text:
\begin{document}

\maketitle

\begin{abstract}

When a gauge-natural invariant variational principle is
assigned, to determine {\em canonical} covariant conservation laws,
the vertical part of
gauge-natural lifts of infinitesimal principal automorphisms --
defining infinitesimal
variations of sections of gauge-natural bundles -- must satisfy
generalized Jacobi equations
for the gauge-natural invariant Lagrangian. {\em Vice versa} all vertical parts
of gauge-natural lifts of infinitesimal principal automorphisms which
are in the kernel of
generalized Jacobi morphisms are generators of canonical covariant currents and
superpotentials. In
particular, only a few gauge-natural lifts can be considered as {\em
canonical} generators of covariant
gauge-natural physical charges.
\medskip

\noindent {\bf 2000 MSC}: 58A20,58A32,58E30,58E40,58J10,58J70.

\noindent {\em Key words}: gauge-natural bundles,
invariant variational principles, variational sequences,
generalized Jacobi morphisms, conservation laws.
\end{abstract}

\section{Introduction}

In
\cite{AnBe51,Vari1} the general program was started of
defining covariant conservation laws for field theories as generators
of infinitesimal
transformations of the basis manifold. In
the classical Lagrangian formulation of field theories the
description of symmetries and conserved quantities
amounts to define suitable (vector) densities  which generate the
conserved currents; in all
relevant physical theories this densities are found to be the
divergence of skew--symmetric
(tensor) densities, which are called {\em superpotentials} for the
conserved currents. It
is also well known that the importance of superpotentials relies on
the fact that they can
be integrated to provide conserved quantities associated with the
conserved currents {\em
via} the Stokes theorem (see \eg \cite{FeFr91} and references quoted
therein). Within such a procedure,
the generalized Bianchi identities for geometric field
theories - introduced by Bergman to get (after an integration by
parts procedure) a consistent
equation between (local) divergences within the first variation formula -
are in fact necessary and (locally) sufficient conditions for the
conserved current
$\epsilon$ to be not only closed but also the divergence of a
skew-symmetric (tensor) density (a superpotential)
along solutions of the  Euler--Lagrange equations.
However, to ``covariantize'' such a derivation of Bianchi
identities and superpotentials, background metrics or
(fibered) connections have to be fixed {\em a priori} (see \eg
\cite{Vari2,FFP01,FeFr91} and the wide literature quoted therein).
The outcoming of such an {\em ad hoc} procedure is, notably, a
triviality result concerning
{\em existence of global superpotentials} for gauge-natural field
theories despite of the topology of the basis of the gauge-natural
bundle (see \eg the review in \cite{Mat03}).

We shall show that, when a gauge-natural invariant variational principle is
assigned, to determine {\em canonical} (\ie completely determined by
the variational problem and its invariance properties) covariant
conservation laws, the vertical part of
gauge-natural lifts of infinitesimal principal automorphisms must
satisfy generalized Jacobi equations
for the gauge-natural invariant Lagrangian. {\em Vice versa} all vertical parts
of gauge-natural lifts of infinitesimal principal automorphisms which
are in the kernel of the
generalized Jacobi morphism are generators of canonical covariant currents and
superpotentials.

This result is the outcoming of the following
facts:
\begin{enumerate}
\item the role played by the invariance of a
given variational problem of order $s$ on a fibered manifold $\bY \to
\bX$  with respect to the contact structure induced by the affine
fiberings $\pi^{s+1}_{s}: J_{s+1}\bY \to J_{s}\bY$ and its encoding
through the Krupka's {\em finite order  variational sequence}
language. It is fundamental to stress that such an invariance has to
reflect too in a precise way on the nature of the conserved Noether
currents associated with a given gauge-natural invariant Lagrangian
(see Remark \ref{fundRem});
\item the Noether Theorems ({\em both of
them}) take a quite particular form in the case of gauge-natural
Lagrangian field theories (see \eg  \cite{FFP01,Mat03}) due to the
fact that the generalized Lie derivative of sections of the
gauge-natural bundles has special linearity properties and it is
related with the vertical part (with respect to the splitting induced
by the contact structure) of gauge-natural lifts of  right-invariant
(also called principal) infinitesimal automorphisms of the underlying
principal bundle (structure bundle);
\item the {\em second variation}
of the action functional can be conveniently represented in the
finite order variational sequence framework in terms of iterated {\em variational
Lie derivatives} (the quotient Lie derivative of {\em variational
morphisms} for  first introduced in \cite{FPV98a}) with respect to
vertical parts  of gauge-natural lifts of principal infinitesimal
automorphisms. In particular, {\em by resorting to the Second Noether
Theorem}, in \cite{FrPa00,FrPa01} the second variation has been
related with the generalized Jacobi morphism and in \cite{PaWi03} the
relation of the kernel of generalized {\em gauge-natural} Jacobi
morphism with the kernel of a fundamental morphism, the {\em
(Bergman-)Bianchi} morphism, has been explicitly clarified in order
to characterize Bianchi identities for geometric field theories in
terms of a special class of gauge-natural lifts of infinitesimal
principal automorphisms, namely those which have their vertical part
in the kernel of the generalized gauge-natural Jacobi
morphism.
\end{enumerate}

Here we claim that the indeterminacy
appearing in the derivation of gauge-natural conserved charges (see
the interesting papers \cite{GoMa03,Mat03}) - \ie the difficulty of
relating in a natural way  infinitesimal gauge transfomations with
infinitesimal transformations of the basis manifold (\eg of
space-time) - can be solved by requiring the second variation to be
zero too. Historically Jacobi equations were related to
the so--called {\em accessory problem} (see, {\em e.g.}
\cite{Car82,Run66}), where they are directly obtained as
the variation of the Euler--Lagrange equations
of a given Lagrangian. Thus they can be characterized {\em via} the
Second Noether
Theorem.

\section{Variational sequences on gauge-natural bundles}

Our framework is a fibered manifold $\pi : \bY \to \bX$,
with $\dim \bX = n$ and $\dim \bY = n+m$.
For $s \geq q \geq 0$ integers we are concerned with the $s$--jet
space $J_s\bY$ of $s$--jet prolongations of
(local) sections
of $\pi$ (see \eg \cite{KMS93,MaMo83a,Sau89}); in particular, we set
$J_0\bY \equiv \bY$. We recall the natural
fiberings
$\pi^s_q: J_s\bY \to J_q\bY$, $s \geq q$, $\pi^s: J_s\bY \to \bX$, and,
among these, the {\em affine\/} fiberings $\pi^{s}_{s-1}$.
We denote with $V\bY$ the vector subbundle of the tangent
bundle $T\bY$ of vectors on $\bY$ which are vertical with respect
to the fibering $\pi$.

Charts on $\bY$ adapted to $\pi$ are denoted by $(x^\sig ,y^i)$.  Greek
indices $\sig ,\mu ,\dots$ run from $1$ to $n$ and they label basis
coordinates, while
Latin indices $i,j,\dots$ run from $1$ to $m$ and label fibre coordinates,
unless otherwise specified.
We denote multi--indices of dimension $n$ by boldface Greek letters such as
$\balp = (\alp_1, \dots, \alp_n)$, with $0 \leq \alp_\mu$,
$\mu=1,\ldots,n$; by an abuse
of notation, we denote with $\sig$ the multi--index such that
$\alp_{\mu}=0$, if $\mu\neq \sig$, $\alp_{\mu}= 1$, if
$\mu=\sig$.
We also set $|\balp| \byd \alp_{1} + \dots + \alp_{n}$ and $\balp ! \byd
\alp_{1}! \dots \alp_{n}!$.
The charts induced on $J_s\bY$ are denoted by $(x^\sig,y^i_{\balp})$, with $0
\leq |\balp| \leq s$; in particular, we set $y^i_{\bf{0}}
\equiv y^i$. The local vector fields and forms of $J_s\bY$ induced by
the above coordinates are denoted by $(\der^{\balp}_i)$ and $(d^i_{\balp})$,
respectively.

In the theory of variational sequences a fundamental role is played by the
{\em contact maps\/} on jet spaces (see \cite{Kru90,MaMo83a,Vit98}).
Namely, for $s\geq 1$, we consider the natural complementary fibered
morphisms over $J_s\bY \to J_{s-1}\bY$
\beq
\mathcal{D} : J_s\bY \ucar{\bX} T\bX \to TJ_{s-1}\bY \,,
\qquad
\vartht : J_{s}\bY \ucar{J_{s-1}\bY} TJ_{s-1}\bY \to VJ_{s-1}\bY \,,
\eeq
with coordinate expressions, for $0 \leq |\balp| \leq s-1$, given by
\beq
\mathcal{D} &= d^\lam\ten {\mathcal{D}}_\lam = d^\lam\ten
(\der_\lam + y^j_{\balp+\lam}\der_j^{\balp}) \,,
\vartht &= \vartht^j_{\balp}\ten\der_j^{\balp} =
(d^j_{\balp}-y^j_{{\balp}+\lam}d^\lam)
\ten\der_j^{\balp} \,.
\eeq
which induce the following natural splitting:
\bEq
\label{jet connection}
J_{s}\bY\ucar{J_{s-1}\bY}T^*J_{s-1}\bY =\left(
J_s\bY\ucar{J_{s-1}\bY}T^*\bX\right) \oplus\cC^{*}_{s-1}[\bY]\,,
\eEq
where $\cC^{*}_{s-1}[\bY] \simeq J_{s}\bY \ucar{J_{s-1}\bY} V^{*}J_{s-1}\bY$.

If $f: J_{s}\bY \to \R$ is a function, then we set
$D_{\sig}f$ $\byd \mathcal{D}_{\sig} f$,
$D_{\balp+\sig}f$ $\byd D_{\sig} D_{\balp}f$, where $D_{\sig}$ is
the standard {\em formal derivative}.
Given a vector field $\Xi : J_{s}\bY \to TJ_{s}\bY$, the splitting
\eqref{jet connection} yields $\Xi \, \com \, \pi^{s+1}_{s} = \Xi_{H}
+ \Xi_{V}$
where, if $\Xi = \Xi^{\gam}\der_{\gam} + \Xi^i_{\balp}\der^{\balp}_i$, then we
have $\Xi_{H} = \Xi^{\gam}D_{\gam}$ and
$\Xi_{V} = (\Xi^i_{\balp} - y^i_{\balp + \gam}\Xi^{\gam})
\der^{\balp}_{i}$. We shall call $\Xi_{H}$ and $\Xi_{V}$ the
horizontal and the vertical part of $\Xi$, respectively.

The splitting
\eqref{jet connection} induces also a decomposition of the
exterior differential on $\bY$,
$(\pi^{s}_{s-1})^*\com \,d = d_H + d_V$, where $d_H$ and $d_V$
are defined to be the {\em horizontal\/} and {\em vertical differential\/}.
The action of $d_H$ and $d_V$ on functions and $1$--forms
on $J_s\bY$ uniquely characterizes $d_H$ and $d_V$ (see, {\em e.g.},
\cite{Sau89,Vit98} for more details).
A {\em projectable vector field\/} on $\bY$ is defined to be a pair
$(\Xi,\xi)$, where $\Xi:\bY \to T\bY$ and $\xi: \bX \to T\bX$
are vector fields and $\Xi$ is a fibered morphism over $\xi$.
If there is no danger of confusion, we will denote simply by $\Xi$ a
projectable vector field $(\Xi,\xi)$.
A projectable vector field $(\Xi,\xi)$
can be prolonged by the flow functor to a projectable vector field
$(j_{s}\Xi, \xi)$, the coordinate expression of which can be found \eg in
\cite{Kru90,MaMo83a,Sau89,Vit98};
in particular, we have the following expressions
$(j_{s}\Xi)_{H} = \xi^{\sig} \, D_{\sig}$,
$(j_{s}\Xi)_{V} = D_{\balp}(\Xi_{V})^i \, \der_i^{\balp}$,
with $(\Xi_{V})^i = \xi^i - \, y^i_{\sig}\xi^{\sig}$, for the
horizontal and the vertical part of $j_{s}\Xi$, respectively.
 From now on, by an abuse of notation, we will write simply $j_{s}\Xi_{H}$ and
$j_{s}\Xi_{V}$.
In particular, we stress that $j_{s}\Xi_{V}$ can be seen as a fibered
morphism: $j_{s}\Xi_{V}: J_{s+1}\bY\ucar{J_{s}\bY}J_{s}\bY\to
J_{s+1}\bY\ucar{J_{s}\bY}J_{s}V\bY$.

\subsection{Gauge-natural bundles}

In the following, we shall develop a suitable geometrical setting
which enables us to
define and investigate the fundamental concept of
conserved quantity in gauge-natural Lagrangian field theories.

An important generalization of natural field theories
\cite{Tra67} to gauge fields theories passed through
the concept of jet prolongation of a principal bundle and the
introduction of a very important geometric construction, namely the
gauge-natural bundle
functor \cite{Ec81,KMS93}.

Let $\bP\to\bX$ be a principal bundle with structure group $\bG$.
Let $r\leq k$ be integers and $\bW^{(r,k)}\bP$ $\byd$
$J_{r}\bP\ucar{\bX}L_{k}(\bX)$,
where $L_{k}(\bX)$ is the bundle of $k$--frames
in $\bX$ \cite{Ec81,KMS93}, $\bW^{(r,k)}\bG \byd J_{r}\bG\odot GL_{k}(n)$
the semidirect product with respect to the action of $GL_{k}(n)$
on $J_{r}\bG$ given by the
jet composition and $GL_{k}(n)$ is the group of $k$--frames
in $\R^{n}$. Here we denote by $J_{r}\bG$ the space of
$(r,n)$-velocities on $\bG$.
The bundle $\bW^{(r,k)}\bP$ is a principal bundle over $\bX$ with
structure group
$\bW^{(r,k)}\bG$.
The right action of $\bW^{(r,k)}\bG$ on the fibers of $\bW^{(r,k)}\bP$
is defined by the composition of jets (see, {\em e.g.},
\cite{KMS93}).

\bDf
The principal bundle $\bW^{(r,k)}\bP$ (resp. the Lie group $\bW^{(r,k)}\bG$)
is said to be the {\em gauge-natural prolongation of order $(r,k)$
of $\bP$ (resp. of $\bG$)}.
\END\eDf

\bDf
We define the {\em vector}
bundle over $\bX$ of right--invariant infinitesimal automorphisms of $\bP$
by setting $\cA = T\bP/\bG$.

For $r\leq k$ we also define the {\em vector} bundle  over $\bX$ of
right invariant
infinitesimal automorphisms of $\bW^{(r,k)}\bP$ by setting
$\cA^{(r,k)} \byd T\bW^{(r,k)}\bP/\bW^{(r,k)}\bG$.
\END\eDf

Let $\bF$ be any manifold and $\zet:\bW^{(r,k)}\bG\ucar{}\bF\to\bF$ be
a left action of $\bW^{(r,k)}\bG$ on $\bF$. There is a naturally defined
right action of $\bW^{(r,k)}\bG$ on $\bW^{(r,k)}\bP \times \bF$ so that
  we can associate in a standard way
to $\bW^{(r,k)}\bP$ the bundle, on the given basis $\bX$,
$\bY_{\zet} \byd \bW^{(r,k)}\bP\times_{\zet}\bF$ \cite{Ec81,KMS93}.

