TITLE: Show that $\sup\left\{\frac{1}{\varepsilon}\int_{\{|f|<\varepsilon\}}|f'(x)|dx:\varepsilon\in(0,1]\right\}$ is finite
QUESTION [7 upvotes]: Let $[a,b]$ be a closed and bounded interval and let $f:[a,b]\to\mathbb{R}$ be a continuously differentiable function $f\in C^1([a,b])$. I am trying to bound from above the following expression $$\frac{1}{\varepsilon}\int_{\{|f|<\varepsilon\}}|f'(x)|dx$$ uniformly as $\varepsilon\downarrow 0$.
I would like to prove $$\sup\left\{\frac{1}{\varepsilon}\int_{\{|f|<\varepsilon\}}|f'(x)|dx:\varepsilon\in(0,1]\right\}$$ is finite but I have no idea of how to do it. Any suggestions?

Olivier made me realise I should probably explain how this came up, so I will explain the original question. Consider the step function approximation of Dirac's delta $\delta^\varepsilon(t)=\frac{1}{2\varepsilon}\textbf{1}_{\{|t|<\varepsilon\}}$. I would like to justify for $f\in C^1(\mathbb{R})$ the equation $$\lim_{\varepsilon\downarrow 0}\int_0^T\delta^\varepsilon(f(t))|f'(t)|dt=\int_0^T\lim_{\varepsilon\downarrow0}\delta^\varepsilon(f(t))|f'(t)|dt$$
Any hints on this would be equally helpful.

REPLY [1 votes]: This is not true.
Suppose $f$ has an infinite number of zeros $z$ for which $f'(x)$ is a constant $M_z\not = 0$ in a small neighborhood of $z$. For any $\varepsilon > 0$, let $z_1, z_2, \dots, z_n$ be zeros of $f$ and $E_i \ni z_i$ be connected components of $\{x : |f(x)| < \varepsilon\}$ such that $f'|_{E_i} = M_i \not = 0$. Denote by $l(E_i)$ the measure of $E_i$. Then $l(E_i) |M_i| = 2\varepsilon$ and
$$
\frac{1}{\varepsilon} \int_{|f| < \varepsilon} |f'(x)| dx \geq \frac{1}{\varepsilon}\sum_{i=1}^n l(E_i)|M_i| = 2n.
$$
Taking $\varepsilon > 0$ sufficiently small, we can find an arbitrarily large number $n$ of such zeros $z_i$ and components $E_i$. The supremum is therefore infinite.
Note: The argument could be refined and applied to $f(x) = x^3 \sin (1/x)$, defined on $[0,1]$, for instance.


TITLE: Does the gradient always point outward of a level surface?
QUESTION [4 upvotes]: Let $f:\mathbb{R}^n\to \mathbb{R}$ be a differentiable function, with $a\in \mathbb{R}$ a regular value of $f$.
Let $M=f^{-1}((-\infty,a])$. Then $M$ is an $n$-manifold with boundary, whose boundary is $\partial M=f^{-1}(a)$.
Let $p\in \partial M$. Then $T_p\partial M=\{\nabla f(p)\}^{\perp}$.

Question: Does $\nabla f(p)$ point outwards of $\partial M$?

I know it's not true if we take the interval the other way around. E.g. $-1$ is a regular value of $f:\mathbb{R}^2\to \mathbb{R}$, $f(x,y)=-x^2-y^2$, but the gradient of $f$ points inwards of the boundary of $f^{-1}([-1,+\infty))$ which is $S^1$.
In fact, by symmetry, if the answer to the question is positive, I suspect that if we take $M=f^{-1}([a,+\infty))$ then $\nabla f(p)$ always points inwards of $\partial M$.

REPLY [0 votes]: I have managed to formulate my own proof of this theorem. I give full detail below.
Definition: Let $M$ be an $n$-manifold with boundary, $p\in \partial M$ and $\varphi:U\subset \mathbb{H}^n\to M$ a parametrization such that $\varphi(q)=p$. We say that $v\in T_pM$ is inward-pointing (resp. outward-pointing) if it is of the form $d\varphi_q(v_0)$ with $v_0\cdot e_n>0$ (resp. $v_0\cdot e_n<0$).
Let's check that this definition does not depend on the choice of parametrization, and at the same time give a useful criterion.
Theorem. Let $M$ be an $n$-manifold with boundary, $p\in \partial M$ and $v\in T_pM$. The following are equivalent:

There exists a parametrization $\varphi:U\to M$ such that, if $\varphi(q)=p$ and $v=d\varphi_q(v_0)$, then $v_0$ is such that $v_0\cdot e_n>0$.
$v\not\in T_p\partial M$ and there exists a curve $\alpha:[0,\epsilon)\to M$ such that $\alpha(0)=p$ and $\alpha'(0)=v$.
For every parametrization $\varphi:U\to M$ we have that, if $\varphi(q)=p$ and $v=d\varphi_q(v_0)$, then $v_0$ is such that $v_0\cdot e_n>0$.

Proof: ($1 \Rightarrow 2$) Define $\alpha:[0,\epsilon)\to M$ as $\alpha(t)=\varphi(q+tv_0)$, taking $\epsilon$ small enough for $q+tv_0$ to be in $U$ for all $t$. Then $\alpha(0)=p$, and by the chain rule, $\alpha'(0)=v$.
($2 \Rightarrow 3$) Let $\varphi:U\to M$ be a parametrization with $\varphi(q)=p$ and $v=d\varphi_q(v_0)$. Since $\alpha(0)=p\in \varphi(U)$, we may suppose that $\alpha([0,\epsilon))\subset \varphi(U)$, taking a smaller $\epsilon$ if necessary. We can define then a curve $\beta:=\varphi^{-1}\circ \alpha:[0,\epsilon)\to U$ with $\beta(0)=q$ and $\beta'(0)=v_0$.
Now, $\beta'(0)=\lim_{t\to 0^+} \frac{\beta(t)-q}{t}$. But $\beta(t)-q\in \mathbb{H}^n$ for all $t$, therefore $\beta'(0)=v_0\in \mathbb{H}^n$. Since we further assumed that $v\not\in T_p\partial M$, then $v_0\not\in \partial \mathbb{H}^n$, whence $v_0\cdot e_n>0$.
($3 \Rightarrow 1$) It is obvious. $\blacksquare$
Theorem. Let $f:M\to N$ be a differentiable function between manifolds with boundary. Let $p\in \partial M$ and $v\in T_pM$. If $v$ is inward-pointing, then $df_p(v)\in T_{f(p)}N$ is inward-pointing.
Proof: Let $\alpha:[0,\epsilon)\to M$ be a curve such that $\alpha(0)=p$, $\alpha'(0)=v$. Consider $f\circ \alpha:[0,\epsilon)\to M$. We have $f(\alpha(0))=f(p)$, and by the chain rule, $(f\circ \alpha)'(0)=df_p(v)$. Therefore $df_p(v)$ is inward-pointing. $\blacksquare$
Theorem: Let $f:\mathbb{R}^n\to \mathbb{R}$ be a differentiable function with $a\in \mathbb{R}$ a regular value. So $M=f^{-1}((-\infty,a])$ is a manifold with boundary such that $\partial M=f^{-1}(a)$. Let $p\in \partial M$. Then $\nabla f(p)$ is outward-pointing.
Proof: We first observe that $\nabla f(p)\not\in T_p\partial M$, because $T_p\partial M=\{\nabla f(p)\}^\perp$. Assume by contradiction that $\nabla f(p)$ is inward-pointing.
By considering $f|_M:M\to (-\infty,a]$, by the previous proposition we conclude that $df_p(\nabla f(p))\in \mathbb{R}$ is inward pointing to $(-\infty,a])$. Explicitly, this means that $df_p(\nabla f(p))<0$.
Indeed, a vector $v\in \mathbb{R}$ is inward-pointing to $(-\infty,a])$ if and only if $v\neq 0$ and there exists a curve $\alpha:[0,\epsilon)\to (-\infty,a]$ such that $\alpha(0)=a, \alpha'(0)=v$. Since $\alpha(t)\leq a$ for all $t$, then $v=\alpha'(0)=\lim_{t\to 0^+} \frac{\alpha(t)-a}{t}\leq 0$, therefore $v<0$.
We have thus that $df_p(\nabla f(p))<0$. We reach a contradiction, since $df_p(\nabla f(p))=\nabla f(p)\cdot \nabla f(p)=\|\nabla f(p)\|^2>0$. $\blacksquare$


