. We say that
$\Sigma$ is $\alpha$-harmonic if $q_{\Sigma}=0$. In particular, any
homogeneous hypersurface, $\langle X\varrho,\nu\rangle=0$, is
$\alpha$-harmonic, as we show in Section 5.1.

######  Theorem 1.1.

Let $\Sigma\subset\mathbb{R}^{n+1}$ be an $\alpha$-harmonic hypersurface of
class $C^{2}$ with $0\in\Sigma$. Any function $f\in C^{2}(\Sigma)$ such that
$\mathscr{L}_{\Sigma}f=0$ satisfies the mean-value formula at $0$

$f(0)=\frac{C_{\Sigma,n,\alpha}}{r^{n+\alpha}}\int_{B_{r}\cap\Sigma}f(\xi)\,|\delta\varrho(\xi)|^{2}d\sigma,$
(1.6)

for all $r\in(0,r_{0})$ and for some $r_{0}>0$ depending on $\Sigma$. The
constant $0<C_{\Sigma,n,\alpha}<\infty$ is defined by

$\frac{1}{C_{\Sigma,n,\alpha}}=\frac{1}{r^{n+\alpha}}\int_{B_{r}\cap\Sigma}|\delta\varrho(\xi)|^{2}d\sigma,$
(1.7)

where the right hand-side does not depend on $r\in(0,r_{0})$.

Above, the balls are

$B_{r}=\\{\xi\in\mathbb{R}^{n+1}:\varrho(\xi)<r\\}.$ (1.8)

and $|\delta\varrho|\leq 1$ is the length of the tangential gradient of
$\varrho$. When $\Sigma$ is homogeneous, the kernel is
$|\delta\varrho|^{2}=|x|^{2\alpha}/\varrho^{2\alpha}$. In the case of
$\alpha$-subharmonic hypersurfaces, $q_{\Sigma}\geq 0$, the statement is
similar.

######  Theorem 1.2.

Let $\Sigma\subset\mathbb{R}^{n+1}$ be an $\alpha$-subharmonic hypersurface of
class $C^{2}$ with $0\in\Sigma$. Any function $f\in C^{2}(\Sigma)$ such that
$\mathscr{L}_{\Sigma}f\geq 0$ satisfies the following sub-mean-value formula
at $0$

$f(0)\leq\frac{C_{\Sigma,n,\alpha}}{r^{n+\alpha}}\int_{B_{r}\cap\Sigma}f(\xi)\,|\delta\varrho(\xi)|^{2}d\sigma,$
(1.9)

for all $r\in(0,r_{0})$ and for some $r_{0}>0$ depending on $\Sigma$. The
constant $0<C_{\Sigma,n,\alpha}<\infty$ is defined by

$\frac{1}{C_{\Sigma,n,\alpha}}=\lim_{r\to
0^{+}}\frac{1}{r^{n+\alpha}}\int_{B_{r}\cap\Sigma}|\delta\varrho(\xi)|^{2}d\sigma.$
(1.10)

The operator $\mathscr{L}$ in (1.2) and the hyper-surface measure $\sigma$ are
invariant with respect to the vertical translations $(x,y)\mapsto(x,y+y_{0})$,
for any fixed $y_{0}\in\mathbb{R}$. Theorem 1.2 can be therefore extended to
get mean value formulas at points $(0,y_{0})\in\Sigma$ also with $y_{0}\neq
0$. Obtaining mean value formulas at points $(x_{0},y_{0})\in\Sigma$ with
$x_{0}\neq 0$ is, instead, difficult because our knowledge of the fundamental
solution of $\mathscr{L}$ with pole at $(x_{0},y_{0})$ with $x_{0}\neq 0$ is
not explicit enough.

Our interest in sub-mean value formulas on hypersurfaces of $\mathbb{R}^{n+1}$
endowed with a system of Hörmander vector fields comes from the theory of
minimal surfaces. One of the key tools in Bombieri-De Giorgi-Miranda’s proof
of the gradient estimate is the sub-mean value property for sub-harmonic
functions on minimal surfaces of the Euclidean space, see [2]. In our setting,
a minimal surface is defined by $H_{\Sigma}=0$. This condition simplifies the
structural function $q_{\Sigma}$, however, this is not sufficient to have
$q_{\Sigma}\geq 0$.

The paper is organized as follows. In Section 2, we recall the basic
definitions of the measure $\sigma$, of the $\alpha$-normal $\nu$ of $\Sigma$,
and of mean curvature $H_{\Sigma}$. In Section 3, we introduce the various
differential operators and we develop a calculus on radial functions. The
explicit computations of second order derivatives of $\varrho$ is crucial,
here. In Section 4, we prove Theorems 1.1 and 1.2. Finally, in Section 5 we
study the structural function $q_{\Sigma}$.

##  2\. Perimeter and mean curvature of hypersurfaces

The _$\alpha$ -perimeter_ of a Lebesgue measurable set
$E\subset\mathbb{R}^{n+1}$ in an open set $A\subset\mathbb{R}^{n+1}$ is

$P_{\alpha}(E;A)=\sup\left\\{\int_{E}\sum_{i=1}^{n+1}X_{i}\varphi_{i}(\xi)\,d\xi\,:\,\varphi\in
C_{c}^{1}(A;\mathbb{R}^{n+1}),\,\displaystyle\max_{\xi\in A}|\varphi(\xi)|\leq
1\right\\}.$ (2.1)

We are using the Lebesgue measure $d\xi=d\mathscr{L}^{n+1}$ in
$\mathbb{R}^{n+1}$. When the boundary of $E$ is locally the graph of a
Lipschitz function, its $\alpha$-perimeter has the following integral
representation (see [6, Proposition 2.1])

$P_{\alpha}(E;A)=\int_{\partial E\cap
A}\sqrt{|\bar{N}|^{2}+|x|^{2\alpha}|N^{\prime}|^{2}}\,d\mathscr{H}^{n},$ (2.2)

where $N(\xi)=(\bar{N}(\xi),N^{\prime}(\xi))\in\mathbb{R}^{n}\times\mathbb{R}$
is the Euclidean outer unit normal to $\partial E$ at the point $\xi$, and
$\mathscr{H}^{n}$ is the standard $n$-dimensional Hausdorff measure in
$\mathbb{R}^{n+1}$. On top of its appearance as a sub-Riemannian perimeter,
the relevance of the perimeter $P_{\alpha}$ is due to its relation with the
Heisenberg perimeter. When $\alpha=1$ and $n$ is even, then the Heisenberg
perimeter of a set with cylindrical symmetry coincides with its
$\alpha$-perimeter, see e.g., [6, Proposition 2.3].

Motivated by (2.2), when $\Sigma\subset\mathbb{R}^{n+1}$ is an orientable
hypersurface that is locally a Lipschitz graph and $N=(\bar{N},N^{\prime})$ is
its Euclidean normal, we call the Borel measure on $\Sigma$

$\sigma=\sqrt{|\bar{N}|^{2}+|x|^{2\alpha}|N^{\prime}|^{2}}\,\mathscr{H}^{n}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Sigma.$ (2.3)

the $\alpha$-perimeter measure of $\Sigma$.

The regular part of $\Sigma$ is the set
$\Sigma^{*}=\\{\xi=(x,y)\in\Sigma:x\neq 0\\}$. At $\mathscr{H}^{n}$-a.e. point
$\xi\in\Sigma^{*}$ we can define the $\alpha$-_normal_ of $\Sigma$ as the
vector field $\nu=\sum_{i=1}^{n+1}\nu_{i}X_{i}$ with

$\begin{split}&\nu_{i}=\frac{N_{i}}{\sqrt{|\bar{N}|^{2}+|x|^{2\alpha}|N^{\prime}|^{2}}},\quad\text{for
}i=1,\dots,n,\\\
&\nu_{n+1}=\frac{aN_{n+1}}{\sqrt{|\bar{N}|^{2}+|x|^{2\alpha}|N^{\prime}|^{2}}}.\end{split}$
(2.4)

With abuse of notation, we identify $\nu$ with the mapping
$\nu:\Sigma^{*}\to\mathbb{R}^{n+1}$ given by the vector of its coordinates
$\nu(\xi)=(\nu_{1}(\xi),\ldots,\nu_{n+1}(\xi))\in\mathbb{R}^{n+1}$ for
$\xi\in\Sigma^{*}$.

######  Remark 2.1 (Comparison with the Riemannian structure).

The hypersurface measure $\sigma$ and the $\alpha$-normal $\nu$ can be
interpreted in the following Riemannian terms. The tensor metric in
$\mathbb{R}^{n+1}\setminus\\{x=0\\}$ making $X_{1},\dots,X_{n+1}$ orthonormal
is

$g_{\alpha}(\xi)=\begin{pmatrix}I_{n}&0\\\ 0&|x|^{-2\alpha}\end{pmatrix},\quad
x\neq 0.$ (2.5)

When $x=0$ the metric is not defined. The Riemannian volume associated with
$g_{\alpha}$ is the measure $\mu=|x|^{-\alpha}\mathscr{L}^{n+1}$ and is
singular at $x=0$. The Riemannian surface area associated with $g_{\alpha}$ of
a hypersurface $\Sigma$ is the measure

$\mu_{\Sigma}=|x|^{-\alpha}\sqrt{|\bar{N}|^{2}+|x|^{2\alpha}|N^{\prime}|^{2}}\mathscr{H}^{n}\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Sigma,$

where $N=(\bar{N},N^{\prime})$ is the Euclidean unit normal. We deduce that
Lebesgue measure and $\alpha$-perimeter are weighted Riemannian volume and
hypersurface measures with the same weight:

$\mathscr{L}^{n+1}=|x|^{\alpha}\mu\quad\textrm{and}\quad\sigma=|x|^{\alpha}\mu_{\Sigma}.$

We now focus on the case of graphs. Let
$\Sigma=\Sigma_{u}=\\{\xi=(x,u(x))\in\mathbb{R}^{n+1}:x\in\Omega\\}$ be the
$y$-graph of a function $u\in C^{1}(\Omega)$ for some open set
$\Omega\subset\mathbb{R}^{n}$. We shall assume that $0\in\Omega$ and let
$\Omega^{*}=\Omega\setminus\\{0\\}$. The $\alpha$-_unit normal_ to
$\Sigma_{u}$ at points in $\Sigma_{u}^{*}$ is the mapping
$\nu=(\bar{\nu},\nu_{n+1}):\Sigma^{*}\to\mathbb{R}^{n+1}$

$\bar{\nu}=\frac{-\nabla u}{\sqrt{|\nabla
u|^{2}+|x|^{2\alpha}}},\quad\nu_{n+1}=\frac{|x|^{\alpha}}{{\sqrt{|\nabla
u|^{2}+|x|^{2\alpha}}}}.$ (2.6)

This normal is pointing upwards. Notice that $\nu=\nu(\xi)$ only depends on
$x$ and not on $y=u(x)$.

From (2.3) and from the area-formula, we deduce that the $\sigma$-area of
$\Sigma$ has the integral representation

$\sigma(\Sigma_{u})=\int_{\Omega}\sqrt{|\nabla
u|^{2}+|x|^{2\alpha}}\;dx=\int_{\Omega}v(x)\;dx,$ (2.7)

where $v$ is the $\sigma$-area element

${v(x)=\sqrt{|\nabla u|^{2}+a^{2}},\qquad a=|x|^{\alpha}.}$ (2.8)

If $\Sigma_{u}$ minimizes the $\sigma$-area with respect to compact
perturbations in $\Omega$ and $u\in C^{2}(\Omega)$, then $u$ satisfies the
partial differential equation of the minimal surface-type

$\operatorname{div}\left(\frac{\nabla u(x)}{\sqrt{|\nabla
u(x)|^{2}+|x|^{2\alpha}}}\right)=0,\quad x\in\Omega^{*}.$ (2.9)

This follows by a standard first variation procedure applied to (2.7). This
suggests the following definition.

######  Definition 2.2 ($\alpha$-mean curvature).

Let $\Sigma$ be the $y$-graph of a function $u\in C^{2}(\Omega)$. We define
the _$\alpha$ -mean curvature_ of $\Sigma$ at the point
$\xi=(x,u(x))\in\Sigma^{*}$ as

$H_{\Sigma}(\xi)=\frac{1}{n}\operatorname{div}\left(\frac{\nabla
u(x)}{\sqrt{|\nabla u(x)|^{2}+|x|^{2\alpha}}}\right).$ (2.10)

We say that $\Sigma$ is an $\alpha$-minimal hypersurface if $H_{\Sigma}=0$ on
$\Sigma^{*}$.

A more geometric definition of $\alpha$-mean curvature will be presented in
the next section.

##  3\. Tangential operators and Laplacians

We introduce tangential differential operators on hypersurfaces in
$\mathbb{R}^{n+1}$ endowed with the Grushin structure. Let
$\Sigma\subset\mathbb{R}^{n+1}$ be an embedded hypersurface of class $C^{2}$
with $\alpha$-normal $\nu:\Sigma^{*}\to\mathbb{R}^{n+1}$.

The $X$-gradient of a function $\varphi\in C^{1}(\mathbb{R}^{n+1})$ is the
vector-field $X\varphi=\sum_{i=1}^{n+1}X_{i}\varphi\,X_{i}$ that we identify,
with abuse of notation, with the vector of its coordinates
$X\varphi=(X_{1}\varphi,\ldots,X_{n+1}\varphi)$. We denote the standard scalar
product on $\mathbb{R}^{n+1}$ by $\langle\cdot,\cdot\rangle$.