\bDf
We say $(\bY_{\zet},\bX,\pi_{\zet};\bF,\bG)$ to be the
{\em gauge-natural bundle} of order
$(r,k)$ associated to the principal bundle $\bW^{(r,k)}\bP$
by means of the left action $\zet$ of the group
$\bW^{(r,k)}\bG$ on the manifold $\bF$.
\END\eDf

\bRm
A principal automorphism $\Phi$ of $\bW^{(r,k)}\bP$ induces an
automorphism of the gauge-natural bundle by:
\bEq
\Phi_{\zet}:\bY_{\zet}\to\bY_{\zet}: [(j^{x}_{r}\gam,j^{0}_{k}t),
\hat{f}]_{\zet}\mto [\Phi(j^{x}_{r}\gam,j^{0}_{k}t),
\hat{f}]_{\zet}\,,
\eEq
where $\hat{f}\in \bF$ and $[\cdot, \cdot]_{\zet}$ is the equivalence class
induced by the action $\zet$.
\END\eRm

Denote by $\cT_{\bX}$ and $\cA^{(r,k)}$ the sheaf of
vector fields on $\bX$ and the sheaf of right invariant vector fields
on $\bW^{(r,k)}\bP$, respectively. A functorial mapping
$\mathfrak{G}$ is defined
which lifts any right--invariant local automorphism $(\Phi,\phi)$ of the
principal bundle $W^{(r,k)}\bP$ into a unique local automorphism
$(\Phi_{\zet},\phi)$ of the associated bundle $\bY_{\zet}$.
Its infinitesimal version associates to any $\bar{\Xi} \in \cA^{(r,k)}$,
projectable over $\xi \in \cT_{\bX}$, a unique {\em projectable} vector field
$\hat{\Xi} \byd \mathfrak{G}(\bar{\Xi})$ on $\bY_{\zet}$ in the
following way:
\bEq
\mathfrak{G} : \bY_{\zet} \ucar{\bX} \cA^{(r,k)} \to T\bY_{\zet} \,:
(\by,\bar{\Xi}) \mto \hat{\Xi} (\by) \,,
\eEq
where, for any $\by \in \bY_{\zet}$, one sets: $\hat{\Xi}(\by)=
\frac{d}{dt} [(\Phi_{\zet \,t})(\by)]_{t=0}$,
and $\Phi_{\zet \,t}$ denotes the (local) flow corresponding to the
gauge-natural lift of $\Phi_{t}$.

This mapping fulfils the following properties:
\begin{enumerate}
\item {\em $\mathfrak{G}$ is linear over $id_{\bY_{\zet}}$};
\item we have $T\pi_{\zet}\circ\mathfrak{G} = id_{T\bX}\circ
\bar{\pi}^{(r,k)}$,
where $\bar{\pi}^{(r,k)}$ is the natural projection
$\bY_{\zet}\ucar{\bX}
\cA^{(r,k)} \to T\bX$;
\item $\mathfrak{G}$ is a homomorphism of Lie algebras: for any pair
$(\bar{\Lam},\bar{\Xi})$ of vector fields in
$\cA^{(r,k)}$, we have
$
\mathfrak{G}([\bar{\Lam},\bar{\Xi}]) = [\mathfrak{G}(\bar{\Lam}),
\mathfrak{G}(\bar{\Xi})]$;
\item in coordinates $
\mathfrak{G} = d^\mu \ten \der_\mu + d^{A}_{\bnu}
\ten (\cZ^{i\bnu}_{A} \der_{i}) + d^{\nu}_{\blam}
\ten (\cZ^{i\blam}_{\nu} \der_{i})$,
with $0<|\bnu|<k$, $1<|\blam|<r$ and
$\cZ^{i\bnu}_{A}$, $\cZ^{i\blam}_{\nu}$ $\in C^{\infty}(\bY_{\zet})$
are suitable functions which depend on the bundle, precisely on the
fibers (see \cite{KMS93}).
\end{enumerate}

\bDf
The map $\mathfrak{G}$ is called the {\em gauge-natural lifting
functor}.
The  {\em projectable} vector field $(\hat{\Xi},\xi)\equiv
\mathfrak{G}((\bar{\Xi},\xi))$ is
called the {\em
gauge-natural lift} of $(\bar{\Xi},\xi)$ to the bundle $\bY_{\zet}$.\END
\eDf

We shall consider  {\em variation vector fields} which are vertical
parts of gauge-natural lifts of infinitesimal principal
automorphisms. We recall that, due to the very definition of
generalized Lie derivative of sections of gauge-natural bundles,
variation vector fields are in fact formal Lie derivatives of
sections with respect to gauge-natural lifts (see the item $4$ in the
following). This will enables us to realize morphisms such as the
Jacobi or the Bianchi morphisms in a very suitable way for our
purposes.

\bDf({\bf Lie derivative of sections.})
Let $\gam$ be a (local)
section of the gauge-natural bundle $\bY_{\zet}$, $\bar{\Xi}$
$\in \cA^{(r,k)}$ and $\hat\Xi$ its gauge-natural lift.
Following \cite{KMS93} we
define the {\em
generalized Lie derivative} of $\gam$ along the vector field
$\hat{\Xi}$ to be the (local) section $\pounds_{\bar{\Xi}} \gam : \bX
\to V\bY_{\zet}$,
by setting:
$\pounds_{\bar{\Xi}} \gam = T\gam \circ \xi - \hat{\Xi} \circ \gam$.
\END\eDf

\bRm
This section is a vertical prolongation of $\gam$, \ie it satisfies
the property: $\nu_{\bY_{\zet}} \circ \pounds_{\Xi} \gam = \gam$,
where $\nu_{\bY_{\zet}}$ is the projection
$\nu_{\bY_{\zet}}: V\bY_{\zet} \to \bY_{\zet}$.
Its coordinate expression is given by
$(\pounds_{\bar{\Xi}}\gam)^{i} = \xi^{\sig} \der_{\sig} \gam ^{i} -
\hat\Xi^{i}(\gam)$. As customary we denote it by $\pounds_{\bar{\Xi}}
\gam$ and not by $\pounds_{\hat{\Xi}} \gam$ because of the functorial
correspondence between $\bar{\Xi}$ and $\hat{\Xi}$.
\eRm

\bRm\label{lie}
The Lie derivative operator acting on sections of gauge-natural
bundles satisfies the following
properties:
\begin{enumerate}\label{lie properties}
\item for any vector field $\bar{\Xi} \in \cA^{(r,k)}$, the
mapping $\gam \mto \pounds_{\bar{\Xi}}\gam$
is a first--order quasilinear differential operator;
\item for any local section $\gam$ of $\bY_{\zet}$, the mapping
$\bar{\Xi} \mto \pounds_{\bar{\Xi}}\gam$
is a linear differential operator;
\item we can regard $\pounds_{\bar{\Xi}}: J_{1}\bY_{\zet} \to V\bY_{\zet}$
as a morphism over the
basis $\bX$. In this case it is meaningful to consider the (standard)
jet
prolongation of $\pounds_{\bar{\Xi}}$, denoted by
$j_{s}\pounds_{\bar{\Xi}}:
J_{s+1}\bY_{\zet} \to VJ_{s}\bY_{\zet}$.
By using the canonical
isomorphism $VJ_{s}\bY_{\zet}\simeq J_{s}V\bY_{\zet}$, we have
$\pounds_{\bar{\Xi}}(j_{s}\gam) = j_{s} (\pounds_{\bar{\Xi}} \gam)$,
for any (local) section $\gam$ of $\bY_{\zet}$ and for any (local)
vector field $\bar{\Xi}\in \cA^{(r,k)}$.

\item as a consequence of linearity properties of gauge-natural
lifts, we have
$j_{s}\hat{\Xi}_{V}(\gam)=-\pounds_{j_{s}\bar{\Xi}}\gam$. In
particular,  we can consider the Lie derivative of sections,
$\pounds$, as a bundle morphism \cite{PaWi03}:
\beq
\pounds: J_{s+1}(\bY_{\zet} \ucar{\bX}
\cA^{(r,k)}) \to J_{s+1}\bY_{\zet}\ucar{J_{s}\bY_{\zet}}VJ_{s}\bY_{\zet}\,.
\eeq

\end{enumerate}
\END\eRm

\subsection{Variational Lie derivative of variational morphisms}

For the sake of simplifying notation, sometimes, we will omit the
subscript $\zet$, so
that all our considerations shall refer to $\bY$ as a gauge-natural
bundle as defined above.

We shall be here concerned with some distinguished sheaves of forms on jet
spaces \cite{Kru90,Sau89,Vit98}. We shall in particular follow
notation given in \cite{Vit98} to which the
reader is referred for details.
  For $s \geq 0$, we consider the standard sheaves $\For^{p}_{s}$
of $p$--forms on $J_s\bY$.
  For $0 \leq q \leq s $, we consider the sheaves $\Hor^{p}_{(s,q)}$ and
$\Hor^{p}_{s}$ of {\em horizontal forms} with respect to the
projections $\pi^s_q$ and $\pi^s_0$, respectively.
  For $0 \leq q < s$, we consider the subsheaves $\Con^{p}_{(s,q)}
\sub \Hor^{p}_{(s,q)}$ and $\Con^{p}{_s} \sub
\Con^{p}_{(s+1,s)}$ of {\em contact forms}, \ie horizontal forms
valued into $\cC^{*}_{s}[\bY]$(they have the property of vanishing
along any section of the gauge-natural bundle).

According to \cite{Kru90,Vit98}, the fibered splitting
\eqref{jet connection} yields the {\em sheaf splitting}
$\Hor^{p}_{(s+1,s)}$ $=$ $\bigoplus_{t=0}^p$
$\Con^{p-t}_{(s+1,s)}$ $\wed\Hor^{t}_{s+1}$, which restricts to the inclusion
$\For^{p}_s$ $\sub$ $\bigoplus_{t=0}^{p}$
$\Con^{p-t}{_s}\wed\Hor^{t,}{_{s+1}^{h}}$,
where $\Hor^{p,}{_{s+1}^{h}}$ $\byd$ $h(\For^{p}_s)$ for $0 < p\leq
n$ and the surjective map
$h$ is defined to be the restriction to $\For^{p}_{s}$ of the projection of
the above splitting onto the non--trivial summand with the highest
value of $t$. By an abuse of notation, let us denote by $d\ker h$ the sheaf
generated by the presheaf $d\ker h$ in the standard way.
We set $\Thd^{*}_{s}$ $\byd$ $\ker h$ $+$
$d\ker h$.

The $s$--th order quotient
{\em variational sequence} associated with the fibered manifold $\bY\to\bX$:
\beq
\diagramstyle[size=1.3em]
\begin{diagram}
0 & \rTo & \R_{Y} & \rTo & \For^{0}_s & \rTo^{\cE_{0}} &
\For^{1}_s/\Thd^{1}_s & \rTo^{\cE_{1}} & \For^{2}_s/\Thd^{2}_s &
\rTo^{\cE_{2}} &
\dots & \rTo^{\cE_{I-1}} & \For^{I}_s/\Thd^{I}_s & \rTo^{\cE_{I}} &
\For^{I+1}_s & \rTo^{d} & 0\,,
\end{diagram}
\eeq
has been introduced by Krupka \cite{Kru90}.

To the aim of characterizing some fundamental morphisms for the
calculus of variations  as sections of quotient sheaves and as
corresponding differential quotient morphisms, let us consider the
truncated variational sequence:
\beq
\diagramstyle[size=1.3em]
\begin{diagram}
0 &\rTo & \R_{Y} &\rTo & \Var^{0}_s & \rTo^{\cE_0} &
\Var^{1}_{s} & \rTo^{\cE_{1}} & \dots  & \rTo^{\cE_{n}} &
\Var^{n+1}_{s}  & \rTo^{\cE_{n+1}} & \cE_{n+1}(\Var^{n+1}_{s})
& \rTo^{\cE_{n+2}} &
0 \,,
\end{diagram}
\eeq
where, following \cite{Vit98}, the sheaves $\Var^{p}_{s}\byd
\Con^{p-n}_{s}\wed\Hor^{n,}{_{s+1}^h}/h(d\ker h)$ with $0\leq p\leq n+2$ are
suitable representations of the corresponding quotient
sheaves in the variational sequence by means of sheaves of sections of tensor
bundles.

Let $\alp\in\Con^{1}_s\wed\Hor^{n,}{_{s+1}^h}
\sub \Var^{n+1}_{s+1}$. Then there is a unique pair of
sheaf morphisms (\cite{Kol83,KoVi03,Vit98})
\bEq\label{first variation}
E_{\alp} \in \Con^{1}_{(2s,0)}\wed\Hor^{n,}{_{2s+1}^{h}} \,,
\qquad
F_{\alp} \in \Con^{1}_{(2s,s)} \wed \Hor^{n,}{_{2s+1}^h} \,,
\eEq
such that
$(\pi^{2s+1}_{s+1})^*\alp=E_{\alp}-F_{\alp}$,
and $F_\alp$ is {\em locally} of the form $F_{\alp} = d_{H}p_{\alp}$,
with $p_{\alp}
\in \Con^{1}_{(2s-1,s-1)}\wed\Hor^{n-1}{_{2s}}$.

\bDf
Let $\gam \in \For^{n+1}_{s}$.
The morphism $E_{h(\gam)}\in\Var^{n+1}_{s}$ is called the
{\em generalized Euler--Lagrange morphism} associated with
$\gam$.
\END
\eDf

Let $\eta\in\Con^{1}_{s}\wed\Con^{1}_{(s,0)}\wed\Hor^{n,}{_{s+1}^{h}}\sub
\Var^{n+2}_{s+1}$,
then there is a unique morphism
\bEq\label{kappa}
K_{\eta} \in
\Con^{1}_{(2s,s)}\otimes\Con^{1}_{(2s,0)}\wed\Hor^{n,}{_{2s+1}^{h}}
\,,
\eEq
such that, for all $\Xi:\bY\to V\bY$,
$
E_{{j_{s}\Xi}\rfloor \eta} = C^{1}_{1} (j_{2s}\Xi\ten K_{\eta})$,
where $C^1_1$ stands for tensor
contraction on the first factor and $\rfloor$ denotes inner product
(see \cite{KoVi03,Vit98}).
Furthermore, there is a unique pair of sheaf morphisms
\bEq\label{second}
H_{\eta} \in
\Con^{1}_{(2s,s)}\wed\Con^{1}_{(2s,0)}\wed\Hor^{n,}{_{2s+1}^{h}} \,,
\quad
G_{\eta} \in \Con^{2}_{(2s,s)}\wed\Hor^{n,}{_{2s+1}^{h}} \,,
\eEq
such that
${(\pi^{2s+1}_{s+1})}^*\eta = H_{\eta} - G_{\eta}$ and $H_{\eta}
= \frac{1}{2} \, A(K_{\eta})$,
where $A$ stands for antisymmetrisation.
Moreover, $G_{\eta}$ is {\em locally} of the type $G_{\eta} = d_H q_{\eta}$,
where
$q_{\eta} \in \Con^{2}_{(2s-1,s-1)}\wed\Hor^{n-1}{_{2s}}$, hence
$[\eta]=[H_{\eta}]$ \cite{KoVi03,Vit98}.

\bDf
Let $\gam \in \For^{n+1}_{s}$.
The morphism $H_{hd\gam}\equiv H_{[\cE_{n+1}(\gam)]}$, where square
brackets denote equivalence class, is called the {\em
generalized Helmholtz  morphism}.
\END
\eDf

The standard Lie derivative of fibered morphisms with respect to a
projectable vector field $j_{s}\Xi$ passes to the quotient in the
variational sequence, thus defining a new quotient operator
(introduced in \cite{FPV98a}), the {\em variational Lie derivative}
$\cL_{j_{s}\Xi}$, acting on equivalence classes of fibered morphisms
which are sections of the quotient sheaves in the variational
sequence. Thus variational Lie derivatives of generalized Lagrangians
or Euler--Lagrange morphisms can be conveniently represented as
equivalence classes in $\Var^{n}_{s}$ and $\Var^{n+1}_{s}$. In
particular, the following two results hold true \cite{FPV98a}.