TITLE: How to determine the minimum of the sum?
QUESTION [0 upvotes]: The sequence $$1,2,3,...n$$ is taken ,if we take all the different possiblities of $+$ or $-$in between all successive terms ,what will be the minimum between all those (as in the minimum of the absolute value) .
My work:
I found that  we can define a vector valued function that :
$$f:\mathbb R^n \mapsto \mathbb R$$ and if we define $\mathrm v $to be the vector:$$(\pm1,\pm1,\pm1...,\pm1)$$ and the vector $\mathrm w$ to be:$$(1,2,3,...,n)$$ the function f is simply the dot product:$$ \vert v \cdot  w\vert$$.
And also i found out that the function is multivalued(using examples),and that being multivalued has to do with number of partitions of a number ,other than that I have no clue on how to analyze this...

REPLY [1 votes]: Lemma. A sequence of $4k$ consecutive positive integers can have $\pm$ signs assigned to its terms so that their sum becomes $0$.
Let the numbers be $m+1,m+2,\cdots,m+4k$. Each pair of symmetric terms $m+j,m+4k+1-j$ adds up to $m+j+(m+4k+1-j) = 2m+4k+1$. There are $2k$ such pairs, so assigning $k$ of them the $+1$ sign, and the other $k$ pairs the $-1$ sign, their sum will be $0$.
Proof. For an arbitrary $n$ let $k = \lfloor \frac{n}{4}\rfloor\,$ and choose signs per previous lemma for the last $4k$ terms so that they add up to $0\,$.
This leaves the first $n \bmod 4$ terms to consider.

If $n \bmod 4 = 0$ then there are no terms left, and the minimum sum is $0$.
If $n \bmod 4 = 1$ then the minimum sum is $+1 = 1$.
If $n \bmod 4 = 2$ then the minimum sum is $-1+2=1$.
If $n \bmod 4 = 3$ then the minimum sum is $-1-2+3=0$.

To prove that $1$ is in fact the minimum sum in the cases $n \bmod 4 = 1,2$ note that each sign change of one term $k$ modifies the sum by $2k$ i.e. an even number, so it preserves the parity of the sum. But the sum $\sum_{k=1}^n k$ is odd when $n \bmod 4 = 1,2$ (as can be easily verified) so there exists no combination of signs to bring the sum down to $0$, therefore $1$ is the best that can be achieved.


text:
\appendices
\section{Analysis for non-integer $\Kavg$}
\label{appendix_groupsAB}
The average number of scheduled users per slot is defined as $\Kavg\defined\Ktot/ \suf$. In case $\Kavg$ is not an integer number, the scheduler must sometimes select more than $\Kavg$ users, sometimes less.
Specifically, the transmitter schedules $K_\aslot=\left\lceil\, \Kavg\, \right\rceil$ in $\suf_\aslot$ time slots, and $K_\bslot=\lfloor \Kavg \rfloor$ in $\suf_\bslot=\suf-\suf_\aslot$ time slots, where $\suf_\aslot$ and $\suf_\bslot$ can be determined from $\suf_\aslot K_\aslot+\suf_\bslot K_\bslot=\Ktot$.
As the power $\Ptot$ is still shared equally between the $K_\aslot$ or $K_\bslot$ scheduled users, each users codeword is transmitted with power $\fixpower_{\aslot}=\Ptot/K_{\aslot}$ or $\fixpower_{\bslot}=\Ptot/K_{\bslot}$. Thus, the users in the type $\bslot$ slots benefit both from a higher beamforming gain and from a higher transmit power.
To maintain fairness between all users, the users are randomly assigned to the $\aslot$ or $\bslot$ slots in each superframe.
The probability of being scheduled in a type $\aslot$ or type $\bslot$ slot is given as $p_{\aslot}=K_{\aslot} \suf_{\aslot}/\Ktot$ and $p_{\bslot}=K_{\bslot} \suf_{\bslot}/\Ktot$, respectively \cite{schiessl2018misoglobecom}.

The Mellin transform of the service $\Ssnr\sufss$ experienced by each user can be obtained by averaging over the Mellin transforms of the service process for the type $\aslot$ or $\bslot$ slots:
\begin{equation}
\Mellin_{\Ssnr\sufss}(1-\s) = p_{\aslot}\Mellin_{\Ssnr\sufss|K_\aslot}(1-\s) + p_{\bslot}\Mellin_{\Ssnr\sufss|K_\bslot}(1-\s)
\label{eq:mellin_service_fixedpowers}
\,,
\end{equation}
where the Mellin transform $\Mellin_{\Ssnr\sufss|K}(\s)$ for a specific number of users $K$ is given by \eqref{eq:mellin_service_constpower_series} for perfect CSI, or by \eqref{eq:mellin_quantized} when considering imperfect CSI and finite blocklength effects.

config: arxiv


TITLE: Maximality of an ideal for showing that an algebra is in fact a field
QUESTION [5 upvotes]: I have an algebra $A$ over the field $F$, with the finite dimensionality $n$ as a vector space over $F$. I can also assume that $A$ is an integral domain. Assuming that $v_1,...,v_n$ is a spanning list of vectors and that $v_1=1$, I believe I can represent $A$ as
$F[v_1,v_2,...,v_n]/I$ where $$I=\left(v_1-1,\ v_iv_j\ \forall\ 2\leq i,j\leq n\right).$$
Now, I'd hope to show that $A$ is a field by proving the maximality of $I$, but I can't figure out how to show that.

REPLY [2 votes]: Here's a different way to show $A$ is a field. We will simply show that if $a\in A$ is nonzero, then $a$ has a multiplicative inverse in $A$. Consider the $F$-linear map $M_a:A\to A$ given by $M_a(b) = a\cdot b$. It is an injective $F$-linear map between $F$-vector spaces of the same (finite) dimension, and so it is also surjective. Therefore, $M_a(b) = 1$ for some $b\in A$, i.e., $a\cdot b = 1$.


TITLE: What's an easy way of proving a subgroup is normal?
QUESTION [19 upvotes]: I think in most situations(for example, in $S_n$ or $D_n$), proving by definition is too complicated because you have to calculate $gng^{-1}$ for every $n$ in $N$ and $g$ in $G$. To prove that all the left cosets are also right cosets is also too complicated because you have to find all those cosets. I wonder if there's a way to do this without having to calculate everything by hand.