######  Definition 3.1 (Tangential gradient).

Let $\nu:\Sigma^{*}\to\mathbb{R}^{n+1}$ be the $\alpha$-normal of $\Sigma$.
The tangential gradient on $\Sigma$ is the mapping $\delta:C^{1}(\Sigma)\to
C(\Sigma^{*};\mathbb{R}^{n+1})$

$\delta\varphi=X\varphi-\langle X\varphi,\nu\rangle\nu.$ (3.1)

For any $i=1,\ldots,n+1$ we also let $\delta_{i}\varphi=X_{i}\varphi-\langle
X\varphi,\nu\rangle\nu_{i}$.

In (3.1), $\varphi$ is extended outside $\Sigma$ in a $C^{1}$ way, and the
definition will be independent of this extension. We are assuming that
$\Sigma$ is oriented and we are fixing a choice of $\alpha$-normal. The
definition does not depend on this choice. When $\Sigma$ is a $y$-graph, we
agree that $\nu$ is pointing upwards. In this case, the $\alpha$-normal $\nu$
can be extended outside $\Sigma$ in a way that is independent of the variable
$y$. In the rest of the paper the surface $\Sigma$ will be always assumed to
be a $y$-graph.

The definition of the tangential operator $\delta$ in (3.1) is extrinsic. A
different possibility could be to define the tangential gradient of functions
on $\Sigma$ using the Riemannian metric $g_{\alpha}$. However, the choice in
(3.1) is the correct one in order to recover the definition in (2.10) of
$\alpha$-mean curvature. Indeed, this definition reads

$H_{\Sigma}=-\frac{1}{n}\sum_{i=1}^{n}X_{i}\nu_{i},$ (3.2)

and it can be rephrased in the following way.

######  Lemma 3.2.

Let $\Sigma\subset\mathbb{R}^{n+1}$ be a $y$-graph of class $C^{2}$. Then on
$\Sigma^{*}$ we have the identity

$H_{\Sigma}=-\frac{1}{n}\sum_{i=1}^{n+1}\delta_{i}\nu_{i}.$

###### Proof.

We use the definition of $\delta$ and observe that
$\sum_{i=1}^{n+1}\nu_{i}X_{k}\nu_{i}=X_{k}(|\nu|^{2}/2)=0$, and
$X_{n+1}\nu_{n+1}=0$, so that

$\begin{split}\sum_{i=1}^{n+1}\delta_{i}\nu_{i}&=\sum_{i=1}^{n}\delta_{i}\nu_{i}+\delta_{n+1}\nu_{n+1}\\\
&=\sum_{i=1}^{n}X_{i}\nu_{i}-\sum_{i=1}^{n}\sum_{k=1}^{n+1}\nu_{i}\nu_{k}X_{k}\nu_{i}+X_{n+1}\nu_{n+1}-\nu_{n+1}\nu_{k}X_{k}\nu_{n+1}\\\
&=\sum_{i=1}^{n}X_{i}\nu_{i}-\sum_{k=1}^{n+1}\left(\nu_{k}\sum_{i=1}^{n+1}\nu_{i}X_{k}\nu_{i}\right)+X_{n+1}\nu_{n+1}\\\
&=\sum_{i=1}^{n}X_{i}\nu_{i}=-nH_{\Sigma}.\qed\end{split}$

Next we introduce the adjoint operators $\delta_{i}^{*}$, integrating by parts
with respect to the measure $\sigma$.

######  Definition 3.3 (Adjoint tangential operators).

For each $i=1,\dots,n+1$, we define the _adjoint tangential operator_
$\delta_{i}^{*}:C^{1}(\Sigma)\to C(\Sigma^{*})$ through the identity

$\int_{\Sigma}\psi\,\delta_{i}\varphi\,d\sigma=-\int_{\Sigma}\varphi\,\delta_{i}^{*}\psi\,d\sigma,\quad\varphi,\psi\in
C^{1}_{c}(\Sigma).$ (3.3)

The explicit formula for adjoint operators is given in the next lemma.

######  Lemma 3.4.

Let $\Sigma\subset\mathbb{R}^{n+1}$ be a hypersurface with $\alpha$-mean
curvature $H_{\Sigma}$. For every $\psi\in C^{1}_{c}(\Sigma)$ and
$i=1,\ldots,n+1$, we have on $\Sigma^{*}$

$\delta_{i}^{*}\psi=-\delta_{i}\psi-\psi\Big{[}\delta_{i}\log|x|^{\alpha}+\frac{1}{\nu_{n+1}}\Big{(}\delta_{n+1}\nu_{i}-\delta_{i}\nu_{n+1}\Big{)}+nH_{\Sigma}\nu_{i}\Big{]}.$
(3.4)

###### Proof.

Let $\varphi,\psi\in C^{1}_{c}(\Sigma^{*})$. Then by the area formula (2.7)
with $v$ as in (2.8)

$\begin{split}\int_{\Sigma}\varphi\,\delta_{i}^{*}\psi\,d\sigma&=\int_{\Sigma}\psi\,\delta_{i}\varphi\,d\sigma=\int_{\Omega}\psi(X_{i}\varphi-\langle
X\varphi,\nu\rangle\nu_{i})vdx\\\
&=-\int_{\Omega}\varphi\Big{[}X_{i}(v\psi)-\sum_{k=1}^{n}X_{k}(\nu_{i}\nu_{k}v\psi)\Big{]}dx\\\
&=-\int_{\Sigma}\varphi\frac{A}{v}d\sigma,\end{split}$ (3.5)

where in the last identity we set
$A=X_{i}(v\psi)-\sum_{k=1}^{n}X_{k}(\nu_{i}\nu_{k}v\psi)$. We are left to
prove that $A/v$ is equal to the right-hand side in (3.4).

We have

$\begin{split}-\frac{A}{v}&=-\frac{1}{v}\left[\psi
X_{i}v+vX_{i}\psi-v\nu_{i}\langle
X\psi,\nu\rangle-\psi\sum_{k=1}^{n}X_{k}(\nu_{i}\nu_{k}v)\right]\\\
&=-\delta_{i}\psi-\frac{\psi}{v}\left[X_{i}v-\nu_{i}\langle
Xv,\nu\rangle-v\sum_{k=1}^{n}X_{k}(\nu_{i}\nu_{k})\right].\end{split}$ (3.6)

In the term within brackets above, we easily recognize $X_{i}v-\nu_{i}\langle
Xv,\nu\rangle=\delta v$. On the other hand, using $X_{n+1}\nu=0$ and (3.2), we
get

$\sum_{k=1}^{n}X_{k}(\nu_{i}\nu_{k})=-n\nu_{i}H_{\Sigma}-\frac{1}{\nu_{n+1}}\Big{[}X_{n+1}\nu_{i}-\nu_{n+1}\langle
X\nu_{i},\nu\rangle\Big{]}=-n\nu_{i}H_{\Sigma}-\frac{1}{\nu_{n+1}}\delta_{n+1}\nu_{i}.$
(3.7)

Summarizing, we have

$\frac{A}{v}=\delta_{i}\psi+\psi\left[\frac{\delta_{i}v}{v}+\frac{\delta_{n+1}\nu_{i}}{\nu_{n+1}}+nH_{\Sigma}\nu_{i}\right].$
(3.8)

To prove our claim it then remains to check the following identity

$\frac{\delta_{i}v}{v}=\frac{\delta_{i}a}{a}-\frac{\delta_{i}\nu_{n+1}}{\nu_{n+1}}=\delta_{i}(\log
a)-\frac{\delta_{i}\nu_{n+1}}{\nu_{n+1}}.$ (3.9)

Indeed, since $\nu_{n+1}=a/v$ we have

$\frac{\delta_{i}a}{a}-\frac{\delta_{i}\nu_{n+1}}{\nu_{n+1}}=\frac{\delta_{i}a}{a}-\frac{v\delta_{i}(\frac{a}{v})}{a}=\frac{\delta_{i}a}{a}-\frac{v}{a}\left(\frac{\delta_{i}a}{v}-\frac{a}{v^{2}}\delta_{i}v\right)=\frac{\delta_{i}v}{v}.$
(3.10)

∎

######  Definition 3.5 (Tangential Laplacians).

Let $\Sigma\subset\mathbb{R}^{n+1}$ be a hypersurface of class $C^{2}$. The
_tangential Laplacian_ of $\Sigma$ is the operator
$\mathscr{L}_{\Sigma}:C^{2}(\Sigma)\to C(\Sigma^{*})$

$\mathscr{L}_{\Sigma}\varphi=-\sum_{i=1}^{n+1}\delta_{i}^{*}\delta_{i}\varphi.$
(3.11)

The relation between $\mathscr{L}_{\Sigma}$ and the non-adjoint Laplacian

$\Delta_{\Sigma}\varphi=\sum_{i=1}^{n+1}\delta_{i}^{2}\varphi.$ (3.12)

is described in the following proposition.

######  Lemma 3.6.

For any $\varphi\in C^{2}(\Sigma)$ we have the identity

$\mathscr{L}_{\Sigma}\varphi=\Delta_{\Sigma}\varphi+\nu_{n+1}^{2}\langle\delta\varphi,\delta\log
a\rangle+\delta_{n+1}\varphi\,\delta_{n+1}\log a$ (3.13)

###### Proof.

We have

$\mathscr{L}_{\Sigma}\varphi=-\sum_{i=1}^{n+1}\delta_{i}^{*}\delta_{i}=\Delta_{\Sigma}\varphi+\langle\delta\varphi,\delta\log
a\rangle-\frac{1}{\nu_{n+1}}\langle\delta\varphi,\delta\nu_{n+1}-\delta_{n+1}\nu\rangle.$
(3.14)

By formula (3.4), for $i=1,\dots,n+1$ we have

$\delta\nu_{n+1}-\delta_{n+1}\nu=\nu_{n+1}\big{[}\delta\log
a-\big{(}\bar{0},\delta_{n+1}\log
a\big{)}\big{]}+\nu_{n+1}^{2}\big{[}\nu\delta_{n+1}\log a-\nu_{n+1}\delta\log
a\big{]}.$ (3.15)

Since $\langle\delta(\cdot),\nu\rangle=0$, we deduce

$\begin{split}-\frac{1}{\nu_{n+1}}\langle\delta\varphi,\delta\nu_{n+1}-\delta_{n+1}\nu\rangle=&-\langle\delta\varphi,\delta\log
a\rangle+\delta_{n+1}\varphi\delta_{n+1}\log a\\\
&+\nu_{n+1}^{2}\langle\delta\varphi,\delta\log a\rangle,\end{split}$ (3.16)

proving the result. ∎

The formal Hessian of $\varphi$ with respect to the vector-fields
$X_{1},\ldots,X_{n+1}$ is the $(n+1)\times(n+1)$ matrix

$X^{2}\varphi=\big{(}X_{i}X_{j}\varphi)_{i,j=1,\ldots,n+1}.$

The non-adjoint Laplacian $\Delta_{\Sigma}$ has a clear representation in
terms of the Grushin operator $\mathscr{L}$ in (1.2), $X^{2}$ and
$\alpha$-mean curvature of $\Sigma$.

######  Lemma 3.7.

Let $\Sigma\subset\mathbb{R}^{n+1}$ be a hypersurface of class $C^{2}$ with
$\alpha$-mean curvature $H_{\Sigma}$. For any $\varphi\in
C^{2}(\mathbb{R}^{n+1})$ we have the identity on $\Sigma^{*}$

$\Delta_{\Sigma}\varphi=\mathscr{L}\varphi-\langle(X^{2}\varphi)\nu,\nu\rangle+nH_{\Sigma}\langle
X\varphi,\nu\rangle.$ (3.17)

###### Proof.

We first compute $\delta_{i}^{2}\varphi$, for $i\geq 1$. We have

$\begin{split}\delta_{i}^{2}\varphi&=\delta_{i}(X_{i}\varphi-\langle
X\varphi,\nu\rangle\nu_{i})\\\ &=X_{i}X_{i}\varphi-\langle
XX_{i}\varphi,\nu\rangle\nu_{i}-\langle
X\varphi,\nu\rangle\delta_{i}\nu_{i}-\delta_{i}(\langle
X\varphi,\nu\rangle)\nu_{i}.\end{split}$ (3.18)

Summing over $i$, we obtain the identities

$\displaystyle\sum_{i=1}^{n+1}\langle
XX_{i}\varphi,\nu\rangle\nu_{i}=\langle(X^{2}\varphi)\nu,\nu\rangle,\quad\sum_{i=1}^{n+1}\delta_{i}\nu_{i}=-nH_{\Sigma},$
(3.19) $\displaystyle\sum_{i=1}^{n+1}\delta_{i}(\langle
X\varphi,\nu\rangle)\nu_{i}=\langle\delta(\langle
X\varphi,\nu\rangle),\nu\rangle=0,$ (3.20)

and this completes the proof. ∎

We specialize the previous formulas to the case when $\varphi$ is a radial
function around $0\in\mathbb{R}^{n+1}$. The symmetry is governed by the gauge
function $\varrho$ in (1.3). Below, we collect the differential identities
concerning first and second order derivatives of $\varrho$. With the notation
$\xi=(x,y)$ and $\varrho=\varrho(\xi)$ we have