\bTh\label{noether I}
Let $[\alp] = h(\alp)$ $\in$ $\Var^{n}_{s}$. Then we
have {\em locally} (up to pull-backs)
\beq

\cL_{j_{s}\Xi}(h(\alp)) =
\Xi_{V} \rfloor \cE_{n}(h(\alp))+
d_{H}(j_{2s}\Xi_{V} \rfloor p_{d_{V}h(\alp)}+ \xi \rfloor h(\alp))\,.
\eeq
\eTh

\bTh\label{GeneralJacobi}
Let $\alp\in\For^{n+1}_{s}$. Then we have {\em globally} (up to pull-backs)
\beq

\cL_{j_{s}\Xi} [\alp] =
\cE_{n}({j_{s+1}\Xi_{V} \rfloor h(\alp)}) +
C^{1}_{1}(j_{s}\Xi_{V}\ten K_{hd\alp}) \,.
\eeq
\eTh

\subsection{Generalized {\em gauge-natural} Jacobi morphisms}

We recall some previous results concerning
the representation of {\em generalized gauge-natural Jacobi morphisms} in
variational sequences and their relation with the second variation of a
generalized gauge-natural invariant Lagrangian \cite{PaWi03}.

\bDf\label{var}
Let $\alp:J_{s}\bY\to
\owed{p}T^*J_{s}\bY$. Let $\psi^{k}_{t_{k}}$, with $1\leq k\leq i$, be the
flows generated by an $i$--tuple
$(\Xi_{1},\ldots,\Xi_{i})$ of (vertical, although actually it is enough that
they are projectable) vector fields on $\bY$ and let $\Gam_{i}$ be the $i$--th
{\em formal variation} generated by the $\Xi_{k}$'s (to which we shall
refer as variation vector fields)  and defined,
for each $\by\in\bY$, by $\Gam_{i}(t_{1},\ldots,t_{i})(\by)=
\psi^{i}_{t_{i}}\circ \ldots \circ \psi^{1}_{t_{1}}(\by)$.
We define the $i$--th formal variation of the morphism $\alp$ to
be
\bEq
\del^{i}\alp \byd \frac{\der^{i}}{\der t_{1}\ldots
\der t_{i}}\big |_{t_{1},\ldots,
t_{i}=0}(\alp\circ j_{s}\Gam_{i}(t_{1},\ldots,t_{i})(\by)) \,.\END
\eEq
\eDf

The following Lemma states the relation between the $i$--th formal variation
of a morphism and its iterated Lie derivative
\cite{FPV02,PaWi03}.

\bLm\label{LIE}
Let $\alp: J_{s}\bY\to
\owed{p}T^*J_{s}\bY$ and $L_{j_{s}\Xi_{k}}$ be the Lie derivative
operator acting on differential fibered morphism.

Let
$\Gam_{i}$ be the $i$--th formal variation generated by variation
vector fields $\Xi_{k}$, $1\leq k\leq i$ on $\bY$. Then we have
\bEq
\del^{i} \alp = L_{j_{s}\Xi_{1}} \ldots L_{j_{s}\Xi_{i}} \alp \,.
\eEq
\eLm

Let  $\alp\in (\Var^{n}_{s})_{\bY}$. The operator $\del^{i}$ passes
to the quotient in the variational sequence. We shall call the
quotient
operator the {\em
$i$--th variational vertical derivative}.  We have
$\del^{i}[\alp]\byd [\del^{i}\alp] =[L_{\Xi_{i}} \ldots
L_{\Xi_{1}}\alp]=\cL_{\Xi_{i}} \ldots
\cL_{\Xi_{1}}[\alp]$.

Let now variation vector fields be vertical parts of  gauge-natural lifts.
By resorting to the Second Noether Theorem, we have the following
characterization of the second variational vertical derivative of a
generalized Lagrangian \cite{PaWi03} which in fact enable us to
relate the second variation with the morphism $K_{\eta}$ defined by
Eq. \eqref{kappa} (for $\eta =hd\del\lam$).

First of all we fix some preliminary properties of gauge-natural lifts.
\bLm
Let $j_{s}\hat{\Xi}$ be the $s$-jet prolongation of $\hat{\Xi}$
which is a vector field on $J_{s}\bY_{\zet}$. It turns out then that
$j_{s}\mathfrak{G}(\bar{\Xi})=\mathfrak{G}(j_{s}\bar{\Xi})$.
\eLm

\bPf
Owing to linearity properties of
the Lie derivative of sections of gauge-natural bundles and since
$j_{s}\hat{\Xi}_{V}=-\pounds_{j_{s}\bar{\Xi}}$, the statement
is a consequence of Proposition 15.5 in \cite{KMS93}. \QED\ePf

Let
$j_{s}{\hat{\Xi}}_{V}$ be the vertical part according to the
splitting \eqref{jet connection}. We shall denote
by $j_{s}\bar{\Xi}_{V}$ the induced section of the vector bundle
$\cA^{(r+s,k+s)}$ following the Lemma above. The set of all sections
of this kind defines a vector subbundle of $J_{s}\cA^{(r,k)}$ which,
by an abuse of notation,  we
shall denote by
$V J_{s}\cA^{(r,k)}$.
Let $j_{s}{\hat{\Xi}}_{V}$ be variation vector fields and let
$\del^{2}_{\mathfrak{G}}\lam$ be the variation of $\lam$ with respect
to such variation vector fields.

\bPr\label{x}
Let $\lam\in (\Var^{n}_{s})_{\bY}$ and let
$\mathfrak{G}(\bar{\Xi})_{V}$ be a variation vector
field. Let $\chi(\lam,\mathfrak{G}(\bar{\Xi})_{V})\byd
C^{1}_{1} (j_{2s}\hat{\Xi}\ten
K_{hd\cL_{j_{2s}\bar{\Xi}_V}\lam})\equiv E_{j_{s}\hat{\Xi}\rfloor
hd\cL_{j_{2s+1}\bar{\Xi}_V}\lam}$; then we have
\bEq
\del^{2}_{\mathfrak{G}}\lam = [\cE_{n}(j_{2s}\Xi \rfloor h\del\lam)
+\cJ(\lam,\mathfrak{G}(\bar{\Xi})_{V})] \,,
\eEq
where $[\, ]$ denotes the equivalence class in the variational
sequence, while \\ $\cJ(\lam,\mathfrak{G}(\bar{\Xi})_{V})$ $\byd$
$E_{\chi(\lam,\mathfrak{G}(\bar{\Xi})_{V})}$.
\ePr

\bDf
Let $\bar{\Xi}\in \cA^{(r,k)}$.
We call the morphism $\cJ(\lam,\mathfrak{G}(\bar{\Xi})_{V})$ the {\em
gauge-natural generalized Jacobi
morphism} associated with the Lagrangian $\lam$ and the gauge-natural
lift $\mathfrak{G}(\bar{\Xi})_{V}$.
\END\eDf

The morphism $\cJ(\lam,\mathfrak{G}(\bar{\Xi})_{V})$ is a
{\em linear} morphism with respect to the projection
$J_{4s}\bY_{\zet}\ucar{\bX}VJ_{4s}\cA^{(r,k)}\to J_{4s}\bY_{\zet}$.
Notice that, {\em seen as a section of
$(\Var^{n}_{s})_{\bY\ucar{\bX}V\bY}$}, the equivalence class
$[\cE_{n}({j_{s}\bar{\Xi}_{V}
\rfloor  h(\del\lam)})]$ vanishes being a
local divergence of higher contact forms. This can also be compared
with analogous results in
\cite{FPV02}.
Thus,
as a consequence Theorem \ref{GeneralJacobi} and Proposition \ref{x},
we have the following.

\bPr\label{comparison}
Let $\del^{2}_{\mathfrak{G}}\lam$ be the variation of $\lam$ with
respect to vertical parts
of gauge-natural lifts of infinitesimal principal automorphisms. We have:
\bEq
\mathfrak{G}(\bar{\Xi})_{V}\rfloor
\cE_{n}(\mathfrak{G}(\bar{\Xi})_{V}\rfloor\cE_{n}(\lam))
=
\del^{2}_{\mathfrak{G}}\lam
=
\cE_{n}(\mathfrak{G}(\bar{\Xi})_{V}\rfloor
h(d\del\lam))\,.
\eEq
\ePr

\section{Canonical covariant conserved currents}

In the following we assume that the field equations are generated by
means of a variational principle from a Lagrangian which is
gauge-natural invariant, \ie invariant with respect to any
gauge-natural lift of infinitesimal right invariant vector fields.

\bDf\label{gn}
Let $(\hat{\Xi},\xi)$ be a projectable vector field on $\bY_{\zet}$.
Let $\lam \in \Var^{n}_{s}$
be a generalized Lagrangian. We say $\hat{\Xi}$ to be a {\em symmetry\/}
of $\lam$ if $\cL_{j_{s+1}\hat{\Xi}}\,\lam = 0$.

We say $\lam$ to be a
{\em gauge-natural invariant Lagrangian} if the gauge-natural lift
$(\hat{\Xi},\xi)$ of {\em any} vector
field $\bar{\Xi} \in \cA^{(r,k)}$ is a  symmetry for
$\lam$, \ie if $\cL_{j_{s+1}\bar{\Xi}}\,\lam = 0$.
In this case the projectable vector field
$\hat{\Xi}\equiv \mathfrak{G}(\bar{\Xi})$ is
called a {\em gauge-natural symmetry} of $\lam$.\END
\eDf

\bRm\label{fundRem}
As well known, the Second Noether Theorem deals
with invariance properties of the Euler-Lagrange equations (so-called
generalized symmetries or also Bessel-Hagen symmetries, see \eg the
fundamental papers \cite{Tra67}). Although symmetries of a Lagrangian
turn out to be also symmetries of the Euler--Lagrange morphism the
converse is not true, in general.

In particular, although for a
gauge-natural invariant Lagrangian $\lam$ we always have
$\cL_{j_{s}\bar{\Xi}}\lam=0$,   $\cL_{j_{s}\bar{\Xi}_V}\lam$  does
not need to be zero in principle; however when the second variation
$\del^{2}_{\mathfrak{G}}\lam$ is required to vanish then
$\cL_{j_{s}\bar{\Xi}_V} \cE_{n}(\lam)$ surely vanishes, \ie
$j_{s}\bar{\Xi}_V$ is a generalized or Bessel--Hagen symmetry. The
symmetries of the Euler--Lagrange morphism (Second Noether Theorem)
impose some constraints on the conserved quantities associated with
gauge-natural symmetries of $\lam$ (see \eg \cite{AnBe51}.

\END\eRm

The First Noether Theorem takes a particularly interesting
form in the case of gauge-natural Lagrangians as shown in the following.

\bPr
\label{symmetry of L}
Let $\lam \in \Var^{n}_{s}$ be a gauge-natural Lagrangian and
$(\hat{\Xi},\xi)$
a gauge-natural symmetry of $\lam$. Then we have
$
0 = - \pounds_{\bar{\Xi}} \rfloor \cE_{n}(\lam)
+d_{H}(-j_{s}\pounds_{\bar{\Xi}}
\rfloor p_{d_{V}\lam}+ \xi \rfloor \lam) $.
Suppose that
$(j_{2s+1}\sig)^{*}(- \pounds_{\bar{\Xi}} \rfloor \cE_{n}(\lam)) = 0$.
Then, the $(n-1)$--form
$\eps = - j_{s}\pounds_{\bar{\Xi}} \rfloor p_{d_{V}\lam}+ \xi \rfloor \lam$
fulfills the equation $d ((j_{2s}\sig)^{*}(\eps)) = 0$.
\ePr

If $\sig$ is a critical section for $\cE_{n}(\lam)$, \ie
$(j_{2s+1}\sig)^{*}\cE_{n}(\lam) = 0$, the above equation
admits a physical interpretation as a so-called {\em weak conservation law}
for the density associated with $\eps$.

\bDf
Let $\lam \in \Var^{n}_{s}$ be a gauge-natural Lagrangian and
$\bar{\Xi} \in \cA^{(r,k)}$. Then the sheaf morphism $\eps:
J_{2s}\bY_{\zet} \ucar{\bX} VJ_{2s}\cA^{(r,k)} \to
\cC_{2s}^{*}[\cA^{(r,k)}]\ten\cC_{0}^{*}[\cA^{(r,k)}]
\wed (\owed{n-1} T^{*}\bX)$
is said to be a {\em gauge-natural weakly conserved current\/}.\END
\eDf

\bRm\label{arbitrary1}
In general, this conserved current is not uniquely defined. In fact,
it depends on the choice of $p_{d_{V}\lam}$, which is not unique, in
general  (see
\cite{Vit98} and references quoted therein).
\END\eRm

In gauge-natural Lagrangian theories it is a well known procedure
to perform suitable integrations by
parts to decompose the conserved current $\eps$ into the sum of
a conserved current vanishing along solutions of the Euler--Lagrange equations,
the so--called {\em reduced current}, and the
formal divergence of a skew--symmetric (tensor) density called a {\em
superpotential} (which is defined modulo a divergence).
Within such a procedure, the generalized Bianchi identities
are in fact necessary and (locally) sufficient conditions for the
conserved current
$\epsilon$ to be not only closed but also the divergence of a
skew-symmetric (tensor) density
along solutions of the  Euler--Lagrange equations.

The following
Lemma is a geometric version of the integration by parts procedure
quoted above and it is based on a global decomposition formula of
vertical morphisms due to Kol\'a\v{r} \cite{Kol83}.

\bLm\label{kol}
Let $\ome(\lam,\mathfrak{G}(\bar{\Xi})_{V})\byd \pounds_{\bar{\Xi}}
\rfloor \cE_{n} (\lam): J_{2s}\bY_{\zet} \ucar{\bX} V
J_{2s}\cA^{(r,k)}
\to
\Con_{2s}^{*}[\cA^{(r,k)}]\ten\Con_{2s}^{*}[\cA^{(r,k)}]\ten\Con_{0}^{*}[\cA^{(r,k)}]\wed
(\owed{n}T^{*}\bX)$. Then we have {\em globally}
\beq
(\pi^{4s+1}_{s+1})^{*}\ome(\lam,\mathfrak{G}(\bar{\Xi})_{V}) =
\bet(\lam,\mathfrak{G}(\bar{\Xi})_{V}) +
F_{\ome(\lam,\mathfrak{G}(\bar{\Xi})_{V})}\,,
\eeq
where
$
\bet(\lam,\mathfrak{G}(\bar{\Xi})_{V})
\equiv
E_{\ome(\lam,\mathfrak{G}(\bar{\Xi})_{V})}:$ \\
$: J_{4s}\bY_{\zet} \ucar{\bX} VJ_{4s}\cA^{(r,k)}
\to\Con_{2s}^{*}[\cA^{(r,k)}]\ten
\Con_{0}^{*}[\cA^{(r,k)}]\ten\Con_{0}^{*}[\cA^{(r,k)}]\wed
(\owed{n}T^{*}\bX) $
and {\em locally}, $F_{\ome(\lam,\mathfrak{G}(\bar{\Xi})_{V})} =
D_{H}M_{\ome(\lam,\mathfrak{G}(\bar{\Xi})_{V})}$,
with
$
M_{\ome(\lam,\mathfrak{G}(\bar{\Xi})_{V})}: $\\
$: J_{4s-1}\bY_{\zet} \ucar{\bX} VJ_{4s-1}\cA^{(r,k)}) \to
\Con_{2s}^{*}[\cA^{(r,k)}]\ten
\Con_{2s-1}^{*}[\cA^{(r,k)}]\ten\Con_{0}^{*}[\cA^{(r,k)}]\wed
(\owed{n-1}T^{*}\bX)$.
\eLm
Coordinate expressions for the morphisms
$\bet(\lam,\mathfrak{G}(\bar{\Xi})_{V}) $ and
$M_{\ome(\lam,\mathfrak{G}(\bar{\Xi})_{V})}$ can be found by a
backwards procedure (see \eg \cite{Kol83}). In particular,
$\bet(\lam,\mathfrak{G}(\bar{\Xi})_{V})$ is nothing but the
Euler--Lagrange morphism associated with the {\em new} Lagrangian
$\ome(\lam,\mathfrak{G}(\bar{\Xi})_{V})$ defined on the fibered
manifold $J_{2s}\bY_{\zet} \ucar{\bX} VJ_{2s}\cA^{(r,k)}\to \bX$.
In particular, we get the following {\em local} decomposition of
$\ome(\lam,\mathfrak{G}(\bar{\Xi})_{V})$:
\bEq
\ome(\lam,\mathfrak{G}(\bar{\Xi})_{V}) =
\bet(\lam,\mathfrak{G}(\bar{\Xi})_{V}) +
D_{H}\tilde{\eps}(\lam,\mathfrak{G}(\bar{\Xi})_{V}) \,,
\eEq
where we put $\tilde{\eps}(\lam,\mathfrak{G}(\bar{\Xi})_{V})
\equiv M_{\ome(\lam,\mathfrak{G}(\bar{\Xi})_{V})}$.