REPLY [3 votes]: If you know a subgroup of a particular order is the ONLY subgroup of that order, then you know it's normal. I know that's a unique case, but just another tool to keep in mind.


text:
\section{Preliminaries}
\label{sec:preliminaries}

Before we give a brief recap on the general hypocoercivity method
and establish several geometrical tools,
we fix
the assumptions on the position manifold.

\begin{comment}
\begin{notation}[Some paradigmas]
\hfill

\begin{itemize}
    \item \textit{blackboard bold:}
        Usually \posfold, \sphere, \basefold, \totalfold etc.\ refer to certain manifolds.
        Exceptions: \indexset is a time interval and \ball signifies some open ball.
    \item \textit{calligraphic:}
    \begin{itemize}[label=\textopenbullet]
        \item We use \verb+\mathcal+ for vector fields or certain bilinear forms.
        \item The other set of calligraphic symbols,
            for instance \posdim, \callidim{b} etc.,
            is reserved exclusively for (positive) dimensions.
    \end{itemize}
    \item \textit{roman:}
        We use roman letters in math mode frequently --
        some might raise objections to this habit.
        Specifically in this paper we denote Riemannian metrics by small roman letters.
        When introducing a weight on a manifold
        the symbol of a Riemannian metric will become boldfaced.
    \item \textit{sans serif:}
        Sans serif letters loosely refer to Lagrangian or Hamiltonian formalism.
    \item \textit{\Leb:}
        Everything with a name involving~\Leb is inherently similar to a Lebesgue measure.
\end{itemize}
\vspace*{-2ex}
\end{notation}
\end{comment}

\begin{condition}[Position manifold~\ref{cond:M}]
\namedlabel{cond:M}{(M)}
\hfill

\begin{enumerate}[label={(M\arabic*)}, ref={(M\arabic*)}]
    \item\label{cond:M:general}
    \emph{general geometry:}\
        Let $(\posfold,\riemannmetric*)$ be a
        real,
        finite dimensional,
        connected
        Riemannian manifold
        with $\posdim\vdef\dim(\posfold)\geq2$.
    \item\label{cond:M:complete}
    \emph{completeness:}\
        Let~\posfold endowed with the intrinsic metric be a complete metric space\footnote{
            The intrinsic metric induces the same topology as given on~\posfold originally.
            This is a consequence of the very definition of a (topological) manifold
            as in~\cite{Lee}
            which we choose to use here.
        }.
\end{enumerate}
\vspace*{-1.5em}
\end{condition}

Note that following a common way of speaking,
the manifold as described in~\ref{cond:M:general} has trivial boundary $\partial\posfold = \emptyset$.
In this paper, we do not address boundary problems for sake of simplicity.
From an analytic point of view,
assumption~\ref{cond:M:complete} is plausible in it's own right
and necessary e.\,g.\ for a discussion of Sobolev spaces on noncompact manifolds,
see~\cite[Chapter~3]{Hebey}.
By the Hopf-Rinow Theorem,
we have that~\ref{cond:M:complete} is equivalent to geodesic completeness of~\posfold.
If we additionally assume an orientation on a manifold
which satisfies~\ref{cond:M:general} and~\ref{cond:M:complete},
then it is well-known that
the usual Laplace-Beltrami operator on smooth test functions is essentially self-adjoint.
See~\cite{Gaffney-harmonic-op,Gaffney-special-Stokes,Gaffney-heat-eq}
for generalisations of the standard case on compact manifolds
as well as~\cite[Section II.5]{LawsonMichelsohn}
for general statements on so-called Dirac operators on spinors.
Moreover, we also want to recommend~\cite{Wolf73}.
If we would assume nontrivial boundary,
then essentially self-adjointness of the Laplace-Beltrami operator is rather delicate.
This challenging problem has been tackled in~\cite{PhD-Perez-Pardo}
and the noteworthy paper~\cite{ILP-JFA}.

\subsection{General hypocoercivity method}
\label{subsec:general-method}

At this point, we want to give an almost criminally brief overview of the hypocoercivity method
in the abstract Hilbert space setting.
If the reader is familiar with this topic,
this section just clarifies some notation.

Originally, the method was algebraically developed
by J.~Dolbeault, C.~Mouhot and C.~Schmeiser
in~\cite{DMS09} and~\cite{DMS15}
studying linear kinetic equations --
algebraically in the sense that
issues of operator domains have been neglected.
Substantial contributions on the general hypocoercivity approach can be found
in~\cite{Villani07} and~\cite{Villani09} --
e.\,g.\ the oncoming condition~\ref{cond:Langevin:P3} on the potentials Hessian appears in~\cite[Theorem~35]{Villani09} first.
Our main reference is~\cite{HypocoercJFA} for two reasons.
On conceptional side,
the authors really tackled the long-time behaviour of solutions of the SDE
via the Kolmogorov backward setting
instead of investigating the Fokker-Planck equation.
In the case of Euclidean position space, normalised velocity,
and the invariant measure having a density wrt.\ Lebesgue measure
there is an isometric isomorphism the Kolmogorov backward to the Fokker-Planck setting,
see~\cite{unpublishedGS}.
However, in a general manifold it is not clear how to map between these two settings.
On technical side,
to gain a rigorous proof of the~\ref{thm:hypocoercivity-thm-named}
and to check its assumptions in applications
several domain issues have to be taken into account,
which are often times just omitted.
For broader discussion and more detailed explanations
we also refer to~\cite{PhD-Stilgenbauer}.
The main result, we aim to apply to our fibre lay-down model,
tells us what microstructure we have to expect in the fleece:
The faster the convergence to the equilibrium state
the more uniform the nonwoven material will appear.
The formal result reads as follows.

\begin{theorem}[Hypocoercivity Theorem]
\namedlabel{thm:hypocoercivity-thm-named}{Hypocoercivity Theorem}\label{thm:hypocoercivity-thm}
Assume the conditions~\ref{cond:D} as well as~\ref{cond:H} (as given below)
and denote by $(T_t)_{t\in[0,\infty)}$
the operator semigroup generated by~$L$ on the Hilbert space~$H$.

Then, there exist constants $\kappa_1,\kappa_2\in(0,\infty)$
computable in terms of the constants $\Lambda_m$, $\Lambda_M$, $c_1$ and $c_2$ appearing in the assumptions
such that it holds
\begin{displaymath}
    \norm{ T_tg-\scalarprod{g}{1}{H} }{H}
    \leq
    \kappa_1 \euler^{-\kappa_2 t}\norm{ g-\scalarprod{g}{1}{H} }{H}
    \qquad\text{for all times } t\geq0
\end{displaymath}
and for all $g\in H$.
\end{theorem}
\begin{proof}
    See~\cite[Theorem~2.18]{HypocoercJFA};
    in its proof one learns
    how to compute the constants~$\kappa_1$ and~$\kappa_2$.
\end{proof}

Now, we are going to explain what the sets of conditions~\ref{cond:D} and~\ref{cond:H} are.
We just mention as a fact that there are more general data assumptions such that
the~\ref{thm:hypocoercivity-thm-named} is still valid,
see~\cite[Section~2.2.3]{PhD-Stilgenbauer}.

\begin{condition}[Data conditions~\ref{cond:D}]
\namedlabel{cond:D}{(D)}
\hfill

\begin{enumerate}[label={(D\arabic*)}, ref={(D\arabic*)}]
    \item\label{cond:D1}
        \emph{model Hilbert space:}\
        Let $(E,\mathfrak{E},\mu)$ be a probability space and
        define the Hilbert space~$H$ to be $\bigL2{E}{\mu}=\bigL2{\mu}$.
    \item\label{cond:D2}
        \emph{Strongly continuous semigroup and its infinitesimal generator:}\
        Let $(L,\opdomain{L})$ be a linear operator on~$H$ and
        $(T_t)_{t\in[0,\infty)}$ be the strongly continuous semigroup generated by $L$, i.\,e.\
        $T_0 =\ID_H$ and $T_tf\rightarrow f$ as $t\downarrow0$ for all $f\in H$.
    \item\label{cond:D3}
        \emph{Core property:}\
        Let $D\subseteq\opdomain{L}$ be dense in~$H$ and
        an operator core of $(L,\opdomain{L})$, i.\,e.\
        the closure of $(L,D)$ coincides with $(L,\opdomain{L})$.
    \item\label{cond:D4}
        \emph{SAD-Decomposition of generator into symmetric and antisymmetric part:}\
        Let $(S,\opdomain{S})$ be symmetric and
        let $(A,\opdomain{A})$ be closed and antisymmetric on $H$ s.\,t.\
        $D \subseteq \opdomain{S}\cap\opdomain{A}$
        and the restriction of $L$ to the core can be decomposed as $L\vert_{D}=S-A$.
    \item\label{cond:D5}
        \emph{Projection:}\
        Let $P\colon\, H\to H$ be an orthogonal projection such that
        $P(H)\subseteq\opdomain{S}$ and $SP=0$ as well as
        $P(D)\subseteq\opdomain{A}$ and $AP(D)\subseteq\opdomain{A}$.
        Define
        \begin{displaymath}
            P_S\colon\ H \longmapsin H,\
            f \longmapsto Pf + \scalarprod{f}{1}{H}
        .
        \end{displaymath}
    \item\label{cond:D6}
        \emph{Invariant measure:}\
        Let~$\mu$ be invariant for $(L,D)$ in the sense that
        \begin{displaymath}
            \scalarprod{Lf}{1}{H}
            =
            \int_E Lf\ \diffd\mu
            =
            0
            \qquad\text{for all } f\in D
        .
        \end{displaymath}
    \item\label{cond:D7}
        \emph{semigroup conservativity:}
        Let $1\in\opdomain{L}$ and $L1=0$.
\end{enumerate}
\vspace*{-1.5em}
\end{condition}