$\displaystyle\nabla_{x}\varrho=x|x|^{2\alpha}\varrho^{-(2\alpha+1)}\quad\textrm{and}\quad\partial_{y}\varrho=(\alpha+1)y\varrho^{-(2\alpha+1)}.$
(3.21)

Then the squared norm of the $X$-gradient of $\varrho$ is

$|X\varrho|^{2}=|\nabla_{x}\varrho|^{2}+|x|^{2\alpha}|\partial_{y}\varrho|^{2}=|x|^{2\alpha}\varrho^{-2\alpha}.$
(3.22)

The second derivatives of $\varrho$ are, with $i,j=1,\dots,n$ and denoting by
$\varepsilon_{ij}$ the Kronecker symbol,

$\begin{split}X_{i}X_{j}\varrho&=|x|^{2\alpha}\Big{[}\varepsilon_{ij}+2\alpha\frac{x_{i}x_{j}}{|x|^{2}}-(2\alpha+1)\frac{x_{i}x_{j}|x|^{2\alpha}}{\varrho^{2(\alpha+1)}}\Big{]},\\\
X_{i}X_{n+1}\varrho&=(\alpha+1)|x|^{\texttrademark
2\alpha}x_{i}y\varrho^{-2\alpha-1}\Big{[}\frac{\alpha}{|x|^{\alpha+2}}-(2\alpha+1)\frac{|x|^{2\alpha}}{\varrho^{2(\alpha+1)}}\Big{]},\\\
X_{n+1}X_{j}\varrho&=-(2\alpha+1)(\alpha+1)|x|^{3\alpha}x_{j}y\varrho^{-4\alpha-3},\\\
X_{n+1}^{2}\varrho&=(\alpha+1)|x|^{2\alpha}\varrho(x)^{-2\alpha-1}\Big{[}1-(2\alpha+1)(\alpha+1)\frac{y^{2}}{\varrho^{2(\alpha+1)}}\Big{]}.\end{split}$
(3.23)

From (3.23), we get the following formulas for the Laplacian $\mathscr{L}$ and
for the quadratic form $\langle(X^{2}\varrho)\nu,\nu\rangle$:

$\mathscr{L}\varrho=(n+\alpha)\frac{|X\varrho|^{2}}{\varrho},$ (3.24)

and

$\begin{split}\langle(X^{2}\varrho)\nu,\nu\rangle=|\bar{\nu}|^{2}&+2\alpha\frac{\langle
x,\bar{\nu}\rangle^{2}}{|x|^{2}}-(2\alpha+1)\frac{|x|^{2\alpha}\langle
x,\bar{\nu}\rangle^{2}}{\varrho^{2(\alpha+1)}}\\\ &+(\alpha+1)\langle
x,\bar{\nu}\rangle^{2}\nu_{n+1}y\left(\frac{\alpha}{|x|^{\alpha+2}}-2(2\alpha+1)\frac{|x|^{2\alpha}\langle
x,\bar{\nu}\rangle^{2}}{\varrho^{2(\alpha+1)}}\right)\\\
&+(\alpha+1)\nu_{n+1}^{2}\left(1-(\alpha+1)(2\alpha+1)\frac{y^{2}}{\varrho^{2(\alpha+1)}}\right).\end{split}$
(3.25)

Let $\Sigma\subset\mathbb{R}^{n+1}$ be a hypersurface with $\alpha$-normal
$\nu$ and $\alpha$-mean curvature $H_{\Sigma}$. The structural function
$q_{\Sigma}:\Sigma^{*}\to\mathbb{R}$ introduced in (1.5) governs the
harmonicity of $\Sigma$ at $0\in\Sigma$ and appears in the following formula.

######  Theorem 3.8 (Tangential Laplacian of radial functions).

For any $\varphi\in C^{2}(\mathbb{R}^{+})$, the function $\psi\in
C^{2}(\mathbb{R}^{n+1}\setminus\\{0\\})$, $\psi=\varphi\circ\varrho$,
satisfies the identity

$\mathscr{L}_{\Sigma}\psi=\Big{\\{}\varphi^{\prime\prime}(\varrho)+\varphi^{\prime}(\varrho)\frac{n+\alpha-1}{\varrho}\Big{\\}}|\delta\varrho|^{2}+q_{\Sigma}\,\varphi^{\prime}(\varrho).$
(3.26)

###### Proof.

In a first step, we prove that formula (3.26) holds with the following
expression for $q_{\Sigma}$:

$\begin{split}q_{\Sigma}=-\frac{n+\alpha-1}{\varrho}|\delta\varrho|^{2}&+(n+\alpha)\frac{|X\varrho|^{2}}{\varrho}-\langle(X^{2}\varrho)\nu,\nu\rangle+nH_{\Sigma}\langle
X\varrho,\nu\rangle\\\
&+\alpha\nu_{n+1}^{2}\langle\delta\varrho,\delta\log|x|\rangle+\alpha\delta_{n+1}\varrho\delta_{n+1}\log|x|.\end{split}$
(3.27)

The proof combines formula (3.13) of Lemma 3.6 and formula (3.17) in Lemma
3.7:

$\displaystyle\Delta_{\Sigma}\psi$
$\displaystyle=\mathscr{L}(\psi\circ\varrho)-\langle
X^{2}(\psi\circ\varrho)\nu,\nu\rangle+nH_{\Sigma}\langle
X(\psi\circ\varrho),\nu\rangle$ (3.28)
$\displaystyle=\varphi^{\prime\prime}(\varrho)|\delta\varrho|^{2}+\varphi^{\prime}(\varrho)\big{(}\mathscr{L}\varrho-\langle(X^{2}\varrho)\nu,\nu\rangle+nH_{\Sigma}\langle
X\varrho,\nu\rangle\big{)},$ (3.29)

where the last identity is a simple computation with the chain rule. On the
other hand, we have

$\mathscr{L}_{\Sigma}\psi=\Delta_{\Sigma}\psi+\alpha\nu_{n+1}^{2}\langle\delta\psi,\delta(\log|x|)\rangle+\alpha\delta_{n+1}\psi\delta_{n+1}(\log|x|).$
(3.30)

Since $\delta\psi=\varphi^{\prime}(\varrho)\delta\varrho$, we deduce

$\displaystyle\mathscr{L}_{\Sigma}\psi=\varphi^{\prime\prime}(\varrho)|\delta\varrho|^{2}+\varphi^{\prime}(\varrho)\big{[}$
$\displaystyle\mathscr{L}\varrho-\langle(X^{2}\varrho)\nu,\nu\rangle+nH_{\Sigma}\langle
X\varrho,\nu\rangle$ (3.31)
$\displaystyle+\alpha\nu_{n+1}^{2}\langle\delta\varrho,\delta(\log|x|)\rangle+\alpha\delta_{n+1}\varrho\delta_{n+1}(\log|x|)\big{]}$
(3.32)

The proof of (3.26) with $q_{\Sigma}$ as in (3.27) is then concluded by adding
and subtracting the quantity $\frac{n+\alpha-1}{\varrho}|\delta\varrho|^{2}$
within squared brackets and using (3.24).

In the next step, we check that $q_{\Sigma}$ in (3.27) is as in (1.5). We
start by observing that an elementary computation gives

$\nu_{n+1}^{2}\langle\delta\varrho,\delta\log|x|\rangle+\delta_{n+1}\varrho\delta_{n+1}\log|x|=\frac{|x|^{2\alpha}}{\varrho^{2\alpha+1}}\Big{(}\nu_{n+1}^{2}-(\alpha+1)\nu_{n+1}|x|^{-2-\alpha}\langle
x,\bar{\nu}\rangle y\Big{)},$ (3.33)

and

$-\frac{n+\alpha-1}{\varrho}|\delta\varrho|^{2}+(n+\alpha)\frac{|X\varrho|^{2}}{\varrho}=\frac{|X\varrho|^{2}}{\varrho}+\frac{n+\alpha-1}{\varrho}\langle
X\varrho,\nu\rangle^{2}.$ (3.34)

Inserting formulas (3.23)–(3.25) into (3.27), we obtain

$\begin{split}q_{\Sigma}=&\frac{n+\alpha-1}{\varrho}\langle
X\varrho,\nu\rangle^{2}+nH_{\Sigma}\langle X\varrho,\nu\rangle\\\
&-\frac{|X\varrho|^{2}}{\varrho}\Big{[}2\alpha\frac{\langle
x,\bar{\nu}\rangle^{2}}{|x|^{2}}-(2\alpha+1)\frac{|x|^{2\alpha}}{\varrho^{2\alpha+2}}\langle
x,\bar{\nu}\rangle^{2}+2\alpha(\alpha+1)\langle x,\bar{\nu}\rangle
y|x|^{-2-\alpha}\nu_{n+1}\\\
&\qquad\qquad\;\;-2(\alpha+1)(2\alpha+1)\frac{|x|^{\alpha}}{\varrho^{2\alpha+2}}\langle
x,\bar{\nu}\rangle y\nu_{n+1}\\\
&\qquad\qquad\;\;-(\alpha+1)^{2}(2\alpha+1)y^{2}\nu_{n+1}^{2}\frac{1}{\varrho^{2\alpha+2}}\Big{]}.\end{split}$
(3.35)

Now we observe that

$\langle
X\varrho,\nu\rangle^{2}=\frac{|x|^{2\alpha}}{\varrho^{4\alpha+2}}\left(|x|^{\alpha}\langle
x,\bar{\nu}\rangle+(\alpha+1)y\nu_{n+1}\right)^{2}$ (3.36)

and so, since $|X\varrho|=(|x|/\varrho)^{\alpha}$,

$\langle
X\varrho,\nu\rangle=\frac{|X\varrho|}{\varrho^{\alpha+1}}\left(|x|^{\alpha}\langle
x,\bar{\nu}\rangle+(\alpha+1)y\nu_{n+1}\right).$ (3.37)

Replacing last identity into (3.35) and after some computations that are
omitted, we obtain (1.5). ∎

##  4\. Mean value formulas

We are ready to prove Theorems 1.1 and 1.2. Let
$\Sigma\subset\mathbb{R}^{n+1}$ be a hypersurface of class $C^{2}$ with
$0\in\Sigma$. We say that:

  * i)

$\Sigma$ is $\alpha$-harmonic at $0$ if $q_{\Sigma}=0$;

  * ii)

$\Sigma$ is $\alpha$-subharmonic at $0$ if $q_{\Sigma}\geq 0$;

  * iii)

$\Sigma$ is $\alpha$-superharmonic at $0$ if $q_{\Sigma}\leq 0$.

###### Proof of Theorem 1.1.

For any $\psi\in C^{\infty}_{c}(\Sigma)$, by the integration by parts formula
(3.3) we have

$0=\int_{\Sigma}\mathscr{L}_{\Sigma}f\,\psi\,d\sigma=-\int_{\Sigma}\langle\delta
f\,\delta\psi\rangle
d\sigma=\int_{\Sigma}f\,\mathscr{L}_{\Sigma}\psi\,d\sigma.$ (4.1)

We shall use this identity for functions $\psi$ with radial structure around
$0$. Let $\chi\in C^{\infty}(\mathbb{R}^{+})$ be a function such that
$\chi(r)=1$ for $0<r<1/2$ and $\chi(r)=0$ for $r>1$. We may also assume that
$\chi^{\prime}\leq 0$. With the notation $m(r)=r^{n+\alpha}$, for $0<s<r$ we
define the function $\vartheta_{s}\in C^{\infty}(\mathbb{R}^{+})$

$\vartheta_{s}(\varrho)=\frac{\partial}{\partial
s}\left(\frac{1}{m(s)}\chi\Big{(}\frac{m(\varrho)}{m(s)}\Big{)}\right),\quad\varrho>0.$
(4.2)

Assume there exists a solution $\varphi_{s}\in C^{\infty}(\mathbb{R}^{+})$ to
the differential problem

$\left\\{\begin{array}[]{l}\varphi_{s}^{\prime\prime}(\varrho)+\frac{m^{\prime\prime}(\varrho)}{m^{\prime}(\varrho)}\varphi_{s}^{\prime}(\varrho)=\vartheta_{s}(\varrho),\quad\varrho>0\\\
\varphi_{s}(\varrho)=0,\quad\varrho>s.\end{array}\right.$ (4.3)

Then we may consider the function $\psi_{s}(\xi)=\varphi_{s}(\varrho(\xi))$
for $\xi\in\Sigma$. By formula (3.26) with $q_{\Sigma}=0$ and (4.3) we have

$\mathscr{L}_{\Sigma}\psi_{s}=\Big{\\{}\varphi_{s}^{\prime\prime}(\varrho)+\varphi_{s}^{\prime}(\varrho)\frac{n+\alpha-1}{\varrho}\Big{\\}}|\delta\varrho|^{2}=\frac{\partial}{\partial
s}\left(\frac{1}{m(s)}\chi\Big{(}\frac{m(\varrho)}{m(s)}\Big{)}\right)|\delta\varrho|^{2},$

and from (4.1) we deduce that

$0=\int_{\Sigma}f\,\mathscr{L}_{\Sigma}\psi_{s}\,d\sigma=\frac{\partial}{\partial
s}\int_{\Sigma}f(\xi)\,\frac{1}{m(s)}\chi\Big{(}\frac{m(\varrho)}{m(s)}\Big{)}|\delta\varrho|^{2}\,d\sigma,$

and thus for any $0<s<r$

$\frac{1}{m(s)}\int_{\Sigma}f(\xi)\,\chi\Big{(}\frac{m(\varrho)}{m(s)}\Big{)}|\delta\varrho|^{2}\,d\sigma=\frac{1}{m(r)}\int_{\Sigma}f(\xi)\,\chi\Big{(}\frac{m(\varrho)}{m(r)}\Big{)}|\delta\varrho|^{2}\,d\sigma.$

We may approximate the characteristic function of the interval
$(0,1)\subset\mathbb{R}$ by a sequence of functions $\chi$ as above. Passing
to the limit in the previous identity, we get

$\frac{1}{m(s)}\int_{B_{s}\cap\Sigma}f(\xi)\,|\delta\varrho|^{2}\,d\sigma=\frac{1}{m(r)}\int_{B_{r}\cap\Sigma}f(\xi)\,|\delta\varrho|^{2}\,d\sigma.$

This formula holds for $f=1$, proving that the right hand-side of (1.7) does
not depend on $r>0$. By continuity of $f$ at $0$, we get (1.6) with
$C_{\Sigma,n,\alpha}$ as in (1.7).