\bDf
We call the global morphism $\bet(\lam,\mathfrak{G}(\bar{\Xi})_{V})
\byd E_{\ome(\lam,\mathfrak{G}(\bar{\Xi})_{V})}$
the {\em generalized} {\em Bianchi morphism} associated
with the Lagrangian $\lam$.
\END\eDf

\bRm
For any $(\bar{\Xi},\xi)\in \cA^{(r,k)}$, as a consequence of the
gauge-natural invariance of the Lagrangian, by
the Noether's First Theorem,  the morphism
$\bet(\lam,\mathfrak{G}(\bar{\Xi})_{V}) \equiv
\cE_{n}(\ome(\lam,\mathfrak{G}(\bar{\Xi})_{V}))$ is {\em locally}
identically vanishing.
We stress that these are just {\em local generalized Bianchi
identities}.
In particular, we have {\em locally}
$\ome(\lam,\mathfrak{G}(\bar{\Xi})_{V})$ $=$
  $D_{H}\tilde{\eps}(\lam,\mathfrak{G}(\bar{\Xi})_{V})$
\cite{Vari1}.\END
\eRm
The form $\tilde{\eps}(\lam,\mathfrak{G}(\bar{\Xi})_{V})\equiv
M_{\ome(\lam,\mathfrak{G}(\bar{\Xi})_{V})}$ is
called a {\em local reduced current}. It vanishes along any critical
section.

The problem of the general covariance of generalized
Bianchi identities for field theories was posed by Anderson and
Bergman already in $1951$ (see \cite{AnBe51}).
Let now $\mathfrak{K} \byd
\textstyle{Ker}_{\cJ(\lam,\mathfrak{G}(\bar{\Xi})_{V})}$
be the {\em kernel} of
the generalized gauge-natural morphism $\cJ(\lam,\mathfrak{G}(\bar{\Xi})_{V})$.
As a consequence of Proposition \ref{comparison} and of considerations
above, we have the following covariant characterization of the kernel
of generalized Bianchi morphism, the detailed proof of which will
appear in \cite{PaWi03}.

\bTh
The generalized Bianchi morphism is globally vanishing if and only
if $\del^{2}_{\mathfrak{G}}\lam\equiv\cJ(\lam,\mathfrak{G}(\bar{\Xi})_{V})=
0$, \ie if and only if
$\mathfrak{G}(\bar{\Xi})_{V}\in\mathfrak{K}$.
\eTh

The gauge-natural invariance of the
variational principle {\em in its whole}  enables us to solve the
{\em intrinsic indeterminacy}  in the conserved charges associated
with gauge-natural symmetries of Lagrangian field theories (in
\cite{Mat03}, for example, the special case of the gravitational
field coupled with fermionic matter is considered and the Kosmann
lift is then invoked as an {\em ad hoc}  choice to recover the well
known expression of the Komar superpotential). By requiring the
second variation to vanish, \ie on the kernel of the Jacobi
morphism, we express gauge-natural lits of infinitesimal principal
automorphism in terms of
the corresponding infinitesimal diffeomorphisms (their projections)
on the basis manifolds (see Theorem \ref{fundtheorem} below). This is
well known to be of great  importance within the theory of Lie
derivative of
sections of a gauge-natural bundle and notably for the Lie derivative
of spinors (see \eg the review given in \cite{Mat03}).

\bTh\label{fundtheorem}
Let $\lam\in \Var^{n}_{s}$ be a gauge-natural invariant generalized
Lagrangian and let
$\mathfrak{G}(\bar{\Xi})$ be a gauge-natural lift of the principal
infinitesimal automorphism
$\bar{\Xi}\in\cA^{r,k}$, \ie a gauge-natural symmetry of $\lam$. Then
$\bar{\Xi}\in\cA^{r,k}$ is related to its
projection $\xi\in\cT_{\bX}$ by the condition
\beq
(-1)^{|\bsig|}D_{\bsig}\,\left(D_{\bmu}
\bar{\Xi}^{j}_{V}\left(\der_{j}(\der^{\bmu}_{i}\lam) - \sum_{|\balp |
= 0}^{s-|\bmu |}
(-1)^{|\bmu +\balp |} \frac{(\bmu +
\balp)!}{\bmu ! \balp !}
D_{\balp}\der^{\balp}_{j}(\der^{\bmu}_{i}\lam)\right)\right)=0\,.
\eeq
\eTh

\bPf
We recall that given a vector field
$j_{s}\hat{\Xi}: J_{s}\bY_{\zet} \to TJ_{r}\bY_{\zet}$, the splitting
\eqref{jet connection} yields $j_{s}\hat{\Xi} \, \com \, \pi^{s+1}_{s} =
j_{s}\hat{\Xi}_{H} + j_{s}\hat{\Xi}_{V}$
where, if $j_{s}\hat{\Xi} = \hat{\Xi}^{\gam}\der_{\gam} +
\hat{\Xi}^i_{\balp}\der^{\balp}_i$, then we
have $j_{s}\hat{\Xi}_{H} = \hat{\Xi}^{\gam}D_{\gam}$ and
$j_{s}\hat{\Xi}_{V} =
D_{\balp}(\hat{\Xi}^i - y^i_{\gam}\hat{\Xi}^{\gam})
\der^{\balp}_{i}$. Analogous considerations
hold true of course also for the unique corresponding invariant vector
field $j_{s}\bar{\Xi}$ on $W^{(r,k)}\bP$.
In particular, the condition $j_{s}\bar{\Xi}_{V}
=D_{\balp}(\bar{\Xi}^{i}_{V})\der^{\balp}_{i}\in \mathfrak{K}$
implies, of course,  that the components $\bar{\Xi}^i_{\balp}$
and
$\bar{\Xi}^{\gam}$ {\em are not} independent, but they are {\em
related} in such a way
that $j_{s}\bar{\Xi}_{V}=
D_{\balp}(\hat{\Xi}^i - y^i_{\gam}\hat{\Xi}^{\gam}) \der^{\balp}_{i}$ must be a
solution of generalized gauge-natural Jacobi equations for the Lagrangian
$\lam$.
\QED
\ePf

\bRm
For each $\bar{\Xi}\in \cA^{(r,k)}$ such that $\bar{\Xi}_{V}\in
\mathfrak{K}$, we have
$\cL_{j_{s}\bar{\Xi}_{H}}\ome(\lam,
\mathfrak{K})=0$; the latter is a naturality condition for the
morphism $\ome(\lam,
\mathfrak{K})$  and says something on the Hamiltonian structure of
the theory itself (see \cite{FrPaWi04} for details).\END
\eRm

The result above reflects in the
theory of conserved currents and superpotentials for gauge-natural
field theories, where the theory of Lie derivatives of sections of
gauge-natural  bundles finds one of its main application.
In the following we shall refer to {\em canonical} globally defined
objects (such as currents or corresponding
superpotentials) by their explicit dependence on $\mathfrak{K}$.

\bCr
Let $\lam \in \Var^{n}_{s}$ be a gauge-natural Lagrangian and
$j_{s}\hat{\Xi}_{V}\in \mathfrak{K}$
a gauge-natural symmetry of $\lam$.
Being $\bet(\lam, \mathfrak{K})\equiv 0$, we have, {\em globally},
$\ome(\lam,\mathfrak{K}) = D_{H}\eps(\lam, \mathfrak{K})$,
then the following holds:
\bEq\label{strong conservation}
D_{H}(\eps(\lam, \mathfrak{K})-\tilde{\eps}(\lam, \mathfrak{K}) = 0\,.
\eEq
\eCr

Eq. \eqref{strong conservation} is referred as a gauge-natural
`strong' conservation law for the {\em global} density $\eps(\lam,
\mathfrak{K}) -\tilde{\eps}(\lam,
\mathfrak{K})$.

We can now state the following fundamental result about the existence and
{\em globality} of canonical gauge-natural superpotentials in the framework
of variational sequences.

\bTh\label{global}
Let $\lam \in \Var^{n}_{s}$ be a gauge-natural Lagrangian and
$(j_{s}\hat{\Xi},\xi)$ a gauge-natural symmetry of $\lam$. Then there exists
a global sheaf morphism
$\nu(\lam, \mathfrak{K})$ $\in$
$\left(\Var^{n-2}_{2s-1}\right)_{\bY_{\zet} \ucar{\bX}
\mathfrak{K}}$
such that
\beq
D_{H}\nu(\lam, \mathfrak{K}) = \eps(\lam, \mathfrak{K}) -\tilde{\eps}(\lam,
\mathfrak{K})\,.
\eeq
\eTh

\bDf
We define the sheaf morphism $\nu(\lam, \mathfrak{K})$ to be a
{\em canonical gauge-natural
superpotential} associated with $\lam$.
\END\eDf

\bEx ({\bf Einstein-Yang--Mills theory})
Let $\lam\in\Var^{n}_s$.
It is known (see {\em e.g.} \cite{Kol83}) that
\beq
& d_{V}\lam = (d_{V}\lam)^{\balp}_{i}\vartht^{i}_{\balp}\wed \ome \,,
\,
E_{d_{V}\lam}
= \cE(\lam)_{i}\vartht^{i}\wed \ome \,,
\,
p_{d_{V}\lam}
= p(\lam)^{\balp\mu}_{i}\vartht^{i}_{\balp}\wed \ome_{\mu}\,,
\\
& p(\lam)^{\bbet\mu}_{i}
= (d_{V}\lam)^{\balp}_{i} \qquad \bbet +\mu= \balp, |\balp|= s  \,,
\\
& p(\lam)^{\bbet\mu}_{i}
= (d_{V}\lam)^{\balp}_{i} - D_{\nu}p(\lam)^{\balp \nu}_{i}
\qquad \bbet+\mu=\balp, |\balp|= s-1  \,,
\\
& \cE(\lam)^{\balp}_{i}
= (d_{V}\lam)^{\balp}_{i}- D_{\nu}p(\lam)^{\balp\nu}_{i}\qquad |\balp|=0 \,.
\eeq
Furthermore,
$\cE(\lam)_{i}=\sum_{|\balp|\leq s}(-1)^{|\balp|}
D_{\balp}(d_{V}\lam)^{\balp}_{i}$.

Let $(\bP,\bX,\pi;\bG)$ be a principal bundle, $g$ a metric on
$\bX$, $\mathfrak{k}$ an $ad$--invariant metric on $\bG$.
Let $\ome$ be a principal connection and $F$ its $\mathfrak{g}$--valued
curvature $2$--form.
Let us now take the {\em gauge--natural} bundle
$\bY = Lor(\bX)\ucar{\bX}\bC$, where $Lor(\bX)$ is the
bundle of Lorentzian metrics
over space--time $\bX$ and $\bC$ is the affine
bundle of principal connections $\ome$ over $\bP$. Local coordinates on $\bY$
are given by $x^{\mu}, g^{\mu\nu},\ome^{i}_{\mu}$.
Let us consider the {\em gauge--natural} Lagrangian $\lam$ defined on
the gauge--natural bundle $J_{2}Lor(\bX)\ucar{\bX}J_{1}\bC$:
\bEq\label{unified}
\lam=\lam_{H}(g^{\mu\nu},R_{\mu\nu})+\lam_{YM}(g^{\mu\nu},F^{i}_{\mu\nu})\,,
\eEq
where $\lam_{H}=-\frac{1}{2\kappa}\sqrt{g}g^{\alp\bet}R_{\alp\bet}$ is
the Einstein Lagrangian, $R_{\alp\bet}$ is the Ricci tensor of the
metric $g$ given by
$R_{\alp\bet}\byd
R^{\mu}_{\alp\mu\bet}=D_{\mu}\gam^{\mu}_{\alp\bet}-D_{\bet}\gam^{\mu}_{\alp\mu}+
\gam^{\mu}_{\nu\mu}\gam^{\nu}_{\alp\bet}-
\gam^{\mu}_{\nu\bet}\gam^{\nu}_{\alp\mu}$, with
$\gam^{\mu}_{\nu\bet}=\frac{1}{2}g^{\mu\alp}(D_{\nu}g_{\bet\alp}-
D_{\alp}g_{\nu\bet}+D_{\bet}g_{\alp\nu})$ the Levi--Civita connection
of $g$, $\sqrt{g}=\sqrt{|det(g^{\mu\nu})|}$, $\kappa$ is a constant and
$\lam_{YM}(g_{\mu\nu},F^{i}_{\mu\nu})=
-\frac{1}{4}\sqrt{g}F^{\lam\gam}_{i}F_{\lam\gam}^{i}$ is the
Yang--Mills Lagrangian. Here
$F^{\lam\gam}_{i}=\mathfrak{k}_{ij}g^{\lam\alp}g^{\gam\bet}F^{j}_{\alp\bet}$.

Notice that in this case $\bY_{\zet}= J_{2}Lor(\bX)\ucar{\bX}J_{1}\bC$
and the order of the gauge-natural bundle is $(r,k)=(3,2)$.
An infinitesimal right-invariant automorphism of $\bP$ is given
in the split form:
\beq
\Xi = \Xi_{h} + \Xi_{v}= \Xi^{\mu}(\der_{\mu} +
\ome^{i}_{\mu}\der_{i}) + (\Xi^{i} -
\ome^{i}_{\mu}\Xi^{\mu})\der_{i} \,,
\eeq
where $\der_{i}$ is a local basis of right-invariant vertical
vector fields on $\bP$.
We shall respectively indicate by $\Xi_{v}$ and $\Xi_{h}$
the vertical and horizontal components of $\Xi$ with
respect to the principal connection $\ome$; we shall write
$\Xi^{i}_{v} = \Xi^{i} - \ome^{i}_{\mu}\Xi^{\mu}$. Notice that
$\Xi^{i}_{v}$  is {\em not} equal to $\Xi^{i}_{V}$, but they are
related in a clear and simple way.
We get $\eps^{\sig}(\lam,\Xi_{h})
=\eps^{\sig}(\lam_{H},\Xi_{h})+\eps^{\sig}(\lam_{YM},\Xi_{h})$,
where
\beq
\eps^{\sig}(\lam_{H},\Xi_{h})=
\frac{1}{\kappa}\sqrt{g}(R^{\sig}_{\bet}-Rg^{\sig}_{\bet})\Xi^{\bet}
+\nabla_{\mu}[\frac{\sqrt{g}}{2\kappa}
(\nabla^{\sig}\Xi^{\mu}-\nabla^{\mu}\Xi^{\sig})]\,.
\eeq
and
\beq
\eps^{\sig}(\lam_{YM},\Xi_{h})=
(2p^{\mu\sig}_{i}\pounds_{\Xi_{h}}\ome^{i}_{\mu}-\lam_{YM}\Xi^{\sig})=
-\sqrt{g}(F^{\mu\sig}_{i}F^{i}_{\mu\nu}-
\frac{1}{4}F^{\mu\rho}_{i}F^{i}_{\mu\rho}\del^{\sig}_{\nu})\Xi^{\nu}\,.
\eeq
$R$ is the scalar curvature and
$p^{\mu\nu}_{i}=-\frac{\sqrt{g}}{2}F^{\mu\nu}_{i}$.
Here and in the sequel $\nabla_{\mu}$ denotes the covariant
metric derivative with respect to $g$.