\begin{condition}[Hypocoercivity conditions~\ref{cond:H}]
\namedlabel{cond:H}{(H)}
\hfill

\begin{enumerate}[label={(H\arabic*)}, ref={(H\arabic*)}]
    \item\label{cond:H1}
        \emph{algebraic relation:}
        \begin{displaymath}
            PAP\vert_D = 0
        \end{displaymath}
    \item\label{cond:H2}
        \emph{microscopic coercivity:}
        \begin{displaymath}
            \exists\,\Lambda_m\in(0,\infty)\,
            \forall\,f\in D\colon\
            \Lambda_m \|(\ID-P_S)f\|_H^2 \leq -\scalarprod{Sf}{f}{H}
        \end{displaymath}
    \item\label{cond:H3}
        \emph{macroscopic coercivity:}
        \begin{displaymath}
            \exists\, \Lambda_M\in(0,\infty)\,
            \forall\, f\in\opdomain{\adjoint{(AP)}(AP)} \colon\
            \Lambda_M \|Pf\|_H^2 \leq \|APf\|_H^2
        \end{displaymath}
    \item\label{cond:H4}
        \emph{boundedness of auxiliary operators:}
        \begin{align*}
            \exists\,c_1, c_2\in(0,\infty)\,\forall\,f\in D\colon{}
            & \|BSf\|_H \leq c_1 \|(\ID-P_j)f\|_H \\
            \land\
            & \|BA(\ID-P)f\|_H \leq c_2 \|(\ID-P_j)f\|_H
        \end{align*}
        for $B\vdef (\ID-(AP)\adjoint{(AP)})\adjoint{(AP)}$ on~$D(\adjoint{(AP)})$
        and projections $P_j\in\{P,P_S\}$, $j\in\{1,2\}$.
\end{enumerate}
\vspace*{-1.5em}
\end{condition}

\subsection{Bundle measures and weighted bundles}
\label{sec:weighting}

We start with defining bundle measures:
Thinking of measures on some manifold~\basefold
we should search for induced measures in bundles over~\basefold,
i.\,e.\ natural measures on the respective total spaces.
Proceeding on the measure theoretic path in the direction of Radon-Nikod\'ym derivatives,
the well-known notion of weighted manifolds is revisited.
The concept of multiplying a density weight function to a volume measure
is not to be confused with conformal transformations of manifolds.
Indeed, conformal transformations affect the geometry;
the difference is the definition of the divergence and
thus the integration by parts formula~\eqref{eq:weighted-ibp} below.
As far as we know,
there is very few literature on measure theory paired with the more geometric concept of fibre bundles.
After developing a notion of bundle measures on our own,
we found the seemingly unknown paper~\cite{Goetz},
where the author concedes more degrees of topological freedom.
Also, there is~\cite[Section~3.4.5]{Nicolaescu}
wherein the author describes a `fibred calculus' just for smoothly indexed families of manifolds.
We give these references for sake of completeness,
even though we stay with our bespoke construction of bundle measures.

Throughout this section,
let $\bundleproj\colon\,\totalfold\longmapsin\basefold$ be a smooth fibre bundle
with standard fibre~$F$
and define $\totalfold_b\vdef\bundleproj^{-1}(\{b\})$ for all $b\in\basefold$.
Furthermore, we suppose that
the fibre is actually a measurable space $(F,\mathfrak{F})$.
The property of local trivialisation yields a natural \sigmaalgebra
$\borel{\basefold}\locprod\mathfrak{F}\vdef\sigma(\mathfrak{G})$ on~\totalfold with generator
\begin{displaymath}
    \mathfrak{G}
    \vdef
    \left\{
        \varphi^{-1}(U\times U_F)
    \ \middle\vert\
        \begin{matrix}
            b\in\basefold,\
            U_F\in\mathfrak{F},\
            W\text{ chart domain at~}b, \\
            U_b\subseteq\basefold \text{ open neighbourhood of~}b \text{ s.\,t.}\\
            \text{diffeomorphism }\varphi\text{ renders the diagram} \\
            \text{in~\autoref{fig:local-trivialisation} commutative},\\
            U\vdef U_b\cap W.
        \end{matrix}
    \right\}
.
\end{displaymath}
One might say that $\borel{\basefold}\locprod\mathfrak{F}$ is the canonical \sigmaalgebra on~\totalfold,
however we will refer to it as the \emph{local product\-/\sigmaalgebra}.
If $\mathfrak{F}=\borel{F}$, then
$
    \borel{\basefold}\locprod\mathfrak{F}
    =
    \borel{\totalfold}
$.
\begin{figure}[ht]\large
    \begin{tikzcd}
	    \arrow{r}{\varphi} \arrow[swap]{d}{\bundleproj} \bundleproj^{-1}(U_b)
        & U_b\times F \arrow{dl}{\proj_{1}} \\
	    U_b &
    \end{tikzcd}
    \caption{Local trivialisation of a fibre bundle}\label{fig:local-trivialisation}
\end{figure}

In the next definition,
we restrict ourselves to probability measures
as this is the sole situation of interest in the present paper.

\begin{definition}[bundle measure]
\label{def:bundle-measure}
    Let $\mu_{\basefold}$ and $\nu_F$ be probability measures
    on $(\basefold,\borel{\basefold})$ and~$(F,\mathfrak{F})$ respectively.
    The \emph{(fibre) bundle measure on the total space \totalfold} is the pullback measure
    $\mu_{\totalfold} \vdef \bundleproj^\ast \mu_{\basefold}$
    of $\mu_{\basefold}$ wrt.~\bundleproj supplemented with the fibre measure $\nu_F$.
    That is the unique probability measure on~\totalfold satisfying
    \begin{displaymath}
        \int_{\totalfold} f\ \diffd \mu_{\totalfold}
        =
        \int_{\basefold} \int_{\bundleproj^{-1}(\{b\})}
            f\vert_{\bundleproj^{-1}(\{b\})}(e)
        \ \diffd \nu_F(e)\,\diffd \mu_{\basefold}(b)
    \end{displaymath}
    for all bounded \measurable{\borel{\basefold}\locprod\mathfrak{F}}{\borel{\Rnum}}
    functions $f\colon\,\totalfold\to\Rnum$.
\end{definition}

\begin{remark}
    In~\hyperref[def:bundle-measure]{Definition~\ref{def:bundle-measure}},
    the measure~$\nu_F$ on the fibre~$\bundleproj^{-1}(\{b\})$ is thought as an independent copy of~$\nu_F$ defined on~$F$.
    By definition a bundle measure yields a disintegration,
    but obviously the former concept is motivated by the local product structure of fibre bundles
    and tries to find an analogue of product measures respecting this structure,
    whereas the latter is kind of the `factorisation' of measures
    that are not necessarily product measures.
    In all generality,
    it's a difficult problem to give conditions for a disintegration to exists;
    in our situation it appears as a byproduct.
\end{remark}

The following lemma is stated as an equivalent characterisation of bundle measures,
however it could be formulated as an existence statement, since its proof is constructive.