We are left to show that problem (4.3) has a solution. A straightforward
computation shows that the function $\vartheta_{s}$ in (4.2) reads

$\begin{split}\vartheta_{s}(\varrho)&=-\frac{m^{\prime}(s)}{m(s)^{2}}\frac{\partial}{\partial\varrho}\left(m(\varrho)\chi\Big{(}\frac{m(\varrho)}{m(s)}\Big{)}\right),\end{split}$
(4.4)

and so the differential equation in (4.3) is equivalent to

$\frac{\partial}{\partial\varrho}\big{(}m^{\prime}(\varrho)\varphi_{s}^{\prime}(\varrho)\big{)}=-\frac{m^{\prime}(s)}{m(s)^{2}}\frac{\partial}{\partial\varrho}\left(m(\varrho)\chi\Big{(}\frac{m(\varrho)}{m(s)}\Big{)}\right).$
(4.5)

Integrating with $\varphi^{\prime}_{s}(\varrho)=0$ for $\varrho>s$ we obtain:

$\varphi_{s}^{\prime}(\varrho)=-\frac{m^{\prime}(s)m(\varrho)}{m(s)^{2}m^{\prime}(\varrho)}\chi\Big{(}\frac{m(\varrho)}{m(s)}\Big{)}=-\frac{\varrho}{s^{n+\alpha+1}}\chi\Big{(}\frac{\varrho^{n+\alpha}}{s^{n+\alpha}}\Big{)}.$
(4.6)

A final integration with $\varphi_{s}(\varrho)=0$ for $\varrho>s$ yields

$\varphi_{s}(\varrho)=\int_{\varrho}^{\infty}\frac{r}{s^{n+\alpha+1}}\chi\Big{(}\frac{r^{n+\alpha}}{s^{n+\alpha}}\Big{)}dr,$
(4.7)

showing that we find a function satisfying as a matter of fact
$\varphi_{s}(\varrho)=0$ for $\varrho>s$.

∎

######  Remark 4.1.

Using the technique of Theorem 1.1, with $m(r)=r^{n+\alpha}$ replaced by
$m(r)=r^{n+\alpha+1}$, one obtains a mean value formula for
$\mathscr{L}$-harmonic functions at $0\in\mathbb{R}^{n+1}$, where
$|\delta\varrho|$ in (1.1) is replaced by $|X\varrho|$, and $\mathscr{L}$ is
the Grushin Laplacian (1.2). The same technique works when
$\mathscr{L}=\sum_{i=1}^{m}X_{j}^{2}$ is the sub-Laplacian of any family
$X_{1},\dots,X_{m}$ of smooth vector fields in $\mathbb{R}^{n+1}$ satisfying
the Hörmander condition, with $2\leq m\leq n+1$ and admitting a global
fundamental solution. The resulting mean-value formulas coincide with the
formulas obtained in [5].

We explain the relation between the two approaches in the case of a Carnot
group of topological dimension $d>2$. Let $\Gamma$ be the fundamental solution
of the corresponding Carnot sub-Laplacian $\mathscr{L}$ with pole at $0$. For
a harmonic function $\mathscr{L}f=0$, the mean value formula (1.4) proved in
[5] reads

$f(0)=\frac{1}{r}\int_{\\{\xi\in\mathbb{R}^{d}:\Gamma(\xi)>\frac{1}{r}\\}}f(\xi)\left|X(\log\Gamma)\right|^{2}\varphi\left(\frac{1}{r\Gamma(\xi)}\right)\;d\xi,\quad
r>0,$ (4.8)

where $\varphi$ is any continuous function on the interval $[0,1]$ with unit
integral.

Let $\varrho(\xi)=\Gamma(\xi)^{\frac{1}{Q-2}}$, $\xi\neq 0$, where
$Q\in\mathbb{N}$ is the homogeneous dimension of the group. The Lebesgue
measure of the balls $B_{s}=\\{\xi\in\mathbb{R}^{d}:\varrho(\xi)<s\\}$
satisfies $m(s)=\mathscr{L}^{d}(B_{s})=Cs^{Q}$ for some constant $C>0$
depending on $n$ and $Q$. Using the technique of Theorem 1.1 we get the mean
value formula

$f(0)=\frac{C_{d,Q}}{m(s)}\int_{B_{s}}f(\xi)\,|X\varrho(\xi)|^{2}d\xi,$ (4.9)

with $C_{d,Q}>0$ fixed on choosing $f=1$. Formula (4.9) is precisely formula
(4.8) with the choice $\varphi(t)=\frac{Q}{Q-2}t^{\frac{2}{Q-2}}$. In fact, in
this case we have

$\left|X(\log\Gamma)\right|^{2}\varphi\left(\frac{1}{r\Gamma}\right)=Q(Q-2)r^{\frac{2}{2-Q}}|X\varrho|^{2},$

and we obtain equivalence with (4.9) by setting $r=s^{Q-2}$.

###### Proof of Theorem 1.2.

The proof is identical to the proof of Theorem 1.1 with minor modifications
that we list below. For any nonnegative $\psi\in C^{\infty}_{c}(\Sigma)$, we
have

$0\leq\int_{\Sigma}f\,\mathscr{L}_{\Sigma}\psi\,d\sigma.$ (4.10)

The function $\varphi_{s}$ is the solution to (4.3) defined in (4.7). Notice
that $\varphi_{s}^{\prime}\leq 0$ if $\chi\geq 0$ in (4.6). Then we have
$q_{\Sigma}(\xi)\varphi_{s}^{\prime}(\varrho(\xi))\leq 0$ for
$\xi\in\Sigma^{*}$. By formula (3.26), the function
$\psi_{s}=\varphi_{s}\circ\varrho$ thus satisfies

$\mathscr{L}_{\Sigma}\psi_{s}\leq\frac{\partial}{\partial
s}\left(\frac{1}{m(s)}\chi\Big{(}\frac{m(\varrho)}{m(s)}\Big{)}\right)|\delta\varrho|^{2},$

and we get, for $0<s<r$,

$\frac{1}{m(s)}\int_{B_{s}\cap\Sigma}f(\xi)\,|\delta\varrho|^{2}\,d\sigma\leq\frac{1}{m(r)}\int_{B_{r}\cap\Sigma}f(\xi)\,|\delta\varrho|^{2}\,d\sigma.$

The choice $f=1$ shows the existence of the limit in (1.10).

∎

######  Remark 4.2.

If $\Sigma$ is $\alpha$-superharmonic, $q_{\Sigma}\leq 0$, then a function $f$
with $\mathscr{L}_{\Sigma}f\leq 0$ satisfies the super-mean-value formula at
$0$:

$f(0)\geq\frac{C_{\Sigma,n,\alpha}}{r^{n+\alpha}}\int_{B_{r}\cap\Sigma}f(\xi)\,|\delta\varrho(\xi)|^{2}d\sigma.$
(4.11)

The proof is the same as in the sub-harmonic case.

##  5\. Analysis of the structural function $q_{\Sigma}$

###  5.1. Homogeneous hypersurfaces are harmonic

In $\mathbb{R}^{n+1}$ with the Grushin structure, we introduce the anisotropic
dilations $d_{\lambda}:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$, $\lambda>0$,

$d_{\lambda}(\xi)=d_{\lambda}(x,y)=(\lambda
x,\lambda^{\alpha+1}y),\quad\xi\in\mathbb{R}^{n+1}.$

We say that a set $\Sigma\subset\mathbb{R}^{n+1}$ is $d_{\lambda}$-homogeneous
if $d_{\lambda}(\Sigma)=\Sigma$ for any $\lambda>0$.

######  Lemma 5.1.

Let $\Sigma\subset\mathbb{R}^{n+1}$ be a hypersurface of class $C^{2}$ with
$0\in\Sigma$. If $\Sigma$ is $d_{\lambda}$-homogeneous then $\Sigma$ is
$\alpha$-harmonic at $0$.

###### Proof.

We check the claim when $\Sigma$ is a $y$-graph
$\Sigma=\\{(x,u(x))\in\mathbb{R}^{n+1}:x\in\mathbb{R}^{n}\\}$ for some
function $u\in\ C^{2}(\mathbb{R}^{n})$ satisfying the identity $u(\lambda
x)=\lambda^{\alpha+1}u(x)$ for any $\lambda>0$ and $x\in\mathbb{R}^{n}$.
Differentiating this identity at $\lambda=1$ we get

$\langle\nabla u(x),x\rangle=(\alpha+1)u(x),\quad x\in\mathbb{R}^{n}.$ (5.1)

Using formulas (2.6) for the $\alpha$-normal $\nu=(\bar{\nu},\nu_{n+1})$,
(5.1) is equivalent to

$|x|^{\alpha}\langle
x,\bar{\nu}\rangle+(\alpha+1)y\nu_{n+1}=0,\quad(x,y)\in\Sigma^{*}.$

By formula (3.37) this is in turn equivalent to $\langle
X\varrho,\nu\rangle=0$ on $\Sigma^{*}$, and this implies $q_{\Sigma}=0$, see
(1.5). ∎

When $\Sigma$ is $d_{\lambda}$-homogeneous we have $\langle
X\varrho,\nu\rangle=0$ and, using (3.22), the kernel $|\delta\varrho|^{2}$
appearing in the mean value formula (1.7) reduces to

$|\delta\varrho|^{2}=|X\varrho|^{2}=\frac{|x|^{2\alpha}}{\varrho^{2\alpha}}.$

This kernel is $0$-homogeneous with respect to the dilations $d_{\lambda}$ and
satisfies $|\delta\varrho|^{2}\leq 1$.

###  5.2. Flat case

The hyperplane $\Sigma=\\{(x,y)\in\mathbb{R}^{n+1}:y=0\\}$ is $d_{\lambda}$
homogeneous and it is therefore $\alpha$-harmonic. The $\alpha$-normal is
constant, $\nu=(0,1)\in\mathbb{R}^{n}\times\mathbb{R}$, and it follows that
$\Sigma$ is also $\alpha$-minimal, $H_{\Sigma}=0$.

The tangential gradient $\delta$ reduces to the standard gradient
$\delta\varphi=(\nabla_{x}\varphi,0)$ and the non-adjoint Laplacian
$\Delta_{\Sigma}$ in (3.12) reduces to the standard Laplacian $\Delta_{x}$ in
$\mathbb{R}^{n}$, see (3.17). From formula (3.13) we deduce that

$\mathscr{L}_{\Sigma}\varphi=\Delta_{x}\varphi+\alpha\langle\nabla_{x}\log|x|,\nabla_{x}\varphi\rangle=\frac{1}{|x|^{\alpha}}\mathrm{div}\Big{(}|x|^{\alpha}\nabla_{x}\varphi\Big{)}.$

Theorem 1.1 states that a function $\varphi\in C^{2}(\mathbb{R}^{n})$
satisfying $\mathscr{L}_{\Sigma}\varphi=0$ has the mean value property at $0$

$\varphi(0)=\frac{n+\alpha}{n\omega_{n}r^{n+\alpha}}\int_{\\{|x|<r\\}}\varphi(x)|x|^{\alpha}dx,$

with $\omega_{n}=\mathscr{L}^{n}(\\{|x|<1\\})$.

###  5.3. $\alpha$-subharmonic surfaces

In this section, we look for sufficient conditions for a hypersurface $\Sigma$
to be $\alpha$-subharmonic at $0\in\Sigma$.

######  Definition 5.2 ($\eta$-flatness).

Let $\eta>0$. We say that a hypersurface $\Sigma\subset\mathbb{R}^{n+1}$ is
$\eta$-flat at $0\in\Sigma$ if there exists $r>0$ such that its
$\alpha$-normal $\nu=(\bar{\nu},\nu_{n+1})$ satisfies

$(\alpha+1)|y||\nu_{n+1}|\leq\eta|x|^{\alpha}|\langle x,\bar{\nu}\rangle|$
(5.2)

for all points $\xi=(x,y)\in\Sigma^{*}\cap B_{r}$.

When $\Sigma$ is the $y$-graph of a function $u$, condition (5.2) reads

$(\alpha+1)|u|\leq\eta\big{|}\langle x,\nabla_{x}u\rangle\big{|}$ (5.3)

holding for points a neighborhood of $0\in\mathbb{R}^{n}$.