A ``horizontal'' superpotential is given by:
$\nu_{h}^{\sig\mu}=\frac{\sqrt{g}}{4\kappa}
(\nabla^{\sig}\Xi^{\mu}-\nabla^{\mu}\Xi^{\sig})$,
which is essentially the Komar superpotential \cite{Vari1}.

Furthermore, we have
$\eps^{\sig}(\lam_{YM},\Xi_{v})=
-2p^{\mu\sig}_{i}\nabla_{\mu}\Xi^{i}_{v}=
-\nabla_{\mu}(-2p^{\mu\sig}_{i}\Xi^{i}_{v})+
2\nabla_{\mu}p^{\mu\sig}_{i}\Xi^{i}_{v}$.
Then there exists a ``vertical'' superpotential, given by:
\beq
\nu_{v}^{\mu\sig}=p^{\mu\sig}_{i}\Xi^{i}_{v}=-\frac{\sqrt{g}}{2}F^{\mu\sig}_{i}
\Xi^{i}_{v}\,.
\eeq

From Theorem \ref{fundtheorem}, we deduce that also the components $\Xi^{i}_{v}$
-- when $\Xi_{V}$ is an element of the kernel of the
gauge-natural generalized Jacobi morphism --
can be expressed invariantly in terms of the components
$\Xi^{\mu}$ of the projection
of $\Xi_{v}$:
\beq
(-1)^{|\bsig|}D_{\bsig}\,(D_{\bmu}
( \hat{\Xi}^{i}_{v} + (\hat{\ome}^{i}_{\gam}-
y^i_{\gam})\hat{\Xi}^{\gam})\psi_{ji}^{\bmu})=0\,,
\eeq
where $\psi_{ji}^{\bmu}=(\der_{j}(\der^{\bmu}_{i}\lam) -
\sum_{|\balp | = 0}^{s-|\bmu |}(-1)^{|\bmu +\balp |}
\frac{(\bmu + \balp)!}{\bmu !\balp !}
D_{\balp}\der^{\balp}_{j}(\der^{\bmu}_{i}\lam))$ and $\hat{\ome}$ is  the gauge
natural prolongation of $\ome$ (see \cite{FFP01}). When such a dependence is expressed
explicitly, we write
$\nu_{v}^{\mu\sig}=\nu_{v}^{\mu\sig}(\lam, \mathfrak{K})$.

Equivalently, given
$\Xi^{i}_{v}$ as above, Theorem
\ref{fundtheorem} says us also that some constraint on the connection $\ome$ is
there. It is maybe noteworthy that $\lam$ is the {\em total} Lagrangian
\eqref{unified}.
\eEx

{\em Acknowledgments.}
The authors wish to thank I. Kol\'a\v r and D. Krupka for
many interesting discussions and the unknown  referee for useful
remarks which lead to improve the text.


TITLE: Clarify about adjoint functors
QUESTION [2 upvotes]: Consider the adjunction $F\dashv G:\mathbf D\to \mathbf C$, with unity $\eta$ and counity $\varepsilon$. Is it true that $\varepsilon _{FGc}=FG\varepsilon_c $ in general? If not,  under what condition is ture? (For example it seems to me that Mac Lane uses this argument in the proof of  the existence of comparison functor, theorem VI$.3.1$)

REPLY [6 votes]: No, this is not true in general. For example, take $F$ to be the free group functor $\mathsf{Set}\to \mathsf{Grp}$ and $G$ the forgetful functor $\mathsf{Grp}\to \mathsf{Set}$. For a group $x$, $\varepsilon_x\colon FGx\to x$ is the "evaluation" map which maps a reduced word in the elements of $x$ (an element of the free group on the underlying set of $x$) to the element of $x$ it evaluates to.
Now $\varepsilon_{FGx}$ and $FG\varepsilon_x$ are both maps $FGFGx\to FGx$, and $FGFGx$ consists of "reduced words in reduced words in elements of $x$". But the map $\varepsilon_{FGx}$ evaluates the outer words, while $FG\varepsilon_x$ evaluates the inner words. For example, if $w$ and $w'$ are elements of $FGx$, then $w\cdot w'$ is an element of $FGFGx$. We have $FG\varepsilon_{x}(w\cdot w') = \varepsilon_x(w)\cdot \varepsilon_x(w')$ (this is a word in $FGx$ consisting of a product of two elements of $x$), but $\varepsilon_{FGx}(w\cdot w')$ is the reduction of the word $ww'$ obtained by concatenating the words $w$ and $w'$, which could be an arbitrarily complicated word in $FGx$.

The point you're missing in Mac Lane's argument is that while it's not true that $FG\varepsilon_x = \varepsilon_{FGx}$, these two arrows are coequalizd by $\varepsilon_x$:  $\varepsilon_x\circ (FG\varepsilon_x) = \varepsilon_x\circ \varepsilon_{FGx}$.
Recall that $\varepsilon$ is a natural transformation $FG\to 1_D$. This means that for any arrow $f\colon x\to y$ in $D$, we have a commutative square: $\require{AMScd}$
\begin{CD}
FGx @>{FGf}>> FGy\\
@V{\varepsilon_x}VV @VV{\varepsilon_y}V\\
x @>{f}>> y
\end{CD}
Now set $x = FGy$ and $f = \varepsilon_y$. Then the naturality square above becomes: \begin{CD}
FGFGy @>{FG\varepsilon_y}>> FGy\\
@V{\varepsilon_{FGy}}VV @VV{\varepsilon_y}V\\
FGy @>{\varepsilon_y}>> y
\end{CD}
Finally, apply $G$ to the above square:
\begin{CD}
GFGFGy @>{GFG\varepsilon_y}>> GFGy\\
@V{G\varepsilon_{FGy}}VV @VV{G\varepsilon_y}V\\
GFGy @>{G\varepsilon_y}>> Gy
\end{CD}
This is exactly the square that expresses the first axiom of algebras.


TITLE: In a theoretical car does gear not affect mileage?
QUESTION [2 upvotes]: Newb to thinking about mileage, and largely a newb to newtonian mechanics!
Theoretical car: Combustion force is in exact direct proportion to amount of air/fuel. No minimum amount of fuel/air needed in cylinder to cause a single combustion, and no cap on amount of fuel/air that will fit into cylinder and combust properly without flooding system.
Not sure how a transmission actually works, whether it’s the “upstream” or “downstream” gear that changes size when you shift, but we’ll say it’s the upstream.
My thinking is that for a fixed downstream gear, to accelerate a fixed amount across a fixed time, the same amount of work is done:
1) In low gear (smaller upstream gear), with a LARGER cumulative downward distance travelled by the piston over many cycles due to the gear ratio, but with each cycle requiring LESS force behind it due to leverage
or
2) In high gear (larger upstream gear), with a SMALLER cumulative downward distance travelled by the piston over less cycles due to the gear ratio, but with each cycle requiring MORE force behind it due to less leverage
I think the Work (Fxd) is the same in both cases – we always say the work done on either side of a lever is the same due to one side exerting more force but the other moving further. I think this is simply a case of that idea. Or is my thinking incorrect?
Maybe fuel use is not actually in direct proportion to work done...
If I’m right, does the real-life need to switch gears at “ideal” ranges have more to do with each combustion requiring a minimum amount of fuel, which would add to the consumption when in low gear?

REPLY [3 votes]: The big reason for having a transmission in a real world internal combustion engine is that IC engines have a fairly limited range of revolution rates at which they work. At revolution rates above their favored speed range friction forces get worse (friction is usually a v squared thing) and at high rates the downstroke doesn't last long enough for combustion to finish, so there is a loss of power. At the low end, the combustion strokes aren't often enough to keep the engine spinning. So you need a way to start the vehicle from zero speed and get up to highway speed (or more) with the IC engine staying in its happy speed range. Hence the multi-gear transmission.


text:
\begin{document}

\begin{abstract}
Given a finite nonnegative Borel measure \(m\) in \(\mathbb{R}^{d}\), we identify the Lebesgue set \(\mathcal{L}(V_{s}) \subset \mathbb{R}^{d}\) of the vector-valued function
\[
V_{s}(x)
= \int_{\mathbb{R}^{d}}\frac{x - y}{|x - y|^{s + 1}} \mathrm{d}m(y),
\]
for any order \(0 < s < d\).{}
We prove that \(a \in \mathcal{L}(V_{s})\) if and only if the integral above has a principal value at \(a\) and
\[
\lim_{r \to 0}{\frac{m(B_{r}(a))}{r^{s}}} = 0.
\]
In that case, the precise representative of \(V_{s}\) at \(a\) coincides with the principal value of the integral.
\end{abstract}

\subjclass[2020]{31B15 (primary); 26B05 (secondary)}

\maketitle

\section{Introduction}

Given \(0 < s < d\) and a finite nonnegative Borel measure \(m\) in \(\R^{d}\), we consider in \(\R^{d}\) the vector-valued function
\begin{equation}
\label{eqRiezDerivative}
V_{s}(x)
\vcentcolon= \int_{\R^{d}} \frac{x - y}{\abs{x - y}^{s + 1}} \dif m(y)
\end{equation}
that is well-defined for every point \(x\) in the set
\[{}
\Dom{V_{s}}
\vcentcolon=  \bigg\{ x \in \R^{d} : \int_{\R^{d}} \frac{\dif m(y)}{\abs{x - y}^{s}} < \infty \bigg\}.
\]
The domain \(\Dom{V_{s}}\) is rather large as its complement \(\R^{d} \setminus \Dom{V_{s}}\) is negligible with respect to the Lebesgue measure.
Indeed, as a consequence of Fubini's theorem one has
\[{}
\int_{\R^{d}} \biggl( \int_{\R^{d}} \frac{\dif m(y)}{\abs{x - y}^{s}}  \biggr) \e^{-\abs{x}^{2}} \dif x
< \infty.
\]
A finer analysis shows that the Hausdorff dimension of  \(\R^{d} \setminus \Dom{V_{s}}\) is actually
not greater that \(s\)\,; see Theorem~1 in Chapter~4 of \cite{Carleson}.

Up to multiplicative constants, \(V_{s}\) is the gradient of the Riesz potential of order \(s - 1\) and, in particular, \(V_{d - 1}\) is the gradient of the Newtonian potential
\[{}
\int_{\R^{d}}  \frac{\dif m(y)}{\abs{x - y}^{d-2}} \quad \text{for \(x \in \R^d\)}
\]
generated by a distribution of mass described by \(m\).{}
Hence, for every function \(u \in L^{1}\loc(\R^{d})\) such that
\begin{equation}
\label{eq-232}
- \Delta u = m
\quad \text{in the sense of distributions in \(\R^{d}\),}
\end{equation}
by Weyl's lemma on weakly harmonic functions it follows that
\[{}
\nabla u = c V_{d - 1} + \Phi{}
\quad \text{almost everywhere in \(\R^{d}\),}
\]
where \(\Phi : \R^{d} \to \R^{d}\) is the gradient of a harmonic function and \(c \in \R\) is a nonzero constant that depends on \(d\).
As a result, a reasonable pointwise definition of \(\nabla u\) for any solution of \eqref{eq-232} can be obtained from a
good understanding of \(V_{d - 1}\).

More generally, we are interested in the existence of the \emph{precise representative} of \(V_{s}\) at a given point \(a \in \R^{d}\).{}
We recall that the precise representative in this case is a vector \(\alpha \in \R^{d}\) that satisfies
\begin{equation}
\label{eq-226}
\lim_{r \to 0}{\fint_{B_{r}(a)}{\abs{V_{s} - \alpha}}}
= 0,
\end{equation}
where the symbol \(\fint\) stands for the average integral with respect to the $d$-dimensional Lebesgue measure and $B_r(a)$
for the ball of radius $r$ centered at $a$.
When such an \(\alpha\) exists, we say that \(a\) is a \emph{Lebesgue point} of \(V_{s}\) and we denote
\[{}
\Lebesgue{V_{s}}(a)
\vcentcolon= \alpha.
\]
It follows from the fundamental property \eqref{eq-226} that
\[{}
\Lebesgue{V_{s}}(a)
= \lim_{r \to 0}{\fint_{B_{r}(a)}{V_{s}}}.
\]
As a consequence of the classical Lebesgue Differentiation Theorem, \(\Lebesgue{V_{s}}(a)\) exists for almost every \(a \in \R^{d}\) with respect to the Lebesgue measure and
\[{}
\Lebesgue{V_{s}} = V_{s}
\quad \text{almost everywhere in \(\R^{d}\).}
\]

When the function one is dealing with has better properties, like being in some Sobolev space,
the exceptional set \(\mathcal{E}\) (that is, the complement of the set \(\mathcal{L}\) of Lebesgue points) is typically smaller
than merely having zero Lebesgue measure; see e.g.~Section~4.8 in \cite{Evans-Gariepy} and Chapter~8 in \cite{Ponce:2016}.
It is then natural to expect that the same property holds for the exceptional set \(\mathcal{E}(V_{s})\) of
the potential \(V_{s}\).

This note stems from the recent work~\cite{Cufi-Verdera} by the first and third authors concerning capacitary differentiability of the Newtonian potential; see also \cite{Verdera}.
In analogy with \cite{Cufi-Verdera}, we show that existence of a principal value for the integral, combined with a density property of \(m\) at \(a\), allows one to decide whether \(a\) is a Lebesgue point of \(V_{s}\).