\begin{lemma}[bundle measures locally are product measures]
\label{lem:loc-prod-measure}
    Let $\mu_{\basefold}$ and $\nu_F$ be probability measures on \basefold and $F$ respectively.
    Denote by $\mu_{\totalfold}\vdef \bundleproj^\ast \mu_{\basefold}$ the (fibre) bundle measure on \totalfold supplemented with fibre measure~$\nu_F$.
    Then, $\mu_{\totalfold}$ is the unique measure on \totalfold
    which locally trivialises to the product measure $\mu_{\basefold}\mathbin{\otimes}\nu_F$ in the following sense:
    By definition of fibre bundles,
    for any $b\in\basefold$ there is a neighbourhood $U_b\subseteq\basefold$ of~$b$
    as well as a diffeomorphism~$\varphi$
    which renders the diagram in~\autoref{fig:local-trivialisation} commutative.
    Let $V\in\borel{\basefold}\locprod\mathfrak{F}$, $W$~a chart domain at~$b$
    and define $U\vdef W\cap U_b$ as well as $V_U\vdef V\cap \bundleproj^{-1}(U)$.
    Then, $\mu_{\totalfold}$ obeys the transformation rule
    \begin{displaymath}
        \mu_{\totalfold}
        =
        \parentheses*{\varphi^{-1}}_\ast
        \parentheses*{
            \mu_{\basefold} \tensorprod \nu_F
        }
        \qquad\text{on }V_U
    .
    \end{displaymath}
    We say that $\mu_{\totalfold}$ is a loc\=/product measure and
    introduce $\mu_{\totalfold}=\mu_{\basefold}\locprod\nu_F$ as the corresponding notation.
\end{lemma}
\begin{proof}
    First, we construct the bundle measure just from the local transformation rule --
    basically, that's the proof of existence of such a measure.
    Note that measurable sets of the same form as~$V_U$
    generate the local product\-/\sigmaalgebra on~\totalfold,
    that they are linked to cylinder sets of the product\-/\sigmaalgebra on $U\times F$ via~$\varphi$,
    and that they form a family stable wrt.\ intersections,
    when the empty set is included of course.
    Since $\varphi$ is continuously invertible, its inverse is measurable.
    As the pushforward measure of~$\mu_{\totalfold}$ wrt.~$\varphi$ should be $\mu_{\basefold}\mathbin{\otimes}\nu_F$,
    we declare~$\mu_{\totalfold}$ to be
    the pushforward measure of $\mu_{\basefold}\mathbin{\otimes}\nu_F$ wrt.~$\varphi^{-1}$.
    It satisfies the desired transformation rule by construction.
    By Fubini-Tonelli this is the particular instance of the integral equation
    \begin{displaymath}
        \int_{\totalfold} f\ \diffd\mu_{\totalfold}
        =
        \int_{\basefold}\int_{\bundleproj^{-1}(\{b\})}
            f\vert_{\bundleproj^{-1}(\{b\})}(e)\
        \diffd\nu_F(e)\,\diffd \mu_{\basefold}(b)
    \end{displaymath}
    with $f=\indicator{V_U}$.
    We use this to obtain the general equation via
    approximating bounded measurable functions with simple functions and
    apply Lebesgue dominated convergence.
    The uniqueness assertion is fulfilled by the general uniqueness theorem for measures
    which finishes the proof.
\end{proof}

\begin{example}[
M\"obius strip]
    A M\"obius strip can be thought as fibre bundle
    with base manifold $\basefold = \sphere{1}$
    and standard fibre~$F$ being an open interval of finite length.
    The natural volume measure on the strip coincides on~$\mathfrak{G}$
    with a bundle measure
    where the base measure~$\mu_{\sphere{1}}$ is the volume measure on the circle
    and the fibre measure is the Lebesgue measure on an interval: $\nu_F=\Leb$.
    Since~$\mathfrak{G}\cup\{\emptyset\}$ is a generator stable wrt.\ intersections,
    the volume measure on the M\"obius strip coincides with the Lebesgue\-/type bundle measure.
    For a set $V_U\vdef V\cap\bundleproj^{-1}(U)$ we have that
    \begin{displaymath}
        \mu_{\totalfold}(V_U)
        =
        \int_{\bundleproj(V_U)\times\proj_2\bincirc\varphi(V_U)}
            \abs*{\det\parentheses*{d\varphi^{-1}}}
        \ \diffd \mu_{\basefold}\tensorprod\nu_F
    ,
    \end{displaymath}
    where $\proj_2$ denotes projection to the second component.
\end{example}

\begin{remark}
\label{rem:L2dense-testfunctions}
    At this point, it should be clear that
    \bigL2{\totalfold}{\mu_{\basefold}\locprod\nu_F}
    is not isomorphic to the Hilbert space tensor product
    $
        \bigL2{\basefold}{\mu_{\basefold}}
        \tensorprod
        \bigL2{F}{\nu_F}
    $.
    Anyway, there might be nice spaces of test functions
    that are dense in \bigL2\=/spaces for bundle measures.
    For sake of simplicity,
    suppose that the base measure is absolutely continuous wrt.\ a given Riemannian volume measure
    and moreover that
    $F\subseteq\Rnum^\callidim{d}$ is a smooth submanifold,
    $\mathfrak{F}\vdef F\cap\borel{\Rnum^\callidim{d}}$
    and $\nu_F$ is absolutely continuous wrt.\ restricted Lebesgue measure.
    Let $\tanproj^{-1}(U) = \varphi^{-1}(U\times F)\in\mathfrak{G}$.
    By~\cite[Theorem~II.10]{ReedSimonI}
    we know that
    $\smoothcompact{U}\tensorprod\smoothcompact{F}$
    is dense in
    $\bigL2{U\times F}{\mu_{\basefold}\tensorprod\nu_F}$,
    hence
    \begin{align*}
        \varphi_\ast\parentheses*{
            \smoothcompact{U}\tensorprod\smoothcompact{F}
        }
        &=
        \varphi_\ast\smoothcompact{U}\tensorprod\varphi_\ast\smoothcompact{F}
        \\
        &\vdef
        \linhull\left\{
            (f\pdot g) \bincirc\varphi
        \ \middle\vert\
            f\in\smoothcompact{U},
            g\in\smoothcompact{F}
        \right\}
    \end{align*}
    is dense in
    $\bigL2{\tanproj^{-1}(U)}{\mu_{\basefold}\locprod\nu_F}$.
    From a partition of unity argument we can infer
    that~\smoothcompact{\totalfold} is dense in
    \bigL2{\totalfold}{\mu_{\basefold}\locprod\nu_F}.
\end{remark}

\begin{definition}[weighted (fibre) bundles]
\label{def:weighted-bundle}
    Consider functions
    $\rho_{\basefold}\colon\,\basefold\to[0,\infty)$
    and $\rho_F\colon\,F\to[0,\infty)$.
    A function~$\rho\colon\,\totalfold\to[0,\infty)$ is called
    \emph{bundle weighting with base weight~$\rho_{\basefold}$ and fibre weight~$\rho_F$}
    if it locally trivialises to the product function
    $\rho_{\basefold}\pdot\rho_F$
    in the following sense:
    Consider the local trivialisation over an open neighbourhood~$U_b\subseteq\basefold$ at~$b$
    with diffeomorphism~$\varphi$ rendering the diagram in~\autoref{fig:local-trivialisation} commutative.
    Then, $\rho$~satisfies
    \begin{displaymath}
        \rho(v)
        =
        \rho_{\basefold}(\bundleproj(v))\pdot\rho_F(\proj_2\bincirc\varphi(v))
        \qquad\text{for all }v\in\bundleproj^{-1}(U_b)
    .
    \end{displaymath}
    A fibre bundle together with a bundle weight is called a \emph{weighted (fibre) bundle}.
\end{definition}