######  Lemma 5.3.

Let $u\in C^{1}(\\{|x|\leq 1\\})$ be a function satisfying (5.3). Then for any
point $|x|\leq 1$ we have

$|u(x)|\leq\Big{(}\max_{|x|=1}|u|\Big{)}|x|^{\frac{\alpha+1}{\eta}}.$ (5.4)

###### Proof.

Let $|x|=1$ be fixed. We prove the claim along the segment $tx$, with
$t\in[0,1]$. Letting $\varphi:[0,1]\to\mathbb{R}$, $\varphi(t)=u(tx)$,
assumption (5.3) reads

$(\alpha+1)|\varphi(t)|\leq\eta t|\varphi^{\prime}(t)|.$ (5.5)

If $\varphi=0$ on $[0,1]$, the claim is trivial. Then we can assume that the
open set $A=\\{t\in(0,1):\varphi(t)\neq 0\\}$ is nonempty. This set is a
finite or countable disjoint union of intervals $(a,b)\subset A$. We always
have $\varphi(a)=0$ and from (5.5) it follows that $\varphi^{\prime}\neq 0$ on
$(a,b)$, say $\varphi^{\prime}(t)>0$ for any $t\in(a,b)$. Then $\varphi$ is
strictly monotone increasing and thus $\varphi(b)>0$, and so $b=1$. It follows
that $A=(a,1)$, for some $a\in[0,1)$, and $\varphi=0$ on $[0,a]$.

We can without loss of generality assume that $a=0$ and conclude the proof in
the following way. We have $\varphi>0$ and, say, $\varphi^{\prime}>0$ on
$(0,1)$. Then (5.5) reads

$\frac{d}{dt}\log(t^{\alpha+1})\leq\eta\frac{d}{dt}\log\varphi(t),$ (5.6)

and integrating this inequality from $t=s$ to $t=1$, $0<s<1$, we get
$\varphi(s)\leq\varphi(1)s^{\frac{\alpha+1}{\eta}}$. This implies (5.4) and
the proof is concluded. ∎

######  Theorem 5.4.

Let $\Sigma\subset\mathbb{R}^{n+1}$ by a hypersurface of class $C^{2}$ with
$\alpha$-normal $\nu=(\bar{\nu},\nu_{n+1})$ and $\alpha$-mean curvature
$H_{\Sigma}$. Assume that:

  * i)

$\Sigma$ is $\eta$-flat at $0\in\Sigma$ for some

$0<\eta<\frac{n+\alpha}{n+3\alpha}.$ (5.7)

  * ii)

We have, with limit restricted to $\xi=(x,y)\in\Sigma$,

$\lim_{\xi\to
0}\frac{|x|^{2}H_{\Sigma}(\xi)}{\langle\bar{\nu}(\xi),x\rangle}=0.$ (5.8)

Then there exists a $\delta>0$ such that $q_{\Sigma}\geq 0$ on $\Sigma^{*}\cap
B_{\delta}$.

###### Proof.

Without loss of generality we may assume that
$\Sigma=\\{(x,u(x))\in\mathbb{R}^{n+1}:|x|\leq 1\\}$ for some function $u\in
C^{2}(\\{|x|\leq 1\\})$, and that (5.2) holds with $r=1$. Letting
$C_{1}=\max_{|x|=1}|u(x)|$, points $(x,y)\in\Sigma$ satisfy by (5.4)

$|y|\leq C_{1}|x|^{\frac{\alpha+1}{\eta}}.$ (5.9)

Formula (1.5) for $q_{\Sigma}$ reads

$q_{\Sigma}(\xi)=\langle
X\varrho,\nu\rangle\Big{[}\frac{n+3\alpha}{\varrho}\langle
X\varrho,\nu\rangle-\frac{2\alpha}{|x|^{2}}\langle
x,\bar{\nu}\rangle+nH_{\Sigma}\Big{]},$

and the inequality $q_{\Sigma}\geq 0$ is thus implied by

$\Big{|}\frac{2\alpha}{|x|^{2}}\langle x,\bar{\nu}\rangle-
nH_{\Sigma}\Big{|}\leq\frac{n+3\alpha}{\varrho}|\langle X\varrho,\nu\rangle|.$
(5.10)

Inserting the identity (3.37) into (5.10) and then using (3.22), we see that
$q_{\Sigma}\geq 0$ is implied by

$\Big{|}\frac{2\alpha}{|x|^{2}}\langle x,\bar{\nu}\rangle-
nH_{\Sigma}\Big{|}\leq\frac{n+3\alpha}{\varrho^{2\alpha+2}}|x|^{\alpha}\big{|}|x|^{\alpha}\langle
x,\bar{\nu}\rangle+(\alpha+1)y\nu_{n+1}\big{|}.$ (5.11)

By the $\eta$-flatness condition (5.2), (5.11) is in turn implied by

$\Big{|}\frac{2\alpha}{|x|^{2}}\langle x,\bar{\nu}\rangle-
nH_{\Sigma}\Big{|}\leq(1-\eta)\frac{n+3\alpha}{\varrho^{2\alpha+2}}|x|^{2\alpha}|\langle
x,\bar{\nu}\rangle|,$ (5.12)

and finally, using assumption (5.8), (5.12) is implied by

$\Big{(}2\alpha+\frac{n|x|^{2}|H_{\Sigma}|}{|\langle
x,\bar{\nu}\rangle|}\Big{)}\leq(1-\eta)(n+3\alpha)\frac{|x|^{2\alpha+2}}{\varrho^{2\alpha+2}}.$
(5.13)

Now observe that, by (5.9),

$\frac{\varrho^{2(\alpha+1)}}{|x|^{2(\alpha+1)}}=1+\frac{(\alpha+1)^{2}y^{2}}{|x|^{2(\alpha+1)}}\leq
1+C_{1}(\alpha+1)^{2}|x|^{2(\alpha+1)(1/\eta-1)}.$ (5.14)

By (5.14) and (5.8), inequality (5.13) is satisfied for $x=0$ as a strict
inequality, by (5.7). Our claim follows by a limiting argument.

∎

## References

  * [1] W. Bauer, K. Furutani, and C. Iwasaki.  Fundamental solution of a higher step Grushin type operator.  Adv. Math., 271:188–234, 2015.
  * [2] E. Bombieri, E. De Giorgi, and M. Miranda.  Una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche.  Arch. Rational Mech. Anal., 32:255–267, 1969.
  * [3] A. Bonfiglioli and E. Lanconelli.  Subharmonic functions in sub-Riemannian settings.  J. Eur. Math. Soc. (JEMS), 15(2):387–441, 2013.
  * [4] G. Citti, N. Garofalo, and E. Lanconelli.  Harnack’s inequality for sum of squares of vector fields plus a potential.  Amer. J. Math., 115(3):699–734, 1993.
  * [5] G. Cupini and E. Lanconelli.  On mean value formulas for solutions to second order linear PDEs.  Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 22(2):777–809, 2021.
  * [6] V. Franceschi and R. Monti.  Isoperimetric problem in $H$-type groups and Grushin spaces.  Rev. Mat. Iberoam., 32(4):1227–1258, 2016.
  * [7] B. Franchi and E. Lanconelli.  Une métrique associée à une classe d’opérateurs elliptiques dégénérés.  Number Special Issue, pages 105–114 (1984). 1983.  Conference on linear partial and pseudodifferential operators (Torino, 1982).
  * [8] N. Garofalo.  Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension.  J. Differential Equations, 104(1):117–146, 1993.
  * [9] N. Garofalo.  Fractional thoughts.  In New developments in the analysis of nonlocal operators, volume 723 of Contemp. Math., pages 1–135. Amer. Math. Soc., [Providence], RI, [2019] ©2019.
  * [10] L. Hörmander.  Hypoelliptic second order differential equations.  Acta Math., 119:147–171, 1967.
  * [11] T. J. Willmore.  Mean value theorems in harmonic Riemannian spaces.  J. London Math. Soc., 25:54–57, 1950.


gauge-invariant hadronic model that reproduces soft and collinear regions of
the one-loop charged-current elastic process. We have performed detailed error
analysis including hadronic and perturbative uncertainties within this model.
The tree-level hard function is the main source of uncertainty in absolute
cross sections. However, this uncertainty largely cancels in important cross-
section ratios, in particular the ratio of electron to muon cross sections.
Remaining uncertainties are at permille level.

Our results confirm a naive estimate of radiative corrections by powers of
leading logarithms. Exclusive cross sections at GeV neutrino energies with
electron flavor can change by $\sim 10-20$ % when only soft photons are
included in the observable and up to $\sim$5 % when both soft and collinear
radiation are included. The corresponding changes in muon (anti)neutrino cross
sections are typically smaller and can reach up to $\sim$5 %. Inclusive cross
sections at fixed hadronic momentum transfer are subject to smaller radiative
corrections, due to the cancellation between virtual and real contributions
after the inclusion of hard photons. However, inclusive results vary by $\sim
10-20$ % ($5-10$ %) for electron (muon) flavor depending on the way that
kinematics is reconstructed from leptons and photons.

An important result from our studies for precision accelerator neutrino
oscillation program is that the total cross section as a function of
(anti)neutrino energy, inclusive of real photon emission, is very similar for
electron and muon (anti)neutrino events, as Figs. 16, 17, and 21 illustrate.
However this simple result is achieved only after summing inclusively over
distinct kinematic configurations. Electron-flavor and muon-flavor cross
sections receive significant, and different, corrections as a function of
kinematics that must be carefully accounted for when experimental cuts and
efficiency corrections are applied in a practical experiment. It is also
important to carefully match the theoretical calculation of radiative
corrections to experimental conditions since radiative corrections depend
strongly on the treatment of real photon radiation.

The double ratio of the neutrino over antineutrino cross sections to the tree-
level result is very close to unity for exclusive observables and for
inclusive observables corresponding to the same value of hadronic momentum
transfer. This situation can be traced to the fact that the collinear function
is the same in neutrino and antineutrino scattering and that enhanced
perturbative contributions to the soft function are similarly the same.
However, differences in the kinematic reconstruction of electron-flavor and
muon-flavor events in practical detectors lead to significant deviations of
this double ratio from the unity.

Our studies have shown that the probability for a muon (anti)neutrino event to
be misidentified as an electron (anti)neutrino event due to the presence of
energetic collinear photons is of order $10^{-4}$ or below. Radiative cross
sections with noncollinear hard photons have typically percent level and
should be accounted for in precise measurements with accelerator
(anti)neutrino beams.

Our work can be directly applied to analyze (anti)neutrino-nucleon scattering
processes, accounting for QED radiative corrections for the first time. An
important application is the extraction of the nucleon axial form factor and
corresponding axial radius from neutrino scattering data at GeV energies.

A primary motivation for our work is the analysis of neutrino oscillation
signals using nuclear targets. Although the study was performed with
(anti)neutrino-nucleon scattering, important cross-section ratios are
insensitive to the explicit form of the nonperturbative hard function and
similar conclusions are valid for scattering on nuclei. First, the radiative
corrections to exclusive cross sections at large momentum transfer, and the
corresponding flavor ratios, are dominated by large perturbative logarithms
that are independent of nuclear or hadronic parameters. Second, the radiative
corrections to inclusive cross sections are governed by an expansion in small
lepton mass [cf. Eq. (59)], which implies small modifications to radiative
corrections from nuclear effects and small theoretical uncertainties on flavor
ratios of inclusive cross sections.

Our work paves the way for the rigorous incorporation of radiative corrections
at accelerator neutrino experiments, an important missing ingredient for the
achievement of the desired precision in extractions of the Pontecorvo-Maki-
Nakagawa-Sakata matrix parameters, squared mass differences, and discovery of
the $CP$ violation in the lepton sector.

Acknowledgements We thank Kaushik Borah for an independent validation of
antineutrino-proton cross-section expressions and Emanuele Mereghetti and Ryan
Plestid for discussions. This work was supported by the U.S. Department of
Energy, Office of Science, Office of High Energy Physics, under Grants No. DE-
SC0019095 and No. DE-SC0008475. Fermilab is operated by Fermi Research
Alliance, LLC under Contract No. DE-AC02-07CH11359 with the United States
Department of Energy. O.T. acknowledges support by the Visiting Scholars Award
Program of the Universities Research Association, and thanks theory groups at
Fermilab and Institute for Nuclear Physics at JGU Mainz for warm hospitality.
O.T. is supported by the U.S. Department of Energy through the Los Alamos
National Laboratory (LANL). Los Alamos National Laboratory is operated by
Triad National Security, LLC, for the National Nuclear Security Administration
of U.S. Department of Energy (Contract No. 89233218CNA000001). This research
is funded by LANL’s Laboratory Directed Research and Development (LDRD/PRD)
program under Project No. 20210968PRD4. Q.C. and O.T. acknowledge KITP
Graduate Fellow program supported by the Heising-Simons Foundation, the Simons
Foundation, and National Science Foundation Grant No. NSF PHY-1748958. R.J.H.
acknowledges support from the Neutrino Theory Network at Fermilab during the
early stages of this work. K.S.M. acknowledges support from a Fermilab
Intensity Frontier Fellowship during the early stages of this work and from
the University of Rochester’s Steven Chu Professorship in Physics. FeynCalc
[230, 231], LoopTools [232], Mathematica [233], and DataGraph were useful in
this work.