Before stating our result, we recall that the principal value of \(V_{s}\) at \(a\) is
\[{}
\mathrm{p.v.}\,V_{s}(a)
\vcentcolon=
\lim_{\epsilon \to 0}{\int_{\R^{d} \setminus B_{\epsilon}(a)} \frac{a - y}{\abs{a - y}^{s + 1}} \dif m(y)}
\]
whenever this limit exists.
We relate the question of existence of a principal value for \(V_{s}\) with that of a precise representative in the following

\begin{theorem}
	\label{theoremRieszLebesgue}
	A point \(a \in \R^{d}\) is a Lebesgue point of \(V_{s}\)
	if and only if the principal value of \(V_{s}\) exists at \(a\) and
	\begin{equation}
	\label{eq-328}
	\lim_{r \to 0}{\frac{m(B_{r}(a))}{r^{s}}} = 0.
	\end{equation}
	One then has
	\[{}
	\Lebesgue{V_{s}}(a)
	= \mathrm{p.v.}\,V_{s}(a).
	\]
\end{theorem}

	We prove Theorem~\ref{theoremRieszLebesgue} in Sections~\ref{sectionConverse} and~\ref{sectionDirect}.
	That every \(a \in \Dom{V_{s}}\) is a Lebesgue point of \(V_{s}\) and \(\Lebesgue{V_{s}}(a) = V_{s}(a)\) can be seen independently from Theorem~\ref{theoremRieszLebesgue} by observing that, for every \(y \in \R^{d}\) and \(r > 0\),
	\[{}
	 \fint_{B_{r}(a)} \frac{\dif x}{|x - y|^{s}}
	 \le \frac{C_{d, s}}{|a - y|^{s}}
	\]
	and then
	\[{}
	\fint_{B_{r}(a)} \biggabs{\frac{x-y}{|x-y|^{s+1}} - \frac{a - y}{|a - y|^{s+1}}} \dif x
	\le \frac{C_{d, s} + 1}{\abs{a - y}^{s}}
	\quad \text{for \(y \ne a\).}
	\]
	An application of Fubini's Theorem and the Dominated Convergence Theorem now gives
	\[{}
	\fint_{B_{r}(a)}{\abs{V_{s} - V_{s}(a)}}
	\le \int_{\R^{d}} \biggl( \fint_{B_{r}(a)} \biggabs{\frac{x-y}{|x-y|^{s+1}} - \frac{a - y}{|a - y|^{s+1}}} \dif x \biggr) \dif m(y)
	\to 0
	\quad \text{as \(r \to 0\).}
	\]
	As \(\Dom{V_{s}}\) is contained in the set of Lebesgue points of \(V_{s}\), we get
	\begin{equation}
	\label{eqHausdorffDimension}
	\dim_{\mathcal{H}}{(\mathcal{E}(V_{s}))} \le{}
	\dim_{\mathcal{H}}{(\R^{d} \setminus \Dom{V_{s}})} \le s.
	\end{equation}

	To obtain a more precise quantification of the size of  \(\mathcal{E}(V_{s})\), we observe that, by Theorem~\ref{theoremRieszLebesgue}, failure of having Lebesgue points may occur by either that the principal value of \(V_{s}\) does not exist at \(a\) or
	\[{}
	\limsup_{r \to 0}{\frac{m(B_{r}(a))}{r^{s}}} > 0.
	\]
	The latter condition holds on a set of points \(a\) with at most \(\sigma\)-finite Hausdorff measure \(\mathcal{H}^{s}\).{}

	The question of existence of principal values is more subtle and has been investigated by Mattila and the third author~\cite{Mattila-Verdera}.
	The answer involves a capacity \(\kappa_{s}\) that is defined on every compact subset \(E \subset \R^{d}\) as
	\[{}
	{\kappa_{s}(E) = \sup{\mu(\R^{d})},}
	\]
	where the supremum is computed over all finite nonnegative Borel measures \(\mu\) supported in \(E\) such that
	\begin{enumerate}[(a)]
		\item \(\mu(A) = 0\) for every Borel set \(A \subset \R^{d}\) with \(\sigma\)-finite \(\cH^{s}\) measure,{}
		\item the maximal Riesz transform \(R_{s}^{*}\) is a bounded operator from \(L^{2}(\mu)\) into itself with
		\[{}
		\norm{R_{s}^{*}(f\mu)}_{L^{2}(\mu)}
		\le \norm{f}_{L^{2}(\mu)}
		\quad \text{for every \(f \in (L^{1} \cap L^{2})(\mu)\)},
		\]
		where
	\[{}
	R^{*}_{s}(f\mu)(x) \vcentcolon= \sup_{\epsilon > 0}{\biggl|\int_{\R^{d} \setminus B_{\epsilon}(x)} \frac{x - y}{\abs{x - y}^{s + 1}} f(y) \dif \mu(y) \biggr|}
	\quad \text{for every \(x \in \R^{d}\).}
	\]
	\end{enumerate}

	As a consequence of~\cite{Mattila-Verdera} and using a straightforward adaptation of the argument in \cite{Cufi-Verdera}, we show that

\begin{theorem}
	\label{theoremCapacity}
	For every compact subset \(E \subset \mathcal{E}(V_{s})\), one has
	\(
	\kappa_{s}(E) = 0.
	\)
\end{theorem}

	Other properties of \(\kappa_{s}\) for any \(0 < s < d\)
	are presented in Section~\ref{sectionCapacity}, where we also prove Theorem~\ref{theoremCapacity} and explain why the latter implies \eqref{eqHausdorffDimension}.
	A variant of $\kappa_s$ was introduced by Prat in \cite{Prat}, where the author obtains analogous estimates for the Hausdorff dimension of sets of zero capacity via comparison between capacity and Hausdorff content.

	A deep result by Ruiz~de~Villa and Tolsa~\cite{RuizdeVilla-Tolsa} ensures that \(\kappa_{d - 1}\) is equivalent to the \(C^{1}\)-harmonic capacity.
	Using this interpretation, the fact that \(\kappa_{d - 1}(E) = 0\) means that every \(C^{1}\)~function in \(\R^{d}\) that is
	harmonic in \(\R^{d} \setminus E\) must be harmonic in the entire space \(\R^{d}\).
	In dimension \(d = 2\), it follows from work by Tolsa~\cite{Tolsa:2004} that the $C^1$-harmonic capacity is equivalent to the classical continuous analytic capacity.
	We then deduce from Theorem~\ref{theoremCapacity} for \(s = 1\) that

\begin{corollary}
	For every finite complex Borel measure \(\nu\) in \(\mathbb{C}\), the exceptional set for Lebesgue points of the Cauchy integral of \(\nu\) defined by
$$
\int_{\mathbb{C}} \frac{\dif\nu(w)}{z-w}{}
\quad \text{for almost every \(z \in \mathbb{C}\)}
$$
has zero continuous analytic capacity.
\end{corollary}

The preceding corollary is sharp in the sense that there exists a compact set $K$ of zero continuous analytic capacity that is exactly the exceptional set of the Cauchy integral of some finite measure.
For instance, take as $K$ the corner quarters planar Cantor set, which has positive and finite one dimensional Hausdorff measure $m$.
Then, the Cauchy integral $f$ of $m$ is continuous in the complement of $K$ and each point of $K$ is an exceptional point of $f$ because \eqref{eq-328} is not satisfied.
It would be interesting to know whether each compact
set with zero continuous analytic capacity can be expressed as the exceptional set of the Cauchy integral of some finite Borel measure.

\section{Proof of the reverse implication of Theorem~\ref{theoremRieszLebesgue}}
\label{sectionConverse}

We first need a couple of estimates related to \(V_{s}\).

\begin{lemma}
	\label{lemmaEstimateAverage}
	For every \(a \in \R^{d}\) and every \(r > 0\),{}
	\[{}
	\fint_{B_{r}(a)} \biggl( \int_{B_{2r}(a)} \frac{ \dif m(y)}{\abs{z - y}^{s}} \biggr) \dif z
	\le C \, \frac{m(B_{2r}(a))}{r^{s}},
	\]
	for some constant \(C > 0\) depending on \(d\).
\end{lemma}

\begin{proof}
	By Fubini's theorem,
	\[{}
	\fint_{B_{r}(a)} \biggl( \int_{B_{2r}(a)} \frac{ \dif m(y)}{\abs{z - y}^{s}} \biggr) \dif z
	=
	\int_{B_{2r}(a)} \biggl( \fint_{B_{r}(a)} \frac{ \dif z}{\abs{z - y}^{s}} \biggr) \dif m(y).
	\]
	For every \(y \in \R^{d}\),
	\[{}
	\fint_{B_{r}(a)} \frac{ \dif z}{\abs{z - y}^{s}}
	\le \fint_{B_{r}(a)} \frac{ \dif z}{\abs{z - a}^{s}}
	= \frac{C}{r^{s}},
	\]
	which gives the conclusion.
\end{proof}

\begin{lemma}
	\label{lemmaEstimateMeanValue}
	For every \(a \in \R^{d}\) and every \(r > 0\),{}
	\[{}
	\fint_{B_{r}(a)}{\biggl( \int_{\R^{d} \setminus B_{2r}(a)} \biggabs{\frac{z - y}{\abs{z - y}^{s + 1}} - \frac{a - y}{\abs{a - y}^{s + 1}}} \dif m(y)}\biggr) \dif z
	\le C' r \int_{2r}^{\infty} \frac{m(B_{t}(a))}{t^{s + 2}} \dif t,
	\]
	for some constant \(C' > 0\) depending on \(d\).
\end{lemma}

\begin{proof}
	To simplify the notation, we may assume that \(a = 0\).
	For \(y \in \R^{d} \setminus B_{2r}(0)\) and \(z \in B_{r}(0)\), by the Mean Value Theorem there exists
	\(0 \le \theta \le 1\) such that
	\[{}
	\biggabs{\frac{z - y}{\abs{z - y}^{s + 1}} + \frac{y}{\abs{y}^{s + 1}}}
	\le \frac{\C \abs{z}}{\abs{\theta z - y}^{s + 1}}
	\le \frac{\Cl{cte-286} r}{\abs{y}^{s + 1}}.
	\]
	We thus have
	\begin{multline*}
	\fint_{B_{r}(0)} \biggl( \int_{\R^{d} \setminus B_{2r}(0)} \biggabs{\frac{z - y}{\abs{z - y}^{s + 1}} +
	\frac{y}{\abs{y}^{s + 1}}} \dif m(y) \biggr) \dif z\\
	\le \Cr{cte-286} r \fint_{B_{r}(0)} \biggl( \int_{\R^{d} \setminus B_{2r}(0)} \frac{ \dif m(y)}{\abs{y}^{s + 1}} \biggr) \dif z = \Cr{cte-286} r \int_{\R^{d} \setminus B_{2r}(0)} \frac{ \dif m(y)}{\abs{y}^{s + 1}}.
	\end{multline*}
	Using Cavalieri's principle, see e.g.~Corollary~2.2.34 in \cite{Willem},  one gets
	\[{}
	\int_{\R^{d} \setminus B_{2r}(0)} \frac{\dif m(y)}{\abs{y}^{s + 1}}
	= (s + 1) \int_{2r}^{\infty} \frac{m(B_{t}(0))}{t^{s + 2}} \dif t
	\]
	and the lemma follows.
\end{proof}

To handle the limit as \(r \to 0\) of the right-hand side in the estimate in Lemma~\ref{lemmaEstimateMeanValue},
we need an elementary fact from Real Analysis:

\begin{lemma}
	\label{lemmaLimit}
	Let \(h : {}[0, \infty){} \to \R\) be a bounded continuous function.
	If \(h(0) = 0\), then, for every \(\beta > 0\),{}
	\[{}
	\lim_{r \to 0}{r^{\beta} \int_{r}^{\infty} \frac{h(t)}{t^{1 + \beta}} \dif t}
	= 0.
	\]
\end{lemma}

\resetconstant
\begin{proof}
	Given \(\epsilon > 0\), take \(\delta > 0\) such that
	\(\abs{h(t)} \le \epsilon\) for every \(0 < t \le \delta\).
	For \(0 < r < \delta\), we then have
	\[{}
	\biggl|\int_{r}^{\infty} \frac{h(t)}{t^{1 + \beta}} \dif t\biggr|
	\le \biggl|\int_{r}^{\delta}\biggr| + \biggl|\int_{\delta}^{\infty}\biggr|
	\le \frac{\epsilon}{\beta} \Bigl(\frac{1}{r^{\beta}} + \frac{\norm{h}_{L^{\infty}}}{\delta^{\beta}} \Bigr).
	\]
	Therefore,
	\[{}
	\limsup_{r \to 0}{r^{\beta} \biggl|\int_{r}^{\infty} \frac{h(t)}{t^{1 + \beta}} \dif t \biggr| }
	\le \frac{\epsilon}{\beta}.
	\]
	Since this inequality holds for every \(\epsilon > 0\), the limit equals zero.
\end{proof}

\resetconstant
\begin{proof}[Proof of Theorem~\ref{theoremRieszLebesgue}] ``\(\Longleftarrow\)''.
	We assume that the principal value \(\mathrm{p.v.}\, V_{s}(a)\) exists and the limit \eqref{eq-328} holds.
	For almost every \(z \in \R^{d}\),{}
	\begin{multline*}
	\biggabs{V_{s}(z) - \int_{\R^{d} \setminus B_{2r}(a)} \frac{a - y}{\abs{a - y}^{s + 1}} \dif m(y)}\\
	\le \biggabs{\int_{\R^{d} \setminus B_{2r}(a)} \frac{z - y}{\abs{z - y}^{s + 1}} \dif m(y) - \int_{\R^{d} \setminus B_{2r}(a)} \frac{a - y}{\abs{a - y}^{s + 1}} \dif m(y) }\\
	+ \biggabs{\int_{B_{2r}(a)} \frac{z - y}{\abs{z - y}^{s + 1}} \dif m(y)}.
	\end{multline*}
	Hence,
	\begin{multline*}
	\fint_{B_{r}(a)}\biggabs{V_{s}(z) - \int_{\R^{d} \setminus B_{2r}(a)} \frac{a - y}{\abs{a - y}^{s + 1}} \dif m(y)} \dif z\\
	\le \fint_{B_{r}(a)} \biggl(\int_{\R^{d} \setminus B_{2r}(a)} \biggabs{\frac{z - y}{\abs{z - y}^{s + 1}} \dif m(y) - \frac{a - y}{\abs{a - y}^{s + 1}}} \dif m(y)  \biggr)\dif z\\
	+ \fint_{B_{r}(a)} \biggl( \int_{B_{2r}(a)} \frac{ \dif m(y)}{\abs{z - y}^{s}} \biggr) \dif z.
	\end{multline*}
	We then have by Lemmas~\ref{lemmaEstimateAverage} and \ref{lemmaEstimateMeanValue},
	\begin{multline}
	\label{eq-334}
	\fint_{B_{r}(a)}\biggabs{V_{s}(z) - \int_{\R^{d} \setminus B_{2r}(a)} \frac{a - y}{\abs{a - y}^{s + 1}} \dif m(y)} \dif z\\
	\le \Cl{cte-335} \biggl( r \int_{2r}^{\infty} \frac{m(B_{t}(a))}{t^{s + 2}} \dif t + \frac{m(B_{2r}(a))}{r^{s}} \biggr).
	\end{multline}
	Denoting by \(\alpha\) the principal value of \(V_{s}\) at \(a\), we get
	\begin{multline*}
	\fint_{B_{r}(a)}\abs{V_{s}(z) - \alpha} \dif z\\
	\le \Cr{cte-335} \biggl( r \int_{2r}^{\infty} \frac{m(B_{t}(a))}{t^{s + 2}} \dif t + \frac{m(B_{2r}(a))}{r^{s}} \biggr)
	 + \biggabs{\int_{\R^{d} \setminus B_{2r}(a)} \frac{a - y}{\abs{a - y}^{s + 1}} \dif m(y) - \alpha}.
	\end{multline*}
	As \(r \to 0\), the last term converges to zero by definition of principal value.
	The quantity in parentheses also converges to zero from the assumption on the measure \(m\), as we can apply Lemma~\ref{lemmaLimit} with \(\beta = 1\) and \(h(t) = m(B_{t}(a))/t^{s}\) for \(t > 0\).
\end{proof}

\section{Proof of the direct implication of Theorem~\ref{theoremRieszLebesgue}}
\label{sectionDirect}

We begin with the following estimate:

\begin{proposition}
	\label{lemmaEstimateDensity}
	Let \(0 < s \le d - 1\).{}
	For every \(\alpha \in \R^{d}\) and every \(r > 0\),
	\[{}
	\frac{m(B_{r}(a))}{r^{s}}
	\le C \fint_{B_{2r}(a)}{\abs{V_{s} - \alpha}},
	\]
	for some constant \(C > 0\) depending on \(d\).
\end{proposition}
\begin{proof}
	We may assume that \(a = 0\).
	We take the inner product of \(V_{s}(x)\) with \(x/\abs{x}\) and integrate over the ball \(B_{2r}(0)\) for any \(r > 0\).{}
	By integrability of the function \((x, y) \mapsto {1}/{\abs{x - y}^{s}}\) with respect to the product measure \(\cH^{d} \otimes m\) in \(B_{2r}(0) \times \R^{d}\) we can interchange the order of integration:
	\begin{equation}
		\label{eq-487}
	\int_{B_{2r}(0)}{V_{s}(x) \cdot \frac{x}{\abs{x}} \dif x}
	= \int_{\R^{d}} \biggl( \int_{B_{2r}(0)} \frac{x - y}{\abs{x - y}^{s + 1}} \cdot \frac{x}{\abs{x}} \dif x \biggr) \dif m(y).
	\end{equation}
	By the integration formula in polar coordinates,
	\begin{equation}
		\label{eq-494}
	\int_{B_{2r}(0)} \frac{x - y}{\abs{x - y}^{s + 1}} \cdot \frac{x}{\abs{x}} \dif x
	= \int_{0}^{2r}\biggl( \int_{\partial B_{\rho}(0)} \frac{x - y}{\abs{x - y}^{s + 1}} \cdot \frac{x}{\abs{x}} \dif\sigma(x) \biggr) \dif \rho,
	\end{equation}
	where \(\sigma\) denotes the surface measure on the sphere \(\partial B_{\rho}(0)\).{}