\begin{lemma}[existence and uniqueness of bundle weightings]
    In the situation of~\hyperref[def:weighted-bundle]{Definition~\ref{def:weighted-bundle}}
    a bundle weight~$\rho$ exists
    and is uniquely determined by base weight and fibre weight.
    Therefore, we introduce the corresponding notation
    $\rho = \rho_{\basefold}\locprod\rho_F$.
\end{lemma}
\begin{proof}
    In the local trivialisation
    the only possible weight function is given via
    \begin{displaymath}
        \rho
        =
        ((\rho_{\basefold}\bincirc\proj_1) \pdot (\rho_F\bincirc\proj_2))\bincirc\varphi
        \qquad\text{on }\bundleproj^{-1}(U_b)
    .
    \end{displaymath}
    Fix an open cover of~\basefold by chart domains,
    then the preimages form an open cover of~\totalfold.
    With a partition of unity subordinate to the latter cover,
    we can glue the definitions of~$\rho$ in the respective local trivialisations together
    which finishes the proof.
\end{proof}

\begin{remark}[local Radon-Nikod\'ym derivatives]
    Suppose some bundle measure $\mu_{\totalfold} = \mu_{\basefold}\locprod\nu_F$ such that
    $\mu_{\basefold}$ has a Radon-Nikod\'ym derivative
    $\rho_{\basefold}\vdef\frac{d\mu_{\basefold}}{dm_{\basefold}}$
    wrt.\ another measure~$m_{\basefold}$ on~$(\basefold,\borel{\basefold})$
    and also $\nu_F$ has a Radon-Nikod\'ym derivative
    $\rho_F\vdef\frac{d\nu_F}{dm_F}$
    wrt.\ another measure~$m_F$ on~$(F,\mathfrak{F})$.
    Then, we think of the induced bundle weighting
    $\rho = \rho_{\basefold}\locprod\rho_F$
    as a local Radon-Nikod\'ym derivative
    in view of it holds
    \begin{align*}
        \mu_{\totalfold} = \mu_{\basefold}\locprod\nu_F
        &=
        (\rho_{\basefold} m_{\basefold})\locprod(\rho_F m_F)
        \\
        &=
        (\rho_{\basefold}\locprod\rho_F) (m_{\basefold}\locprod m_F)
        =
        \rho (m_{\basefold}\locprod m_F)
    .
    \end{align*}
    For sake of brev'ty, we call~$\rho$ \emph{loc\=/density of~$\mu_{\totalfold}$}.
\end{remark}

\begin{remark}[Ehresmann connection and weighting]
    We emphasise again that weighting does not effect geometry:
    The Ehresmann connection induced by the metric~\riemannmetric* on~\posfold,
    see~\autoref{subsec:Ehresmann-Sasaki} below,
    does not change in the course of the weighting procedure.
    Indeed, the corresponding Levi-Civita connection \connection{}{\riemannmetric*}
    is not affected by reweighting as one can easily check
    using the Leibniz rule when checking the metric compatibility condition.
    Also several geometric objects associated to this connection
    just depend on the original metric~\riemannmetric* and not on the weight function.
\end{remark}

\begin{example}[trivial bundles]
    If the fibre bundle is trivial,
    then bundle measures are pushforwards of product measures
    wrt.\ the trivialisation isomorphism.
    Loc\=/densities basically are products of the densities for the respective measures.

    One particularly simple example arises with the standard fibre being a singleton.
    Then, every fibre measure is absolutely continuous wrt.\ the Dirac measure for the single point in~$F$
    and fibre weightings reduce to multiplication with a constant factor.
\end{example}

\begin{example}[weighted manifolds]
\label{ex:weighted-manifold}
    Consider an orientable Riemannian manifold~$(\basefold,\basemetric*)$
    and a strictly positive, nonconstant, smooth\footnote{
        Smoothness is actually not required,
        we could use e.\,g.\ a loc\=/Lipschitzian weight function instead.
        But in order to have~\hyperref[eq:weighted-ibp]{Equation~\eqref{eq:weighted-ibp}}
        we need that at least locally a weak gradient of~$\rho_{\basefold}$ exists
        and a weak version of Stokes Theorem.
    } weight function $\rho_{\basefold}\in\smoothfunc{\basefold}$.
    The \emph{Riemannian metric weighted by~$\rho_{\basefold}$}
    or just \emph{$\rho_{\basefold}$\=/weighted metric} on~\basefold
    is given as
    \begin{displaymath}
        \wbasemetric{v}{w}
        \vdef
        \basemetric{\rho_{\basefold}v}{\rho_{\basefold}w}
    \end{displaymath}
    for all $v,w\in\tanbundle{\basefold}$ with $\tanproj(v)=\tanproj(w)$.
    We do not keep the usual definition of gradients for this weighted metric,
    but in fact choose the \emph{$\rho_{\basefold}$\=/weighted gradient} as
    $\grad{\wbasemetric*} \vdef \frac1{\rho_{\basefold}}\grad{\basemetric*}$.
    I.\,e.\ for all smooth vector fields~$\Xfield$ on~\basefold
    and functions $f\in\smoothfunc{\basefold}$ holds
    $
        \wbasemetric{\Xfield}{\grad{\wbasemetric*}f}
        =
        \rho_{\basefold}\pdot\partial_{\Xfield}f
    $.
    The \emph{$\rho_{\basefold}$\=/weighted divergence} is to be defined as
    \begin{displaymath}
        \divergence{\wbasemetric*}
        \vdef
        \frac1{\rho_{\basefold}}\, \divergence{\basemetric*}(\rho_{\basefold}^2 \pdot\ID)
        ,
    \end{displaymath}
    since the \emph{$\rho_{\basefold}$\=/weighted Laplace-Beltrami operator}
    \begin{displaymath}
        \laplace{\wbasemetric*}
        \vdef
        \divergence{\wbasemetric*}(\grad{\wbasemetric*})
        =
        \frac1{\rho_{\basefold}}\,
        \divergence{\basemetric*}(\rho_{\basefold}\grad{\basemetric*})
    \end{displaymath}
    shall obey the following integration by parts formula:
    \begin{equation}
    \label{eq:weighted-ibp}
        \int_{\basefold}
            \laplace{\wbasemetric*}f \pdot g
        \ \diffd\Leb{\wbasemetric*}
        =
        - \int_{\basefold}
            \wbasemetric{\grad{\wbasemetric*}f}{\grad{\wbasemetric*}g}
        \ \diffd\Leb{\wbasemetric*}
        =
        - \int_{\basefold}
            \basemetric{\grad{\basemetric*}f}{\grad{\basemetric*}g}
        \ \diffd\Leb{\wbasemetric*}
        =
        \int_{\basefold}
            f \pdot \laplace{\wbasemetric*}g
        \ \diffd\Leb{\wbasemetric*}
    \end{equation}
    for all $f,g\in\smoothcompact{\posfold}$,
    where~$\Leb{\basemetric*}$ and $\Leb{\wbasemetric*}=\rho_{\basefold}\Leb{\basemetric*}$
    refer to Riemannian volume measures.
    If the weight function is constant,
    then the definition of the gradient is not to be changed:
    $\grad{\wbasemetric*}=\grad{\basemetric*}$.
\end{example}

\begin{notation}
\label{not:Lp-measure}
    For sake of readability,
    we write~\bigL{p}{\basefold}{\wbasemetric*}
    instead of~\bigL{p}{\basefold}{\Leb{\wbasemetric*}}.
    Similar notation is used for other spaces depending on a weighted Riemannian volume measure.
\end{notation}