## References

  * De Rujula  _et al._ [1979] A. De Rujula, R. Petronzio, and A. Savoy-Navarro, Nucl. Phys. B 154, 394 (1979).
  * Day and McFarland [2012] M. Day and K. S. McFarland, Phys. Rev. D 86, 053003 (2012), arXiv:1206.6745 [hep-ph].
  * Tomalak  _et al._ [2022] O. Tomalak, Q. Chen, R. J. Hill, and K. S. McFarland, Nature Commun. 13, 5286 (2022).
  * Seltzer [1987] S. Seltzer, “Xcom-photon cross sections database, nist standard reference database 8,”  (1987).
  * Zyla  _et al._ [2020] P. Zyla _et al._ (Particle Data Group), PTEP 2020, 083C01 (2020).
  * Antonello  _et al._ [2015] M. Antonello _et al._ (MicroBooNE, LAr1-ND, ICARUS-WA104), CERN Document Server (2015), arXiv:1503.01520 [physics.ins-det].
  * Abe  _et al._ [2014] K. Abe _et al._ , Nucl. Instrum. Meth. A 737, 253 (2014), arXiv:1307.0162 [physics.ins-det].
  * McKinley and Feshbach [1948] W. A. McKinley and H. Feshbach, Phys. Rev. 74, 1759 (1948).
  * Dalitz [1951] R. H. Dalitz, Proc. Roy. Soc. Lond. A206, 509 (1951).
  * Caswell and Lepage [1986] W. E. Caswell and G. P. Lepage, Phys. Lett. B 167, 437 (1986).
  * Pachucki [1996] K. Pachucki, Phys. Rev. A 53, 2092 (1996).
  * Pineda and Soto [1998] A. Pineda and J. Soto, Nucl. Phys. B Proc. Suppl. 64, 428 (1998), arXiv:hep-ph/9707481.
  * Jentschura  _et al._ [2005] U. D. Jentschura, A. Czarnecki, and K. Pachucki, Phys. Rev. A 72, 062102 (2005), arXiv:physics/0603123.
  * Hill  _et al._ [2013] R. J. Hill, G. Lee, G. Paz, and M. P. Solon, Phys. Rev. D 87, 053017 (2013), arXiv:1212.4508 [hep-ph].
  * Neubert [1994] M. Neubert, Phys. Rept. 245, 259 (1994), arXiv:hep-ph/9306320.
  * Manohar and Wise [2000] A. V. Manohar and M. B. Wise,  _Heavy quark physics_ , Vol. 10 (Camb.Monogr.Part.Phys.Nucl.Phys.Cosmol., 2000).
  * Hill [2017] R. J. Hill, Phys. Rev. D 95, 013001 (2017), arXiv:1605.02613 [hep-ph].
  * Fukugita and Kubota [2004] M. Fukugita and T. Kubota, Acta Phys. Polon. B 35, 1687 (2004), arXiv:hep-ph/0403149.
  * Raha  _et al._ [2012] U. Raha, F. Myhrer, and K. Kubodera, Phys. Rev. C 85, 045502 (2012), [Erratum: Phys.Rev.C 86, 039903 (2012)], arXiv:1112.2007 [hep-ph].
  * Bloch and Nordsieck [1937] F. Bloch and A. Nordsieck, Phys. Rev. 52, 54 (1937).
  * Yennie  _et al._ [1961] D. Yennie, S. C. Frautschi, and H. Suura, Annals Phys. 13, 379 (1961).
  * Kinoshita [1962] T. Kinoshita, J. Math. Phys. 3, 650 (1962).
  * Lee and Nauenberg [1964] T. Lee and M. Nauenberg, Phys. Rev. 133, B1549 (1964).
  * Tomalak and Hill [2020] O. Tomalak and R. J. Hill, Phys. Rev. D 101, 033006 (2020), arXiv:1907.03379 [hep-ph].
  * Sudakov [1956] V. Sudakov, Sov. Phys. JETP 3, 65 (1956).
  * ’t Hooft and Veltman [1979] G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B 153, 365 (1979).
  * Korchemsky and Radyushkin [1987] G. Korchemsky and A. Radyushkin, Nucl. Phys. B 283, 342 (1987).
  * Kilian  _et al._ [1993] W. Kilian, P. Manakos, and T. Mannel, Phys. Rev. D 48, 1321 (1993).
  * Burgers [1985] G. Burgers, Phys. Lett. B 164, 167 (1985).
  * Kniehl [1990] B. A. Kniehl, Phys. Lett. B 237, 127 (1990).
  * Mastrolia and Remiddi [2003] P. Mastrolia and E. Remiddi, Nucl. Phys. B 664, 341 (2003), arXiv:hep-ph/0302162.
  * Arbuzov and Kopylova [2015] A. B. Arbuzov and T. V. Kopylova, Eur. Phys. J. C 75, 603 (2015), arXiv:1510.06497 [hep-ph].
  * Hoang  _et al._ [1995] A. Hoang, J. H. Kuhn, and T. Teubner, Nucl. Phys. B 452, 173 (1995), arXiv:hep-ph/9505262.
  * Bernreuther  _et al._ [2005] W. Bernreuther, R. Bonciani, T. Gehrmann, R. Heinesch, T. Leineweber, P. Mastrolia, and E. Remiddi, Nucl. Phys. B 706, 245 (2005), arXiv:hep-ph/0406046.
  * Becher and Melnikov [2007] T. Becher and K. Melnikov, JHEP 06, 084 (2007), arXiv:0704.3582 [hep-ph].
  * Gribov and Lipatov [1972a] V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys. 15, 438 (1972a).
  * Gribov and Lipatov [1972b] V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys. 15, 675 (1972b).
  * Lipatov [1975] L. N. Lipatov, Sov. J. Nucl. Phys. 20, 94 (1975).
  * Dokshitzer [1977] Y. L. Dokshitzer, Sov. Phys. JETP 46, 641 (1977).
  * Altarelli and Parisi [1977] G. Altarelli and G. Parisi, Nucl. Phys. B 126, 298 (1977).
  * Dokshitzer  _et al._ [1980] Y. L. Dokshitzer, D. Diakonov, and S. I. Troian, Phys. Rept. 58, 269 (1980).
  * Peskin and Schroeder [1995] M. E. Peskin and D. V. Schroeder,  _An Introduction to quantum field theory_ (Addison-Wesley, Reading, USA, 1995).
  * Becher and Neubert [2009a] T. Becher and M. Neubert, JHEP 06, 081 (2009a), [Erratum: JHEP 11, 024 (2013)], arXiv:0903.1126 [hep-ph].
  * Becher and Neubert [2009b] T. Becher and M. Neubert, Phys. Rev. D 79, 125004 (2009b), [Erratum: Phys.Rev.D 80, 109901 (2009)], arXiv:0904.1021 [hep-ph].
  * Hill and Tomalak [2020] R. J. Hill and O. Tomalak, Phys. Lett. B 805, 135466 (2020), arXiv:1911.01493 [hep-ph].
  * Llewellyn Smith [1972] C. Llewellyn Smith, Phys. Rept. 3, 261 (1972).
  * Kuzmin  _et al._ [2008] K. S. Kuzmin, V. V. Lyubushkin, and V. A. Naumov, Eur. Phys. J. C 54, 517 (2008), arXiv:0712.4384 [hep-ph].
  * Dudelzak [1965] B. Dudelzak, Ph.D. thesis, University of Paris (1965).
  * Janssens  _et al._ [1966] T. Janssens, R. Hofstadter, E. B. Hughes, and M. R. Yearian, Phys. Rev. 142, 922 (1966).
  * Bartel  _et al._ [1966] W. Bartel _et al._ , Phys. Rev. Lett. 17, 608 (1966).
  * Albrecht  _et al._ [1966] W. Albrecht _et al._ , Phys. Rev. Lett. 17, 1192 (1966).
  * Frèrejacque  _et al._ [1966] D. Frèrejacque, D. Benaksas, and D. Drickey, Phys. Rev. 141, 1308 (1966).
  * Albrecht  _et al._ [1967] W. Albrecht _et al._ , Phys. Rev. Lett. 18, 1014 (1967).
  * Goitein  _et al._ [1967] M. Goitein, J. R. Dunning, and R. Wilson, Phys. Rev. Lett. 18, 1018 (1967).
  * Litt  _et al._ [1970] J. Litt _et al._ , Phys. Lett. B 31, 40 (1970).
  * Goitein  _et al._ [1970] M. Goitein _et al._ , Phys. Rev. D 1, 2449 (1970).
  * Berger  _et al._ [1971] C. Berger _et al._ , Phys. Lett. B 35, 87 (1971).
  * Price  _et al._ [1971] L. E. Price _et al._ , Phys. Rev. D 4, 45 (1971).
  * Ganichot  _et al._ [1972] D. Ganichot, B. Grossetête, and D. Isabelle, Nucl. Phys. A 178, 545 (1972).
  * Bartel  _et al._ [1973] W. Bartel _et al._ , Nucl. Phys. B 58, 429 (1973).
  * Kirk  _et al._ [1973] P. N. Kirk _et al._ , Phys. Rev. D 8, 63 (1973).
  * Borkowski  _et al._ [1974] F. Borkowski _et al._ , Nucl. Phys. A222, 269 (1974).
  * Murphy  _et al._ [1974] J. J. Murphy, Y. M. Shin, and D. M. Skopik, Phys. Rev. C 9, 2125 (1974).
  * Borkowski  _et al._ [1975] F. Borkowski _et al._ , Nucl. Phys. B93, 461 (1975).
  * Stein  _et al._ [1975] S. Stein _et al._ , Phys. Rev. D 12, 1884 (1975).
  * Simon  _et al._ [1980] G. G. Simon, C. Schmitt, F. Borkowski, and V. H. Walther, Nucl. Phys. A333, 381 (1980).
  * Simon  _et al._ [1981] G. G. Simon, C. Schmitt, and V. H. Walther, Nucl. Phys. A364, 285 (1981).
  * Bosted  _et al._ [1990] P. E. Bosted _et al._ , Phys. Rev. C 42, 38 (1990).
  * Rock  _et al._ [1992] S. Rock _et al._ , Phys. Rev. D 46, 24 (1992).
  * Sill  _et al._ [1993] A. F. Sill _et al._ , Phys. Rev. D 48, 29 (1993).
  * Walker  _et al._ [1994] R. C. Walker _et al._ , Phys. Rev. D 49, 5671 (1994).
  * Andivahis  _et al._ [1994] L. Andivahis _et al._ , Phys. Rev. D 50, 5491 (1994).
  * Dutta  _et al._ [2003] D. Dutta _et al._ , Phys. Rev. C 68, 064603 (2003).
  * Christy  _et al._ [2004] M. E. Christy _et al._ (E94110), Phys. Rev. C70, 015206 (2004), arXiv:nucl-ex/0401030 [nucl-ex].
  * Qattan  _et al._ [2005] I. A. Qattan _et al._ , Phys. Rev. Lett. 94, 142301 (2005), arXiv:nucl-ex/0410010 [nucl-ex].
  * Bernauer  _et al._ [2014] J. C. Bernauer _et al._ (A1), Phys. Rev. C90, 015206 (2014), arXiv:1307.6227 [nucl-ex].
  * Milbrath  _et al._ [1998] B. D. Milbrath _et al._ (Bates FPP), Phys. Rev. Lett. 80, 452 (1998), [Erratum: Phys. Rev. Lett.82,2221(1999)], arXiv:nucl-ex/9712006 [nucl-ex].
  * Pospischil  _et al._ [2001] T. Pospischil _et al._ (A1), Eur. Phys. J. A12, 125 (2001).
  * Gayou  _et al._ [2001] O. Gayou _et al._ , Phys. Rev. C64, 038202 (2001).
  * Gayou  _et al._ [2002] O. Gayou _et al._ (Jefferson Lab Hall A), Phys. Rev. Lett. 88, 092301 (2002), arXiv:nucl-ex/0111010 [nucl-ex].
  * Strauch  _et al._ [2003] S. Strauch _et al._ (Jefferson Lab E93-049), Phys. Rev. Lett. 91, 052301 (2003), arXiv:nucl-ex/0211022 [nucl-ex].
  * Punjabi  _et al._ [2005] V. Punjabi _et al._ , Phys. Rev. C71, 055202 (2005), [Erratum: Phys. Rev.C71,069902(2005)], arXiv:nucl-ex/0501018 [nucl-ex].
  * MacLachlan  _et al._ [2006] G. MacLachlan _et al._ , Nucl. Phys. A764, 261 (2006).
  * Jones  _et al._ [2006] M. K. Jones _et al._ (Resonance Spin Structure), Phys. Rev. C74, 035201 (2006), arXiv:nucl-ex/0606015 [nucl-ex].
  * Crawford  _et al._ [2007] C. B. Crawford _et al._ , Phys. Rev. Lett. 98, 052301 (2007), arXiv:nucl-ex/0609007 [nucl-ex].
  * Ron  _et al._ [2011] G. Ron _et al._ (Jefferson Lab Hall A), Phys. Rev. C84, 055204 (2011), arXiv:1103.5784 [nucl-ex].
  * Zhan  _et al._ [2011] X. Zhan _et al._ , Phys. Lett. B705, 59 (2011), arXiv:1102.0318 [nucl-ex].