	We claim that
	\begin{equation}
		\label{eq-588}
	\int_{\partial B_{\rho}(0)} \frac{x - y}{\abs{x - y}^{s + 1}} \cdot \frac{x}{\abs{x}} \dif\sigma(x){}
	\ge \epsilon \, \rho^{d - s - 1} \chi_{B_{\rho}(0)}(y),
	\end{equation}
	for some \(\epsilon > 0\) independent of \(\rho\) and \(y\).{}
	To this end, note that
	\[{}
	\frac{x - y}{\abs{x - y}^{s + 1}} = \nabla g(x)
	\quad \text{where \(g(x) \vcentcolon= - \frac{1}{(s - 1)} \frac{1}{\abs{x - y}^{s - 1}}\)}.
	\]
	When \(s < d - 1\), it follows from the Divergence Theorem in \(B_{\rho}(0)\) that
	\[{}
	\int_{\partial B_{\rho}(0)} \frac{x - y}{\abs{x - y}^{s + 1}} \cdot \frac{x}{\abs{x}} \dif\sigma(x){}
	= \int_{B_{\rho}(0)} \Delta g
	= (d - s - 1) \int_{B_{\rho}(0)} \frac{\dif x}{\abs{x - y}^{s + 1}},
	\]
	which implies \eqref{eq-588}.
	When \(s = d - 1\), the function \(g\) is harmonic in \(\R^{d} \setminus \{y\}\).{}
	An application of the Divergence Theorem on \(B_{\rho}(0) \setminus B_{\eta}(y)\) with \(\eta \to 0\) yields
	\[{}
	\int_{\partial B_{\rho}(0)} \frac{x - y}{\abs{x - y}^{s + 1}} \cdot \frac{x}{\abs{x}} \dif\sigma(x){}
	=
	\begin{cases}
		\cH^{d - 1}(\partial B_{1}(0)) & \text{if \(y \in {B_{\rho}(0)}\),}\\
		0 & \text{if \(y \not\in \overline{B_{\rho}(0)}\),}
	\end{cases}
	\]
	which also gives \eqref{eq-588} and completes the proof of the claim.

	Combining \eqref{eq-494} and \eqref{eq-588}, we get
	\[{}
	\int_{B_{2r}(0)} \frac{x - y}{\abs{x - y}^{s + 1}} \cdot \frac{x}{\abs{x}} \dif x
	\ge c \, r^{d - s}\chi_{B_{r}(0)}(y),
	\]
	which by \eqref{eq-487} then gives
	\[{}
	\int_{B_{2r}(0)}{V_{s}(x) \cdot \frac{x}{\abs{x}} \dif x}
	\ge c \, r^{d - s} m(B_{r}(0)).
	\]
	Since \(x/\abs{x}\) has zero average we can subtract any constant vector \(\alpha\) in the integrand in the left-hand side without changing the value of the integral.
	The conclusion readily follows since \(x/\abs{x}\) is a unit vector.
\end{proof}

	When \(s > d - 1\), one has
	\[{}
	\int_{\partial B_{\rho}(0)} \frac{x - y}{\abs{x - y}^{s + 1}} \cdot \frac{x}{\abs{x}} \dif\sigma(x){}
	=
	\begin{cases}
	\displaystyle (s + 1 - d) \int_{\R^{d} \setminus B_{\rho}(0)} \frac{\dif x}{\abs{x - y}^{s + 1}}	 & \text{if \(y \not\in \overline{B_{\rho}(0)}\),}\\
	\displaystyle (d - s - 1) \int_{B_{\rho}(0)} \frac{\dif x}{\abs{x - y}^{s + 1}}	 & \text{if \(y \in {B_{\rho}(0)}\).}
	\end{cases}
	\]
	In particular, these values have opposite signs, in contrast with the previous proof.
	We prove in that case the following counterpart of Proposition~\ref{lemmaEstimateDensity} that is enough for our purposes:

\begin{proposition}
	\label{lemmaEstimateDensityNew}
	Let \(d - 1 < s < d\).{}
	For every \(\alpha \in \R^{d}\) and every \(r > 0\),
	\[{}
	\frac{m(B_{r}(a))}{r^{s}}
	\le C' \biggl( \fint_{B_{r}(a)}{\abs{V_{s} - \alpha}}
	+ r^{d - s} \int_{\R^{d} \setminus B_{r}(a)}{ \frac{\abs{V_{s}(x) - \alpha}}{\abs{x - a}^{2d - s}} \dif x } \biggr),
	\]
	for some constant \(C' > 0\) depending on \(s\) and \(d\).
\end{proposition}

As the vector-field \((x - a)/\abs{x - a}\) does not seem to be a convenient choice to test \(V_{s}\) in this range of \(s\), we rely on a different one that is provided by our next

\begin{lemma}
	\label{lemmaInversion}
	Let \(d - 1 < s < d\).{}
	For every \(\varphi \in C_{c}^{\infty}(\R^{d})\), there exists a summable smooth vector-field \(\Psi : \R^{d} \to \R^{d}\) such that \(\int_{\R^{d}}{\Psi} = 0\),
	\[{}
	\varphi(x)
	= \int_{\R^{d}}{\frac{x - y}{\abs{x - y}^{s + 1}} \cdot \Psi(y) \dif y}
	\quad \text{for every \(x \in \R^{d}\),}
	\]
	and
	\[{}
	\abs{\Psi(x)}
	\le \frac{C''}{(1 + \abs{x})^{2d - s}}
	\quad \text{for every \(x \in \R^{d}\),}
	\]
	for some constant \(C'' > 0\) depending on \(s\), \(d\) and \(\varphi\).
\end{lemma}

In terms of the Fourier transform, the vector-field \(\Psi\) satisfies
\[{}
\widehat{\varphi}
= \widehat{\frac{z}{\abs{z}^{s + 1}}} \cdot \widehat{\Psi}
= c \,  \frac{\xi}{\abs{\xi}^{d - s + 1}}  \cdot \widehat{\Psi}
\]
for a constant \(c \in \mathbb{C} \setminus \{0\}\).{}
This identity is satisfied for example with
\begin{equation}
	\label{eq-636}
\widehat{\Psi}
= \frac{1}{c} \abs{\xi}^{d - s - 1} \xi \, \widehat{\varphi},
\end{equation}
which indicates a convenient choice of \(\Psi\).

\resetconstant
\begin{proof}[Proof of Lemma~\ref{lemmaInversion}]
	Take \(\Psi : \R^{d} \to \R^{d}\) defined for \(x \in \R^{d}\) by
	\[{}
	\Psi(x)
	= c' \int_{\R^{d}}{\frac{\nabla \varphi(y)}{\abs{x - y}^{2d - s - 1}} \dif y},
	\]
	with \(c'\in \R\).{}
	Since \(s > d - 1\), the integral above is well-defined and one shows that \(\Psi\) satisfies identity \eqref{eq-636}
	for a suitable choice of \(c'\).
	That \(\Psi\) satisfies the desired relation with \(\varphi\) can also be checked without the Fourier transform, by an explicit computation of the convolution between the homogeneous functions \(z/\abs{z}^{\ell}\) and \(1/\abs{z}^{j}\)\,; see Lemma~15.10 and Exercise~15.4 in \cite{Ponce:2016} for dimensions \(d \ge 2\) and \(d = 1\), respectively.

	We observe that \(\Psi\) is bounded and summable.
	Moreover, smoothness of \(\Psi\) follows from a standard change of variable,
	\begin{equation}
		\label{eqPsi}
	\Psi(x)
	= c' \int_{\R^{d}}{\frac{\nabla \varphi(x - z)}{\abs{z}^{2d - s - 1}} \dif z}.
	\end{equation}
	Since \(\int_{\R^{d}}{\nabla \varphi} = 0\), we also get by integration on both sides of \eqref{eqPsi} and Fubini's theorem that
	\[{}
	\int_{\R^{d}}{\Psi} = 0.
	\]

	We now take \(R > 0\) such that \(\supp{\varphi} \subset B_{R}(0)\).{}
	To conclude the proof, it suffices to show that
	\begin{equation}
		\label{eqPsiEstimate}
	\abs{\Psi(x)}
	\le \frac{\C}{\abs{x}^{2d - s}}
	\quad \text{for every \(\abs{x} \ge 2R\).}
	\end{equation}
	To this end, we use again that \(\int_{\R^{d}}{\nabla \varphi} = 0\) to write
	\[{}
	\Psi(x)
	= c' \int_{\R^{d}}{\nabla \varphi(y) \biggl(\frac{1}{\abs{x - y}^{2d - s - 1}} - \frac{1}{\abs{x}^{2d - s - 1}} \biggr) \dif y}.
	\]
	For \(y \in B_{R}(0)\) and \(x \in \R^{d} \setminus B_{2R}(0)\), an application of the Mean Value Theorem gives \(0 \le \theta \le 1\) with
	\[{}
	\biggabs{\frac{1}{\abs{x - y}^{2d - s - 1}} - \frac{1}{\abs{x}^{2d - s - 1}}}
	\le \frac{\C \abs{y}}{\abs{x - \theta y}^{2d - s}}
	\le \frac{\Cl{eq-660}}{\abs{x}^{2d - s}},
	\]
	where \(\Cr{eq-660} > 0\) depends on \(s\), \(d\) and \(R\).
	Hence, when \(\abs{x} \ge 2R\) we get
	\[{}
	\abs{\Psi(x)}
	\le \int_{B_{R}(0)}{\abs{\nabla \varphi(y)} \biggabs{\frac{1}{\abs{x - y}^{2d - s - 1}} - \frac{1}{\abs{x}^{2d - s - 1}}} \dif y}
	\le \biggl( \int_{\R^{d}}{\abs{\nabla \varphi}} \biggr) \frac{\Cr{eq-660}}{\abs{x}^{2d - s}},
	\]
	which implies \eqref{eqPsiEstimate}.
	The estimate of \(\abs{\Psi(x)}\) in \(\R^{d}\) then follows from the boundedness of \(\Psi\).
\end{proof}

\resetconstant
\begin{proof}[Proof of Proposition~\ref{lemmaEstimateDensityNew}]
	We may assume that \(a = 0\).
	Fix a nonnegative function \(\varphi \in C_{c}^{\infty}(\R^{d})\) such that \(\varphi \ge 1\) in \(B_{1}(0)\) and let \(\Psi : \R^{d} \to \R^{d}\) be the vector-field given by Lemma~\ref{lemmaInversion}.
	For \(r > 0\),  one has by scaling that
	\[{}
	\varphi\Bigl(\frac{x}{r}\Bigr)
	= \int_{\R^{d}}{\frac{x - y}{\abs{x - y}^{s + 1}} \cdot \Psi_{r}(y) \dif y}
	\quad \text{for every \(x \in \R^{d}\),}
	\]
	where \(\Psi_{r}(x) \vcentcolon= \frac{1}{r^{d - s}} \Psi(\frac{x}{r})\).{}
		Since \(\varphi \ge 1\) on \(B_{1}(0)\) and \(m\) is a nonnegative measure,{}
	we have
	\[{}
	m(B_{r}(0))
	\le \int_{\R^{d}}{\varphi\Bigl(\frac{x}{r}\Bigr) \dif m(x)}
	= \int_{\R^{d}}{\biggl( \int_{\R^{d}}{\frac{x - y}{\abs{x - y}^{s + 1}} \cdot \Psi_{r}(y) \dif y} \biggr) \dif m(x)}.
	\]
	Then, by Fubini's theorem,
	\[{}
	m(B_{r}(0))
	\le - \int_{\R^{d}} V_{s} \cdot \Psi_{r} \,.
	\]
	Since \(\int_{\R^{d}}{\Psi_{r}} = 0\), for every \(\alpha \in \R^{d}\) we get
	\begin{equation}
		\label{eq-697}
	m(B_{r}(0))
	\le - \int_{\R^{d}} (V_{s} - \alpha) \cdot \Psi_{r}
	\le \int_{\R^{d}} \abs{V_{s} - \alpha} \abs{\Psi_{r}}.
	\end{equation}
	The pointwise estimate satisfied by \(\Psi\) gives
	\begin{equation}
		\label{eq-708}
	\abs{\Psi_{r}(x)} = \frac{1}{r^{d - s}} \Bigabs{\Psi\Bigl(\frac{x}{r}\Bigr)}
	\le \C \Bigl( \frac{1}{r^{d - s}} \chi_{B_{r}(0)}(x) + \frac{r^{d}}{\abs{x}^{2d - s}} \chi_{\R^{d} \setminus B_{r}(0)}(x) \Bigr).
	\end{equation}
	The conclusion follows by inserting \eqref{eq-708} in \eqref{eq-697}.
\end{proof}

To handle the additional term that appears in Proposition~\ref{lemmaEstimateDensityNew},
compared to Proposition~\ref{lemmaEstimateDensity}, we need the following

\begin{lemma}
	\label{lemmaComputation}
	If \(f \in L^{1}(\R^{d})\) is such that
	\[{}
	\lim_{r \to 0}{\fint_{B_{r}(a)}{f}}
	= 0,
	\]
	then, for every \(\beta > 0\),{}
	\[{}
	\lim_{r \to 0}{r^{\beta} \int_{\R^{d} \setminus B_{r}(a)} \frac{f(x)}{\abs{x - a}^{d + \beta}} \dif x}
	= 0.
	\]
\end{lemma}

\resetconstant
\begin{proof}
	We first prove that, for every \(r > 0\),{}
	\begin{equation}
	\label{eq-738}
	\int_{\R^{d} \setminus B_{r}(a)}{\frac{f(x)}{\abs{x - a}^{d + \beta}} \dif x}
	= \Cl{cte-803} \int_{r}^{\infty} \frac{1}{\rho^{1 + \beta}} \biggl( \fint_{B_{\rho}(a)}{f} \biggr) \dif\rho -  \frac{\Cl{cte-803-bis}}{r^{\beta}}\fint_{B_{r}(a)}{f},
	\end{equation}
	for some constants \(\Cr{cte-803}, \Cr{cte-803-bis} > 0\).
	By the integration formula in polar coordinates,
	\[{}
	\begin{split}
	\int_{\R^{d} \setminus B_{r}(a)}{\frac{f(x)}{\abs{x - a}^{d + \beta}} \dif x}
	& = \int_{r}^{\infty}\biggl( \int_{\partial B_{\rho}(a)} \frac{f(x)}{\abs{x - a}^{d + \beta}} \dif\sigma(x)  \biggr) \dif\rho{}\\
	& = \int_{r}^{\infty} \frac{1}{\rho^{d + \beta}} \biggl( \frac{\dif}{\dif\rho} \int_{B_{\rho}(a)} f \biggr) \dif\rho.
	\end{split}
	\]
	One then gets \eqref{eq-738} by integration by parts.
	To conclude, it suffices to apply Lemma~\ref{lemmaLimit} with \(h(t) = \fint_{B_{t}(a)}{f}\) for \(t > 0\).
\end{proof}

\resetconstant
\begin{proof}[Proof of Theorem~\ref{theoremRieszLebesgue}] ``\(\Longrightarrow\)''.
	Let \(a\) be a Lebesgue point of \(V_{s}\).{}
	We first show that the limit \eqref{eq-328} holds.
	To this end, denote by \(\alpha\) the precise representative of \(V_{s}\) at \(a\).{}
	When \(0 < s \le d - 1\), we deduce from Proposition~\ref{lemmaEstimateDensity} that
	\[
	\limsup_{r \to 0}{\frac{m(B_{r}(a))}{r^{s}}}
	\le C \lim_{r \to 0}{\fint_{B_{2r}(a)}{\abs{V_{s} - \alpha}}}
	= 0,
	\]
	which implies \eqref{eq-328}.
	When \(d - 1 < s < d\)\,, we apply Proposition~\ref{lemmaEstimateDensityNew}.
	In this case, Lemma~\ref{lemmaComputation} with \(f = |V_{s} - \alpha|\) and \(\beta = d - s\) handles the additional term in the estimate as \(r \to 0\) and we get \eqref{eq-328} as before.