\begin{remark}[adjoint vector fields wrt.\ weighted metric]
\label{rem:adjoint-sec}
    Let $(\basefold,\basemetric*)$ be an orientable Riemannian manifold.
    The general form of an adjoint vector field is given via the Divergence Theorem:
    If \Xfield is a vector field on~\basefold,
    then its adjoint wrt.\ the Riemannian metric,
    i\,e.\ wrt.\ the \bigL2{\basefold}{\basemetric*}\=/scalar product,
    is $\adjoint{\Xfield} = - \Xfield - \divergence_{\basemetric*}(\Xfield)$.
    Thus, solenoidal vector fields could be viewed as antisymmetric operators.
    Introducing a smooth, nonconstant weight function~$\rho_{\basefold}$ on~\basefold as above yields
    \begin{align*}
        \int_{\basefold} \Xfield f\pdot g \ \diffd\Leb{\wbasemetric*}
        &=
        \int_{\basefold}
            - f\Xfield(\rho_{\basefold} g)
            - \divergence{\basemetric*}(\Xfield)\rho_{\basefold} fg
        \ \diffd\Leb{\basemetric*}
        \\
        &=
        \int_{\basefold}
            - \rho_{\basefold} f \Xfield g
            - fg \Xfield\rho_{\basefold}
            - \divergence{\basemetric*}(\Xfield)\rho_{\basefold} fg
        \ \diffd\Leb{\basemetric*}
        \\
        &=
        \int_{\basefold}
            - f \Xfield g
            - fg \,\frac1{\rho_{\basefold}}\, \Xfield\rho_{\basefold}
            - \divergence{\basemetric*}(\Xfield) fg
        \ \diffd\Leb{\wbasemetric*}
    \end{align*}
    for all $f,g\in\smoothcompact{\basefold}$.
    Hence, the adjoint of~\Xfield wrt.~\wbasemetric* reads as
    \begin{displaymath}
        \adjoint{\Xfield}
        =
        - \Xfield
        - \divergence_{\basemetric*}(\Xfield)
        - \frac1{\rho_{\basefold}}\Xfield\rho_{\basefold}
    .
    \end{displaymath}
    Therefore, we would correct the differential operator~\Xfield
    for a vector field~\Xfield solenoidal wrt.~\basemetric*
    by the \emph{logarithmic derivative of $\rho$ along~\Xfield} to become antisymmetric again,
    i.\,e.\ replace~\Xfield by~$\Xfield+\frac1{\rho_{\basefold}}\Xfield\rho_{\basefold}$.
\end{remark}

\begin{lemma}[weighted Laplace-Beltrami in terms of logarithmic derivative]
\label{lem:weighted-Laplace}
    Again, let $(\basefold,\basemetric*)$ be an orientable Riemannian manifold
    weighted by~$\rho_{\basefold}$ strictly positive, nonconstant and smooth.
    The weighted Laplace-Beltrami is written as
    \begin{displaymath}
        \laplace{\wbasemetric*}
        =
        \laplace{\basemetric*}
        +
        \frac1{\rho_{\basefold}}\,\basemetric{\grad{\basemetric*}\rho_{\basefold}}{\grad{\basemetric*}}
        =
        \laplace{\basemetric*}
        +
        \frac1{\rho_{\basefold}}\,\grad{\basemetric*}\rho_{\basefold}
    .
    \end{displaymath}
    We call the second summand the \emph{logarithmic derivative of $\rho_{\basefold}$}.
    In particular, for $\rho_{\basefold}=\exp(-\psi)$ with $\psi\in\smoothfunc{\basefold}$
    we obtain
    \begin{displaymath}
        \laplace{\wbasemetric*}
        =
        \laplace{\basemetric*} - \grad{\basemetric*}\psi
    .
    \end{displaymath}
\end{lemma}
\begin{proof}
    Substantially, the proof looks like in the Euclidean case
    as we need just the Leibniz rule, Stokes Theorem and the defining characterisation of the gradient.
\end{proof}

Later on, we will have to talk about Poincar\'e inequalities.
For this purpose among others,
we shall fix some notions concerning sections in general.
In a nutshell, we are relying on the assumption that
smooth vector fields viewed as first order differential operators
could equivalently seen as smooth sections.
This is true in all applications we are interested in,
but we are aware of counterexamples like noncommutative tori,
see~\cite{Rosenberg}.
In this paper, we just leave issues of noncommutative geometry aside
and won't mention them again.

\begin{notation}[space of sections]
    We denote by~\measurablesec{\basefold}{\totalfold} the space of measurable sections.
    If no confusion is possible,
    we omit the base manifold writing just~\measurablesec{\totalfold}.
    Moreover, we denote the space of $m$-times continuously differentiable sections
    by~\contidiffsec{m}{\basefold}{\totalfold} or just~\contidiffsec{m}{\totalfold}.
    As usual the differentiability parameter equals the regularity of the differentiable structure,
    thus it is $m=\infty$ in our context for sake of simplicity.
\end{notation}

For the rest of this subsection,
we consider~$F$ to be a Banach space, i.\,e.\
the fibre bundle is a Banach bundle,
see e.\,g.~\cite[Chapter~III]{SergeLang}.
This gives us a section~$\abs{\cdot}{\totalfold}$
in the bundle of functions $\totalfold\to\Rnum$ over~\basefold such that
$
    \abs{\cdot}{\totalfold, b}\vdef\abs{\cdot}{\totalfold}(b)\colon\,
    \totalfold_b\to\Rnum
$
is a norm
and $(\totalfold_b,\abs{\cdot}{\totalfold, b})$ is a Banach space
for all $b\in\basefold$.
One might think of~$\abs{\cdot}{\totalfold}$ as a `Riemannian norm',
but should be careful since it is not necessarily related to a Riemannian metric.
Furthermore, we assume a $\sigma$\=/finite measure $\mu_{\basefold}$  on~$(\basefold,\borel{\basefold})$.

\begin{definition}[Integrable sections]
    Let $p\in[1,\infty)$ and $\Xfield\in\measurablesec{\totalfold}$.
    We call~\Xfield \emph{$p$\=/integrable wrt.~$\mu_{\basefold}$},
    if the integral
    $
        \int_{\basefold}
            \abs{ \Xfield(b) }{\totalfold,b}^p
        \ \diffd\mu_{\basefold}(b)
    $
    is finite.
    By~\bigL{p}{\basefold\to\totalfold}{\mu_{\basefold}}
    we denote the set of equivalence classes of $p$\=/integrable sections
    wrt.\ equality {$\mu_{\basefold}$}\=/almost everywhere.
    Clearly, we endow~\bigL{p}{\basefold\to\totalfold}{\mu_{\basefold}} with the norm
    \begin{displaymath}
        \norm{\cdot}{\bigL{p}{\basefold\to\totalfold}{\mu_{\basefold}}}
        \vdef
        \parentheses*{
            \int_{\basefold}
                \abs{ \cdot }{\totalfold,b}^p
            \ \diffd\mu_{\basefold}(b)
        }^{\sfrac1p}
    .
    \end{displaymath}
    As usual for the case $p=\infty$, we define~\bigL\infty{\basefold\to\totalfold}{\mu_{\basefold}}
    as the space of measurable sections $\Xfield\in\measurablesec{\totalfold}$ which are bounded almost everywhere, i.\,e.\
    there is a $c\in(0,\infty)$ such that
    $\abs{ \Xfield(b) }{\totalfold,b} \leq c$
    for $\mu_{\basefold}$\=/almost all $b\in\basefold$.
    The norm of such a $\Xfield\in\bigL\infty{\basefold\to\totalfold}{\mu_{\basefold}}$
    is the infimum of all such bounds~$c$.
\end{definition}

Completeness of these spaces of integrable sections is shown
like in the Fischer-Riesz Theorem for usual \bigL{p}\=/spaces.
One might say that we defined~\bigL{p}{\basefold\to\totalfold}{\mu_{\basefold}}
as a `direct integral of Banach spaces'.
This is merely a verbalisation of a more general concept of
integrating homoousios fibres wrt.\ some ($\sigma$\=/finite) measure on the base space.
This belongs to mathematical folklore and
we can not discuss this construction here in exhausting detail.