  * Puckett  _et al._ [2010] A. J. R. Puckett _et al._ , Phys. Rev. Lett. 104, 242301 (2010).
  * Paolone  _et al._ [2010] M. Paolone _et al._ , Phys. Rev. Lett. 105, 072001 (2010), arXiv:1002.2188 [nucl-ex].
  * Puckett  _et al._ [2012] A. J. R. Puckett _et al._ , Phys. Rev. C85, 045203 (2012), arXiv:1102.5737 [nucl-ex].
  * Puckett  _et al._ [2017] A. J. R. Puckett _et al._ , Phys. Rev. C96, 055203 (2017), [erratum: Phys. Rev.C98,no.1,019907(2018)], arXiv:1707.08587 [nucl-ex].
  * Meyerhoff  _et al._ [1994] M. Meyerhoff _et al._ , Phys. Lett. B327, 201 (1994).
  * Eden  _et al._ [1994] T. Eden _et al._ , Phys. Rev. C50, R1749 (1994).
  * Passchier  _et al._ [1999] I. Passchier _et al._ , Phys. Rev. Lett. 82, 4988 (1999), arXiv:nucl-ex/9907012 [nucl-ex].
  * Herberg  _et al._ [1999] C. Herberg _et al._ , Eur. Phys. J. A5, 131 (1999).
  * Rohe  _et al._ [1999] D. Rohe _et al._ , Phys. Rev. Lett. 83, 4257 (1999).
  * Golak  _et al._ [2001] J. Golak, G. Ziemer, H. Kamada, H. Witala, and W. Gloeckle, Phys. Rev. C63, 034006 (2001), arXiv:nucl-th/0008008 [nucl-th].
  * Schiavilla and Sick [2001] R. Schiavilla and I. Sick, Phys. Rev. C64, 041002 (2001), arXiv:nucl-ex/0107004 [nucl-ex].
  * Zhu  _et al._ [2001] H. Zhu _et al._ (E93026), Phys. Rev. Lett. 87, 081801 (2001), arXiv:nucl-ex/0105001 [nucl-ex].
  * Bermuth  _et al._ [2003] J. Bermuth _et al._ , Phys. Lett. B564, 199 (2003), arXiv:nucl-ex/0303015 [nucl-ex].
  * Madey  _et al._ [2003] R. Madey _et al._ (E93-038), Phys. Rev. Lett. 91, 122002 (2003), arXiv:nucl-ex/0308007 [nucl-ex].
  * Warren  _et al._ [2004] G. Warren _et al._ (Jefferson Lab E93-026), Phys. Rev. Lett. 92, 042301 (2004), arXiv:nucl-ex/0308021 [nucl-ex].
  * Glazier  _et al._ [2005] D. I. Glazier _et al._ , Eur. Phys. J. A24, 101 (2005), arXiv:nucl-ex/0410026 [nucl-ex].
  * Geis  _et al._ [2008] E. Geis _et al._ (BLAST), Phys. Rev. Lett. 101, 042501 (2008), arXiv:0803.3827 [nucl-ex].
  * Riordan  _et al._ [2010] S. Riordan _et al._ , Phys. Rev. Lett. 105, 262302 (2010), arXiv:1008.1738 [nucl-ex].
  * Schlimme  _et al._ [2013] B. S. Schlimme _et al._ , Phys. Rev. Lett. 111, 132504 (2013), arXiv:1307.7361 [nucl-ex].
  * Rock  _et al._ [1982] S. Rock _et al._ ,  _New Horizons in Electromagnetic Physics, Charlottesville, Virginia, April 21-24, 1982_ , Phys. Rev. Lett. 49, 1139 (1982).
  * Lung  _et al._ [1993] A. Lung _et al._ , Phys. Rev. Lett. 70, 718 (1993).
  * Gao  _et al._ [1994] H. Gao _et al._ , Phys. Rev. C50, R546 (1994).
  * Anklin  _et al._ [1998] H. Anklin _et al._ , Phys. Lett. B428, 248 (1998).
  * Kubon  _et al._ [2002] G. Kubon _et al._ , Phys. Lett. B524, 26 (2002), arXiv:nucl-ex/0107016 [nucl-ex].
  * Anderson  _et al._ [2007] B. Anderson _et al._ (Jefferson Lab E95-001), Phys. Rev. C75, 034003 (2007), arXiv:nucl-ex/0605006 [nucl-ex].
  * Lachniet  _et al._ [2009] J. Lachniet _et al._ (CLAS), Phys. Rev. Lett. 102, 192001 (2009), arXiv:0811.1716 [nucl-ex].
  * Xiong  _et al._ [2019] W. Xiong _et al._ , Nature 575, 147 (2019).
  * Mohr  _et al._ [2016] P. J. Mohr, D. B. Newell, and B. N. Taylor, Rev. Mod. Phys. 88, 035009 (2016), arXiv:1507.07956 [physics.atom-ph].
  * Schneider  _et al._ [2017] G. Schneider _et al._ , Science 358, 1081 (2017).
  * Ye  _et al._ [2018] Z. Ye, J. Arrington, R. J. Hill, and G. Lee, Phys. Lett. B777, 8 (2018), arXiv:1707.09063 [nucl-ex].
  * Pohl  _et al._ [2010] R. Pohl _et al._ , Nature 466, 213 (2010).
  * Antognini  _et al._ [2013] A. Antognini _et al._ , Science 339, 417 (2013).
  * Kopecky  _et al._ [1995] S. Kopecky, P. Riehs, J. A. Harvey, and N. W. Hill, Phys. Rev. Lett. 74, 2427 (1995).
  * Kopecky  _et al._ [1997] S. Kopecky, M. Krenn, P. Riehs, S. Steiner, J. A. Harvey, N. W. Hill, and M. Pernicka, Phys. Rev. C56, 2229 (1997).
  * Lee  _et al._ [2015] G. Lee, J. R. Arrington, and R. J. Hill, Phys. Rev. D92, 013013 (2015), arXiv:1505.01489 [hep-ph].
  * Mann  _et al._ [1973] W. A. Mann _et al._ , Phys. Rev. Lett. 31, 844 (1973).
  * Barish  _et al._ [1977] S. J. Barish _et al._ , Phys. Rev. D 16, 3103 (1977).
  * Miller [1981] K. L. Miller,  _A Study of the Weak Axial Vector form Factor in the Quasi-Elastic Reaction $\nu_{\mu}d\to\mu^{-}pp_{s}$_, Ph.D. thesis, Carnegie Mellon U. (1981).
  * Baker  _et al._ [1981] N. J. Baker, A. M. Cnops, P. L. Connolly, S. A. Kahn, H. G. Kirk, M. J. Murtagh, R. B. Palmer, N. P. Samios, and M. Tanaka, Phys. Rev. D 23, 2499 (1981).
  * Kitagaki  _et al._ [1983] T. Kitagaki _et al._ , Phys. Rev. D 28, 436 (1983).
  * Belikov  _et al._ [1985] S. V. Belikov _et al._ , Z. Phys. A 320, 625 (1985).
  * Bodek  _et al._ [2008] A. Bodek, S. Avvakumov, R. Bradford, and H. S. Budd, Eur. Phys. J. C 53, 349 (2008), arXiv:0708.1946 [hep-ex].
  * Bhattacharya  _et al._ [2011] B. Bhattacharya, R. J. Hill, and G. Paz, Phys. Rev. D 84, 073006 (2011), arXiv:1108.0423 [hep-ph].
  * Meyer  _et al._ [2016] A. S. Meyer, M. Betancourt, R. Gran, and R. J. Hill, Phys. Rev. D 93, 113015 (2016), arXiv:1603.03048 [hep-ph].
  * Alvarez-Ruso  _et al._ [2019] L. Alvarez-Ruso, K. M. Graczyk, and E. Saul-Sala, Phys. Rev. C 99, 025204 (2019), arXiv:1805.00905 [hep-ph].
  * Megias  _et al._ [2020] G. D. Megias, S. Bolognesi, M. B. Barbaro, and E. Tomasi-Gustafsson, Phys. Rev. C 101, 025501 (2020), arXiv:1910.13263 [hep-ph].
  * Green  _et al._ [2017] J. Green, N. Hasan, S. Meinel, M. Engelhardt, S. Krieg, J. Laeuchli, J. Negele, K. Orginos, A. Pochinsky, and S. Syritsyn, Phys. Rev. D 95, 114502 (2017), arXiv:1703.06703 [hep-lat].
  * Alexandrou  _et al._ [2017] C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, and A. Vaquero Aviles-Casco, Phys. Rev. D 96, 054507 (2017), arXiv:1705.03399 [hep-lat].
  * Capitani  _et al._ [2019] S. Capitani, M. Della Morte, D. Djukanovic, G. M. von Hippel, J. Hua, B. Jäger, P. M. Junnarkar, H. B. Meyer, T. D. Rae, and H. Wittig, Int. J. Mod. Phys. A 34, 1950009 (2019), arXiv:1705.06186 [hep-lat].
  * Gupta  _et al._ [2017] R. Gupta, Y.-C. Jang, H.-W. Lin, B. Yoon, and T. Bhattacharya, Phys. Rev. D 96, 114503 (2017), arXiv:1705.06834 [hep-lat].
  * Bali  _et al._ [2019] G. S. Bali, S. Collins, M. Gruber, A. Schäfer, P. Wein, and T. Wurm, Phys. Lett. B 789, 666 (2019), arXiv:1810.05569 [hep-lat].
  * Jang  _et al._ [2020] Y.-C. Jang, R. Gupta, B. Yoon, and T. Bhattacharya, Phys. Rev. Lett. 124, 072002 (2020), arXiv:1905.06470 [hep-lat].
  * Alexandrou  _et al._ [2021] C. Alexandrou _et al._ , Phys. Rev. D 103, 034509 (2021), arXiv:2011.13342 [hep-lat].
  * Djukanovic  _et al._ [2021] D. Djukanovic, T. Harris, G. von Hippel, P. M. Junnarkar, H. B. Meyer, D. Mohler, K. Ottnad, T. Schulz, J. Wilhelm, and H. Wittig, Phys. Rev. D 103, 094522 (2021), arXiv:2102.07460 [hep-lat].
  * Maximon and Tjon [2000] L. C. Maximon and J. A. Tjon, Phys. Rev. C62, 054320 (2000), arXiv:nucl-th/0002058 [nucl-th].
  * Blunden  _et al._ [2003] P. Blunden, W. Melnitchouk, and J. Tjon, Phys. Rev. Lett. 91, 142304 (2003), arXiv:nucl-th/0306076.
  * Graczyk [2014] K. M. Graczyk, Phys. Lett. B 732, 315 (2014), arXiv:1308.5581 [hep-ph].
  * Tomalak and Vanderhaeghen [2014] O. Tomalak and M. Vanderhaeghen, Phys. Rev. D 90, 013006 (2014), arXiv:1405.1600 [hep-ph].
  * Galster  _et al._ [1971] S. Galster, H. Klein, J. Moritz, K. Schmidt, D. Wegener, and J. Bleckwenn, Nucl. Phys. B 32, 221 (1971).
  * Lepage and Brodsky [1980] G. Lepage and S. J. Brodsky, Phys. Rev. D 22, 2157 (1980).
  * Kelly [2004] J. Kelly, Phys. Rev. C 70, 068202 (2004).
  * Weinberg [1958] S. Weinberg, Phys. Rev. 112, 1375 (1958).
  * Shiomi [1996] H. Shiomi, Nucl. Phys. A 603, 281 (1996), arXiv:hep-ph/9601329.
  * Govaerts and Lucio-Martinez [2000] J. Govaerts and J.-L. Lucio-Martinez, Nucl. Phys. A 678, 110 (2000), arXiv:nucl-th/0004056.
  * Minamisono  _et al._ [2002] K. Minamisono _et al._ , Phys. Rev. C 65, 015501 (2002).
  * Behrends and Sirlin [1960] R. Behrends and A. Sirlin, Phys. Rev. Lett. 4, 186 (1960).
  * Dmitrasinovic and Pollock [1995] V. Dmitrasinovic and S. Pollock, Phys. Rev. C 52, 1061 (1995), arXiv:hep-ph/9504414.
  * Miller [1998] G. A. Miller, Phys. Rev. C 57, 1492 (1998), arXiv:nucl-th/9711036.
  * Lewis [2007] R. Lewis, Eur. Phys. J. A 32, 409 (2007), arXiv:nucl-th/0608079.
  * Kubis and Lewis [2006] B. Kubis and R. Lewis, Phys. Rev. C 74, 015204 (2006), arXiv:nucl-th/0605006.
  * Borah  _et al._ [2020] K. Borah, R. J. Hill, G. Lee, and O. Tomalak, Phys. Rev. D 102, 074012 (2020), arXiv:2003.13640 [hep-ph].
  * Tomalak and Vanderhaeghen [2016] O. Tomalak and M. Vanderhaeghen, Phys. Rev. D 93, 013023 (2016), arXiv:1508.03759 [hep-ph].
  * Sterman and Weinberg [1977] G. F. Sterman and S. Weinberg, Phys. Rev. Lett. 39, 1436 (1977).
  * Mukhi and Sterman [1982] S. Mukhi and G. F. Sterman, Nucl. Phys. B 206, 221 (1982).
  * Curci and Greco [1978] G. Curci and M. Greco, Phys. Lett. B 79, 406 (1978).
  * Smilga [1979] A. V. Smilga, Nucl. Phys. B 161, 449 (1979).
  * Grindhammer  _et al._ [1990] G. Grindhammer, M. Rudowicz, and S. Peters, Nucl. Instrum. Meth. A 290, 469 (1990).
  * Seltzer [2020] S. Seltzer, “Atomic and nuclear properties,”  (2020).