	We next recall that
	\[{}
	\alpha = \lim_{r \to 0}{\fint_{B_{r}(a)}{V_{s}}}.
	\]
	Moreover, by estimate \eqref{eq-334} in the proof of Theorem~\ref{theoremRieszLebesgue}, we have
	\begin{multline*}
	\biggabs{\fint_{B_{r}(a)} V_{s} - \int_{\R^{d} \setminus B_{2r}(a)} \frac{a - y}{\abs{a - y}^{s + 1}} \dif m(y)}\\
	\le \C \biggl( r \int_{2r}^{\infty} \frac{m(B_{t}(a))}{t^{s + 2}} \dif t + \frac{m(B_{2r}(a))}{r^{s}} \biggr){}.
	\end{multline*}
	As \(r \to 0\), using \eqref{eq-328} we deduce from Lemma~\ref{lemmaLimit} that the right-hand side converges to zero and then
	\[{}
	\alpha = \lim_{r \to 0}{\fint_{B_{r}(a)}{V_{s}}} = \lim_{r \to 0}{\int_{\R^{d} \setminus B_{2r}(a)} \frac{a - y}{\abs{a - y}^{s + 1}} \dif m(y)}.
	\]
	Thus, by definition, \(\alpha\) is the principal value of \(V_{s}\) at \(a\).
\end{proof}

\section{The capacity $\kappa_{s}$ and proof of Theorem~\ref{theoremCapacity}}
\label{sectionCapacity}

	We begin by showing that \(\kappa_{s}\) is subadditive:

\begin{proposition}
	\label{propositionSubadditivity}
	For every compact subsets \(E_{1}, E_{2} \subset \R^{d}\),{}
	\[{}
	\kappa_{s}(E_{1} \cup E_{2})
	\le \kappa_{s}(E_{1}) + \kappa_{s}(E_{2}).
	\]
\end{proposition}

The capacity \(\kappa_{s}\) is equivalent to another one that was known to be semi-additive (that is, the estimate above is verified with a constant \(C \ge 1\)); see p.~3643 in \cite{RuizdeVilla-Tolsa} and Section~2 in \cite{Prat:2012}.

Before proving the proposition, we start with the following observation:
	If \(\mu\) is a Borel measure in \(\R^{d}\) that is admissible in the definition of \(\kappa_{s}(E)\) and if \(F \subset E\) is a compact subset, then, for any Borel subset \(A \subset F\), the Borel measure \(\mu\lfloor_{A}\) defined by
	\[{}
	\mu\lfloor_{A}(B) = \mu(A \cap B)
	\]
	is admissible for \(\kappa_{s}(F)\).{}
	Indeed, it suffices to verify the \(L^{2}\) boundedness of the maximal Riesz transform.
	Since \(R_{s}^{*}\) is bounded in \(L^{2}(\mu)\), we can estimate
	\[{}
	\begin{split}
	\int_{\R^{d}}{\abs{R_{s}^{*}(f\mu\lfloor_{A})}^{2} \dif \mu\lfloor_{A}}
	& \le \int_{\R^{d}}{\abs{R_{s}^{*}(f\chi_{A} \mu)}^{2} \dif \mu}\\
	& \le \int_{\R^{d}}{\abs{f\chi_{A}}^{2} \dif \mu}
	= \int_{\R^{d}}{\abs{f}^{2} \dif \mu\lfloor_{A}},
	\end{split}
	\]
	which justifies our assertion.

\begin{proof}[Proof of Proposition~\ref{propositionSubadditivity}]
	Let \(\mu\) be a nonnegative Borel measure supported in \(E_{1} \cup E_{2}\) that satisfies (a) and (b)
	in the definition of $\kappa_s$ for this set.
	By the computation above, the measure \(\mu\lfloor_{E_{1}}\) is admissible for \(\kappa_{s}(E_{1})\) and \(\mu\lfloor_{E_{2} \setminus E_{1}}\) is admissible for \(\kappa_{s}(E_{2})\).{}
	Thus,
	\[{}
	\mu(\R^{d})
	= \mu(E_{1} \cup E_{2})
	= \mu(E_{1}) + \mu(E_{2} \setminus E_{1})
	\le \kappa_{s}(E_{1}) + \kappa_{s}(E_{2})
	\]
	and it suffices to take the supremum with respect to \(\mu\).
\end{proof}

It follows readily from (a) in the definition of \(\kappa_{s}\) that
\(\kappa_{s}(E) = 0\) for every compact set \(E\) with \(\cH^{s}(E) < \infty\).{}
Dimension \(s\) is critical for \(\kappa_{s}\)\,:{}

\begin{proposition}
	\label{propositionCapacityHausdorffDimension}
	If \(E \subset \R^{d}\) is compact and \(\kappa_{s}(E) = 0\), then \(\dim_{\mathcal{H}}{(E)} \le s\).
\end{proposition}

\resetconstant
\begin{proof}
	It suffices to prove that if \(E \subset \R^{d}\) is a compact set with \(\cH^{t}(E) > 0\) for some \(t > s\), then \(\kappa_{s}(E) > 0\).{}
	By Frostman's lemma, there exists a nontrivial finite nonnegative Borel measure \(\mu\) supported by \(E\) such that
	\begin{equation}
	\label{eq-546}
	\mu(B_{r}(x))
	\le r^{t}
	\quad \text{for every \(x \in \R^{d}\) and \(r > 0\).}
	\end{equation}
	Hence, \(\mu \le \C \cH^{t}_{\infty}\)\,, where \(\cH^{t}_{\infty}\) is the Hausdorff content of dimension \(t\).{}
	For every set \(A \subset \R^{d}\) with \(\sigma\)-finite \(\cH^{s}\) measure, we have \(\cH^{t}_{\infty}(A) = 0\).
	Then, \(\mu(A) = 0\) and the first requirement in the definition of \(\kappa_{s}(E)\) is satisfied.

	Next,
	for every \(f \in (L^{1} \cap L^{2})(\mu)\) and every \(x \in \R^{d}\),
	\[{}
	\abs{R_{s}^{*}(f\mu)(x)}
	\le \int_{\R^{d}} \frac{\abs{f(y)}}{\abs{x - y}^{s}} \dif \mu(y).
	\]
	Then, by Young's inequality,
	\[{}
	\norm{R_{s}^{*}(f\mu)}_{L^{2}(\mu)}
	\le \biggl(\int_{\R^{d}} \frac{\dif \mu(z)}{\abs{z}^{s}} \biggr) \norm{f}_{L^{2}(\mu)}.
	\]
	By Cavalieri's principle,
	\[{}
	\int_{\R^{d}}{\frac{\dif \mu(z)}{\abs{z}^{s}}}
	= s \int_{0}^{\infty} \frac{\mu(B_{r}(0))}{r^{s + 1}} \dif r.
	\]
	Since \(\mu\) is finite and satisfies \eqref{eq-546} with exponent \(t > s\),{}
	\[{}
	\int_{0}^{\infty} \frac{\mu(B_{r}(0))}{r^{s + 1}} \dif r
	\le \int_{0}^{1} \frac{r^{t}}{r^{s + 1}} \dif r + \int_{1}^{\infty} \frac{\mu(\R^{d})}{r^{s + 1}} \dif r
	< \infty.
	\]
	Therefore,
	\[{}
	\norm{R_{s}^{*}(f\mu)}_{L^{2}(\mu)}
	\le \Cl{cte-423} \norm{f}_{L^{2}(\mu)},
	\]
	which implies that \(\mu/\Cr{cte-423}\) is a nontrivial admissible measure in the definition of \(\kappa_{s}(E)\).
	In particular, \(\kappa_{s}(E) > 0\).
\end{proof}

The proof of Theorem~\ref{theoremCapacity} relies on Theorem~1.6 of \cite{Mattila-Verdera} concerning the existence of principal values for the Riesz transform.
We recall that from \cite{Mattila-Verdera} one knows that, for every finite nonnegative Borel measure \(\mu\) in \(\R^{d}\) such that
\begin{equation}
\label{eq-692}
	\lim_{r \to 0}{\frac{\mu(B_{r}(x))}{r^{s}}}
	= 0
	\quad \text{for \(\mu\)-almost every \(x \in \R^{d}\)}
\end{equation}
and
\begin{equation}
\label{eq-698}
	\norm{R_{s}^{*}(f\mu)}_{L^{2}(\mu)}
	\le C \norm{f}_{L^{2}(\mu)}
	\quad \text{for every \(f \in (L^{1} \cap L^{2})(\mu)\)},
\end{equation}
the principal value of
\begin{equation}
\label{eq-705}
	x \longmapsto \int_{\R^{d}} \frac{x - y}{\abs{x - y}^{s + 1}} \dif\mu(y)
\end{equation}
	exists \(\mu\)-almost everywhere.

	We observe that in Theorem~1.6 of \cite{Mattila-Verdera} one also assumes
	\[{}
	\mu(B_{r}(x))
	\le C' r^{s} \quad \text{for every \(x \in \R^{d}\) and \(r > 0\),}
	\]
	but such a property is a consequence of the uniform boundedness of the Riesz transform given by \eqref{eq-698} and the
	fact that, by \eqref{eq-692}, the measure \(\mu\) cannot charge points; see Proposition~1.4 in Part~III of \cite{David}.

	We also need a standard property from Measure Theory:

\begin{proposition}
	\label{propositionMattilaVerdera}
	If \(\mu\) is a finite nonnegative Borel measure in \(\R^{d}\) such that \(\mu(A) = 0\) for every Borel set \(A \subset \R^{d}\) with \(\sigma\)-finite \(\cH^{s}\) measure, then \eqref{eq-692} holds.
\end{proposition}

\begin{proof}
	Since \(\mu\) does not charge sets with finite \(\mathcal{H}^{s}\) measure, by Proposition~3.2 in \cite{Ponce} for every \(\epsilon > 0\) one finds a compact set \(K \subset \R^{d}\) with \(\mu(\R^{d} \setminus K) \le \epsilon\) such that, for every \(c > 0\), there exists \(\delta > 0\) with
	\begin{equation*}
	\mu\lfloor_{K}(B_{r}(x))
	\le c r^{s} \quad \text{for every \(x \in \R^{d}\) and \(0 < r \le \delta\).}
	\end{equation*}
	Thus,
\begin{equation}
\label{eq-1009}
	\lim_{r \to 0}{\frac{\mu\lfloor_{K}(B_{r}(x))}{r^{s}}}
	= 0
	\quad \text{for every \(x \in \R^{d}\).}
\end{equation}
	On the other hand, by the Besicovitch Differentiation Theorem,
\begin{equation}
\label{eq-1016}
	\lim_{r \to 0}{\frac{\mu\lfloor_{K}(B_{r}(x))}{\mu(B_{r}(x))}}
	= 1
	\quad \text{for \(\mu\)-almost every \(x \in K\).}
\end{equation}
	Combining \eqref{eq-1009} and \eqref{eq-1016}, one gets the limit in \eqref{eq-692} for \(\mu\)-almost every \(x \in K\).
	Since \(\mu(\R^{d} \setminus K) \le \epsilon\) and \(\epsilon > 0\) is arbitrary, the conclusion follows.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{theoremCapacity}]
	Let \(\mu\) be a finite nonnegative Borel measure that is admissible for \(\kappa_{s}(E)\), where \(E\) is any compact subset of \(\mathcal{E}(V_{s})\).{}
	By Theorem~\ref{theoremRieszLebesgue}, we can write this set as \(E = E_{1} \cup E_{2}\), where
	\[{}
	E_{1}
	\vcentcolon= \biggl\{ x \in E : \limsup_{r \to 0}{\frac{\mu(B_{r}(x))}{r^{s}}} > 0 \biggr\}
	\]
	and
	\[{}
	E_{2}
	\vcentcolon= \bigl\{ x \in E : \mathrm{p.v.}\,V_{s}(x) \text{ does not exist} \bigr\}.
	\]
	On one hand, since \(E_{1}\) is \(\sigma\)-finite for \(\mathcal{H}^{s}\) and \(\mu\) does not charge those sets,
	\(
	\mu(E_{1}) = 0.
	\)
	On the other hand, from Proposition~\ref{propositionMattilaVerdera} above and Theorem~1.6 of \cite{Mattila-Verdera} we know that the principal value of \(V_{s}\) exists \(\mu\)-almost everywhere in \(\R^{d}\).
	Hence,
	\(
	\mu(E_{2}) = 0.
	\)
	We thus have
	\[{}
	\mu(\R^{d})
	= \mu(E)
	\le \mu(E_{1}) + \mu(E_{2})
	= 0,
	\]
	for every measure \(\mu\) that is admissible for \(\kappa_{s}(E)\).
	Therefore, \(\kappa_{s}(E) = 0\).
\end{proof}

	As a final observation, \eqref{eqHausdorffDimension} can also be deduced as a consequence of
	Theorem~\ref{theoremCapacity}.
	Indeed, if we had \(\dim {\mathcal{E}(V_{s})} > s\), then for any \(s < t < \dim{\mathcal{E}(V_s)}\) one could find a
	compact subset \(F \subset \mathcal{E}(V_{s})\) such that \(\mathcal{H}^{t}(F) > 0\); see Theorem~8.13 in \cite{Mattila}.
	Such a property would then contradict Proposition~\ref{propositionCapacityHausdorffDimension} above.

\section*{Acknowledgements}

The authors would like to thank L.~Grafakos for an interesting correspondence.
J.~Cuf\'i acknowledges partial support from grant 2017SGR358 (Generalitat de Catalunya) and J.~Verdera from grants 2017SGR395
(Generalitat de Catalunya) and  MTM2016--75390 (Mi\-nis\-terio de
Educaci\'{o}n y Ciencia).
A.~C. Ponce is grateful for the invitation, hospitality and support of the Departament de Matemàtiques of the Universitat Autònoma de Barcelona.
He also acknowledges support of the Fonds de la Recherche scientifique (F.R.S.--FNRS) from grant J.0020.18.

\end{document}

config: arxiv


TITLE: solution to an ODE separable variable
QUESTION [0 upvotes]: I have this ODE: $du/u = dt$ which I think the solution is $\ln|u| = t + c \leftrightarrow |u| = e^{t+c} = ce^t$. Is this correct? Is $u$ really a function then, since each value of $t$ would be mapped onto two values of $u$?

REPLY [0 votes]: Here are the steps
$$ \frac{1}{u}du=dt $$
$$ \int \frac{1}{u}du=\int dt $$
$$ \ln |u|+C_1=t +C_2 $$
$$ \ln |u|=t +C_2 -C_1=t+C $$
$$ e^{\ln |u|}=e^{t+C} $$
$$ u=e^{t+C} $$
Note that the domain of $\frac{1}{u}$ is $\{u\in\mathbb{R}:u\ne 0\}$, which includes negative numbers. Also the domain of $\ln u$ is $\{u\in\mathbb{R}:u\gt 0\}$, which does not include negative numbers. This is why when we integrate $\frac{1}{u}$, we get $\ln|u|+C$. Once we get rid of the $\ln|u|$, we can also get rid of the absolute value bars because $u$ in this case, is allowed to be negative.