  * Rossi [1952] B. Rossi,  _High Energy Particles_ (New York: Prentice-Hall, 1952).
  * Day  _et al._ [1990] D. B. Day, J. S. McCarthy, T. W. Donnelly, and I. Sick, Ann. Rev. Nucl. Part. Sci. 40, 357 (1990).
  * Capuzzi  _et al._ [1991] F. Capuzzi, C. Giusti, and F. D. Pacati, Nucl. Phys. A 524, 681 (1991).
  * Donnelly and Sick [1999] T. W. Donnelly and I. Sick, Phys. Rev. Lett. 82, 3212 (1999), arXiv:nucl-th/9809063.
  * De Pace  _et al._ [2003] A. De Pace, M. Nardi, W. M. Alberico, T. W. Donnelly, and A. Molinari, Nucl. Phys. A 726, 303 (2003), arXiv:nucl-th/0304084.
  * De Pace  _et al._ [2004] A. De Pace, M. Nardi, W. M. Alberico, T. W. Donnelly, and A. Molinari, Nucl. Phys. A 741, 249 (2004), arXiv:nucl-th/0403023.
  * Amaro  _et al._ [2011] J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W. Donnelly, and C. F. Williamson, Phys. Lett. B 696, 151 (2011), arXiv:1010.1708 [nucl-th].
  * Meucci  _et al._ [2011] A. Meucci, M. B. Barbaro, J. A. Caballero, C. Giusti, and J. M. Udias, Phys. Rev. Lett. 107, 172501 (2011), arXiv:1107.5145 [nucl-th].
  * Bodek  _et al._ [2011] A. Bodek, H. S. Budd, and M. E. Christy, Eur. Phys. J. C 71, 1726 (2011), arXiv:1106.0340 [hep-ph].
  * Martini  _et al._ [2011] M. Martini, M. Ericson, and G. Chanfray, Phys. Rev. C 84, 055502 (2011), arXiv:1110.0221 [nucl-th].
  * Martini and Ericson [2013] M. Martini and M. Ericson, Phys. Rev. C 87, 065501 (2013), arXiv:1303.7199 [nucl-th].
  * Nieves  _et al._ [2013] J. Nieves, I. Ruiz Simo, and M. J. Vicente Vacas, Phys. Lett. B 721, 90 (2013), arXiv:1302.0703 [hep-ph].
  * Carlson  _et al._ [2015] J. Carlson, S. Gandolfi, F. Pederiva, S. C. Pieper, R. Schiavilla, K. E. Schmidt, and R. B. Wiringa, Rev. Mod. Phys. 87, 1067 (2015), arXiv:1412.3081 [nucl-th].
  * Benhar  _et al._ [2017] O. Benhar, P. Huber, C. Mariani, and D. Meloni, Phys. Rept. 700, 1 (2017), arXiv:1501.06448 [nucl-th].
  * Benhar  _et al._ [2015] O. Benhar, A. Lovato, and N. Rocco, Phys. Rev. C 92, 024602 (2015), arXiv:1502.00887 [nucl-th].
  * Rocco  _et al._ [2016] N. Rocco, A. Lovato, and O. Benhar, Phys. Rev. Lett. 116, 192501 (2016), arXiv:1512.07426 [nucl-th].
  * Van Cuyck  _et al._ [2016] T. Van Cuyck, N. Jachowicz, R. González-Jiménez, M. Martini, V. Pandey, J. Ryckebusch, and N. Van Dessel, Phys. Rev. C 94, 024611 (2016), arXiv:1606.00273 [nucl-th].
  * Pastore  _et al._ [2018] S. Pastore, A. Baroni, J. Carlson, S. Gandolfi, S. C. Pieper, R. Schiavilla, and R. B. Wiringa, Phys. Rev. C 97, 022501 (2018), arXiv:1709.03592 [nucl-th].
  * Alvarez-Ruso  _et al._ [2018] L. Alvarez-Ruso _et al._ (NuSTEC), Prog. Part. Nucl. Phys. 100, 1 (2018), arXiv:1706.03621 [hep-ph].
  * Lynn  _et al._ [2019] J. E. Lynn, I. Tews, S. Gandolfi, and A. Lovato, Ann. Rev. Nucl. Part. Sci. 69, 279 (2019), arXiv:1901.04868 [nucl-th].
  * Andreopoulos  _et al._ [2010] C. Andreopoulos _et al._ , Nucl. Instrum. Meth. A 614, 87 (2010), arXiv:0905.2517 [hep-ph].
  * Alvarez-Ruso  _et al._ [2021] L. Alvarez-Ruso _et al._ (GENIE), Eur. Phys. J. ST 230, 4449 (2021), arXiv:2106.09381 [hep-ph].
  * Buss  _et al._ [2012] O. Buss, T. Gaitanos, K. Gallmeister, H. van Hees, M. Kaskulov, O. Lalakulich, A. B. Larionov, T. Leitner, J. Weil, and U. Mosel, Phys. Rept. 512, 1 (2012), arXiv:1106.1344 [hep-ph].
  * Hayato [2009] Y. Hayato, Acta Phys. Polon. B 40, 2477 (2009).
  * Hayato [2002] Y. Hayato, Nucl. Phys. B Proc. Suppl. 112, 171 (2002).
  * Golan  _et al._ [2012] T. Golan, C. Juszczak, and J. T. Sobczyk, Phys. Rev. C 86, 015505 (2012), arXiv:1202.4197 [nucl-th].
  * Gran  _et al._ [2013] R. Gran, J. Nieves, F. Sanchez, and M. J. Vicente Vacas, Phys. Rev. D 88, 113007 (2013), arXiv:1307.8105 [hep-ph].
  * Megias  _et al._ [2016] G. D. Megias, J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W. Donnelly, and I. Ruiz Simo, Phys. Rev. D 94, 093004 (2016), arXiv:1607.08565 [nucl-th].
  * Smith and Moniz [1972] R. A. Smith and E. J. Moniz, Nucl. Phys. B 43, 605 (1972), [Erratum: Nucl.Phys.B 101, 547 (1975)].
  * Martini  _et al._ [2016] M. Martini, N. Jachowicz, M. Ericson, V. Pandey, T. Van Cuyck, and N. Van Dessel, Phys. Rev. C 94, 015501 (2016), arXiv:1602.00230 [nucl-th].
  * Ankowski [2017] A. M. Ankowski, Phys. Rev. C 96, 035501 (2017), arXiv:1707.01014 [nucl-th].
  * Nikolakopoulos  _et al._ [2019] A. Nikolakopoulos, N. Jachowicz, N. Van Dessel, K. Niewczas, R. González-Jiménez, J. M. Udías, and V. Pandey, Phys. Rev. Lett. 123, 052501 (2019), arXiv:1901.08050 [nucl-th].
  * Cai  _et al._ [2021] T. Cai, M. Moore, _et al._ , “Measurements of nuclear effects and the $\bar{\nu}_{\mu}+\mathrm{H}\to\mu^{+}+n$ cross section in MINERvA with neutron tagging, Ph.D. thesis, University of Rochester, Department of Physics and Astronomy,”  (2021).
  * Miller  _et al._ [1982] K. L. Miller _et al._ , Phys. Rev. D 26, 537 (1982).
  * Allasia  _et al._ [1990] D. Allasia _et al._ , Nucl. Phys. B 343, 285 (1990).
  * Kitagaki  _et al._ [1990] T. Kitagaki _et al._ , Phys. Rev. D 42, 1331 (1990).
  * Gran  _et al._ [2006] R. Gran _et al._ (K2K Collaboration), Phys. Rev. D 74, 052002 (2006), arXiv:hep-ex/0603034 [hep-ex].
  * Aguilar-Arevalo  _et al._ [2010] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Rev. D 81, 092005 (2010), arXiv:1002.2680 [hep-ex].
  * Fiorentini  _et al._ [2013] G. A. Fiorentini _et al._ (MINERvA), Phys. Rev. Lett. 111, 022502 (2013), arXiv:1305.2243 [hep-ex].
  * Walton  _et al._ [2015] T. Walton _et al._ (MINERvA), Phys. Rev. D 91, 071301 (2015), arXiv:1409.4497 [hep-ex].
  * Adamson  _et al._ [2015] P. Adamson _et al._ (MINOS), Phys. Rev. D 91, 012005 (2015), arXiv:1410.8613 [hep-ex].
  * Abe  _et al._ [2015a] K. Abe _et al._ (T2K), Phys. Rev. D 92, 112003 (2015a), arXiv:1411.6264 [hep-ex].
  * Abe  _et al._ [2015b] K. Abe _et al._ (T2K), Phys. Rev. D 91, 112002 (2015b), arXiv:1503.07452 [hep-ex].
  * Abe  _et al._ [2016] K. Abe _et al._ (T2K), Phys. Rev. D 93, 112012 (2016), arXiv:1602.03652 [hep-ex].
  * Abe  _et al._ [2018a] K. Abe _et al._ (T2K), Phys. Rev. D 97, 012001 (2018a), arXiv:1708.06771 [hep-ex].
  * Betancourt  _et al._ [2017] M. Betancourt _et al._ (MINERvA), Phys. Rev. Lett. 119, 082001 (2017), arXiv:1705.03791 [hep-ex].
  * Lu  _et al._ [2018] X. G. Lu _et al._ (MINERvA), Phys. Rev. Lett. 121, 022504 (2018), arXiv:1805.05486 [hep-ex].
  * Ruterbories  _et al._ [2019] D. Ruterbories _et al._ (MINERvA), Phys. Rev. D 99, 012004 (2019), arXiv:1811.02774 [hep-ex].
  * Abe  _et al._ [2018b] K. Abe _et al._ (T2K), Phys. Rev. D 98, 032003 (2018b), arXiv:1802.05078 [hep-ex].
  * Cai  _et al._ [2020] T. Cai _et al._ (MINERvA), Phys. Rev. D 101, 092001 (2020), arXiv:1910.08658 [hep-ex].
  * Carneiro  _et al._ [2020] M. F. Carneiro _et al._ (MINERvA), Phys. Rev. Lett. 124, 121801 (2020), arXiv:1912.09890 [hep-ex].
  * Abe  _et al._ [2020a] K. Abe _et al._ (T2K), Phys. Rev. D 101, 112004 (2020a), arXiv:2004.05434 [hep-ex].
  * Abe  _et al._ [2020b] K. Abe _et al._ (T2K), Phys. Rev. D 101, 112001 (2020b), arXiv:2002.09323 [hep-ex].
  * Abratenko  _et al._ [2020a] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. D 102, 112013 (2020a), arXiv:2010.02390 [hep-ex].
  * Abratenko  _et al._ [2020b] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. Lett. 125, 201803 (2020b), arXiv:2006.00108 [hep-ex].
  * Hayato and Pickering [2021] Y. Hayato and L. Pickering, Eur. Phys. J. ST 230, 4469 (2021), arXiv:2106.15809 [hep-ph].
  * Andreopoulos  _et al._ [2015] C. Andreopoulos, C. Barry, S. Dytman, H. Gallagher, T. Golan, R. Hatcher, G. Perdue, and J. Yarba, Preprint (2015), arXiv:1510.05494 [hep-ph].
  * Abed Abud  _et al._ [2021] A. Abed Abud _et al._ (DUNE), Instruments 5, 31 (2021), arXiv:2103.13910 [physics.ins-det].
  * Stowell  _et al._ [2017] P. Stowell _et al._ , JINST 12, P01016 (2017), arXiv:1612.07393 [hep-ex].
  * Patrick  _et al._ [2018] C. E. Patrick _et al._ (MINERvA), Phys. Rev. D 97, 052002 (2018), arXiv:1801.01197 [hep-ex].
  * Bashyal [2021] A. Bashyal, “DUNE and MINERvA Flux Studies and a Measurement of the Charged-Current Quasielastic Antineutrino Scattering Cross Section with $<$$E_{\nu}$ $>$$\sim$ 6 GeV on a CH Target,” Ph.D. thesis, Oregon State University (2021).
  * Bashyal  _et al._ [2022] A. Bashyal _et al._ (MINERvA), “High-Statistics Measurement of Antineutrino Quasielastic-like scattering at $E_{\nu}\sim 6$ GeV on a Hydrocarbon Target,” In preparation (2022).
  * Rodrigues  _et al._ [2016] P. A. Rodrigues _et al._ (MINERvA), Phys. Rev. Lett. 116, 071802 (2016), arXiv:1511.05944 [hep-ex].
  * Ascencio  _et al._ [2021] M. V. Ascencio _et al._ (MINERvA), preprint (2021), arXiv:2110.13372 [hep-ex].
  * Mertig  _et al._ [1991] R. Mertig, M. Bohm, and A. Denner, Comput. Phys. Commun. 64, 345 (1991).
  * Shtabovenko  _et al._ [2016] V. Shtabovenko, R. Mertig, and F. Orellana, Comput. Phys. Commun. 207, 432 (2016).
  * Hahn and Perez-Victoria [1999] T. Hahn and M. Perez-Victoria, Comput. Phys. Commun. 118, 153 (1999).
  * Inc. [2022] W. R. Inc., “Mathematica, Version 12.2.0.0,” (2022), champaign, IL, 2022\.