\begin{example}[direct integral of Hilbert space fibres]
    Let \basefold be a {\callidim{b}}\=/dimensional Riemannian manifold,
    the standard fibre~$F$ be $\Rnum^{\callidim{d}}$ for some natural number~\callidim{d}
    and let a section~\basemetric*
    in the bundle of symmetric bilinear forms
    $\mathrm{Sym}(\totalfold)\rightarrow\basefold$
    such that it is pointwise positive semidefinite.
    Then, \basemetric* induces a section of norms via
    \begin{displaymath}
        \abs{ e }{\basemetric*}
        \vdef
        \basemetric{e}{e}{\bundleproj(e)}^{\frac12}
        \qquad\text{for all }e\in\totalfold
    \end{displaymath}
    and all fibres are Hilbert spaces,
    since all norms on~$\Rnum^{\callidim{d}}$ are equivalent.
    Thus, we can describe the space of square-integrable sections
    as the direct integral
    $\int_{\basefold}^{\oplus} \totalfold_b \ \diffd\mu_{\basefold}(b)$
    wrt.~$\mu_{\basefold}$.
    This gives us a Hilbert space again.
    Of course, $(\basefold,\basemetric*)$ being a Riemannian manifold
    would be the most interesting case, i.\,e.\
    $\totalfold=\tanbundle\basefold$ with $F=\Rnum^{\callidim{b}}$
    and the section~\basemetric* being a Riemannian metric.
    Then, embracing~\hyperref[not:Lp-measure]{Notation~\ref{not:Lp-measure}} we have that
    \begin{displaymath}
        \bigL2{\basefold\to\tanbundle\basefold}{\basemetric*}
        =
        \int_{\basefold}^{\oplus} \tanbundle{\basefold}{b} \ \diffd\Leb{\basemetric*}(b)
    .
    \end{displaymath}
\end{example}

In principle, a function $u\in\Lloc1{\basefold}{\basemetric*}$ on a Riemannian manifold $(\basefold,\basemetric*)$
is weakly differentiable
if the composition $u\bincirc b$ is weakly differentiable for every chart~$b$.
For the more formal (topological) definition of Sobolev spaces~\sobolev{m}{p}{\basefold} on~\basefold
we refer to~\cite[Section~2.2]{Hebey}.
As in the case of Sobolev spaces,
we are not content with characterising a weak gradient in charts
as vector field with locally integrable components,
but from the existence proof for Sobolev spaces as function spaces
we know the coordinate-free description of weak gradients:
A function~$u$ is weakly differentiable
if there is an element
$\Ufield\in\Lloc1{\basefold\to\tanbundle\basefold}{\basemetric*}$ satisfying
\begin{displaymath}
    \int_{\basefold}
        \basemetric{\Xfield}{\Ufield}
    \ \diffd\Leb{\basemetric*}
    =
    -
    \int_{\basefold}
        \divergence{\basemetric*}(\Xfield) \pdot u
    \ \diffd\Leb{\basemetric*}
    \qquad\text{for all }\Xfield\in\smoothcompactsec{\tanbundle\basefold}
.
\end{displaymath}
We denote such a weak gradient as~$\Ufield = \grad{\basemetric*}u$ in analogy to the usual gradient.
In case of $u\in\sobolev1{p}{\basefold}$
we have that
$\grad{\basemetric*}u\in\bigL{p}{\basefold\to\tanbundle\basefold}{\basemetric*}$.
The previous characterisation instances to
\begin{displaymath}
    \int_{\basefold}
        \Ufield\varphi
    \ \diffd\Leb{\basemetric*}
    =
    \int_{\basefold}
        \basemetric{\grad{\basemetric*}\varphi}{\Ufield}
    \ \diffd\Leb{\basemetric*}
    =
    -\int_{\basefold}
        u\laplace{\basemetric*}\varphi
    \ \diffd\Leb{\basemetric*}
    \qquad\text{for all }\varphi\in\smoothcompact{\basefold}
.
\end{displaymath}
If
$
    \int_{\basefold}
        u \laplace{\basemetric*}\varphi
    \,\diffd\Leb{\basemetric*}
    =
    0
$ for all test functions $\varphi\in\smoothcompact{\basefold}$,
then we call~$u$ \emph{weakly harmonic}.
Using this definition
we bypass the problem to extend the divergence operator --
which is taking the trace of the covariant derivative --
with domain~\smoothsec{\tanbundle\basefold}
to all the possible weak gradients in the Banach space \bigL{p}{\basefold\to\tanbundle\basefold}.
In view of Weyl's Lemma characterising weakly harmonic functions
those functions can be thought as harmonic functions,
since we don't have boundaries in this paper.

\begin{example}[loc-Lipschitz potentials]
\label{ex:loc-Lipschitz}
    We know that
    if~$\psi$ is Lipschitzian\footnote{
        We say that functions~$f\colon\,\basefold\to\Rnum$ are Lipschitzian if
        there is a positive constant~$C$ such that
        $\vert f(x)-f(y) \vert \leq C d_{\basemetric*}(x,y)$
        for all $x,y\in\basefold$,
        where we denote by~$d_{\basemetric*}$ the intrinsic metric induced by the Riemannian metric~\basemetric*.
        As a consequence,
        if we carelessly embed~$\basefold$ into some~$\Rnum^{\callidim{d}}$ large enough,
        the embedding might change the structure of the metric space and
        thus the class of Lipschitz functions.
    }
    with compact support,
    then $\psi\in\sobolev1p{\basefold}$ for all $p\in[1,\infty]$
    meaning that~$\grad{\basemetric*}\psi$ exists as a weak gradient,
    see~\cite[Proposition~2.4]{Hebey}.
    Indeed, if~\basefold is a compact manifold,
    then the proposition applies to any Lipschitzian~$\psi$;
    in general, we have not a compact manifold,
    but consider arbitrary compact subsets instead.
\end{example}

\subsection{Ehresmann connections and Sasaki metric}
\label{subsec:Ehresmann-Sasaki}

In this subsection,
we introduce Ehresmann connections
as a decomposition of the double tangent bundle in terms of a Whitney sum
as well as the Sasaki metric on the tangent space
as the Riemannian metric respecting the entire Ehresmann connection.

\begin{definition}[vertical bundle]
    Let $\bundleproj\colon\,\totalfold\longmapsin\basefold$ be a (smooth) fibre bundle.
    The \emph{space of vertical (tangent) vectors} is
    $\vertbundle{\totalfold}\vdef\nullspace{d\bundleproj}$,
    the nullspace of the differential
    $d\bundleproj\colon\,\tanbundle{\totalfold}\to\tanbundle{\basefold}$.
    Vertical vectors are thought as being tangent to the fibres of~\bundleproj.
    This yields the so-called \emph{vertical bundle}
    $\tanproj\colon\,\vertbundle{\totalfold}\longmapsin\totalfold$.
    As $d_v\bundleproj$ is surjective for all $v\in\totalfold$,
    the vertical bundle is a smooth subbundle.
    Smooth sections in this bundle are called \emph{vertical}.

    Additionally, if $\bundleproj\colon\,\totalfold\longmapsin\basefold$ happens to be a vector bundle,
    than we can define the \emph{vertical lift at $v$}
    $\vertlift_v\colon\
    \totalfold_{\bundleproj(v)} \to \tanbundle{\totalfold}{v}$
    for $v\in\totalfold$ fixed
    via the action on arbitrary test functions $f\in\smoothfunc{\totalfold_{\bundleproj(v)}}$ as
    $
        \dualpair{\vertlift_v(w)}{d_vf}
        =
        \frac{d}{dt} f(v+tw)\bigr\vert_{t=0}
    $.
    If $\totalfold=\tanbundle\basefold$, then
    \begin{displaymath}
        \dualpair{\vertlift_v(w)}{d_v(df)}
        =
        \dualpair{w}{df}
        \qquad\text{for all }f\in\smoothfunc{\basefold}
    \end{displaymath}
    determines the lift uniquely.
    The smooth section
    $\canonfield\in\smoothsec{\totalfold}{\vertbundle{\totalfold}}$ given as
    $v\mapsto\canonfield(v)\vdef\vertlift_v(v)$
    is called \emph{canonical vector field}.
    Furthermore,
    the \emph{vertical projection \vertproj} is given at $v\in\totalfold$ as
    the projection mapping $\tanbundle{\totalfold}{v}\to\vertbundle{\totalfold}{v}$.
\end{definition}

Now, we restrict ourselves to the case of vector bundles, but
a definition of Ehresmann connections for fibre bundles can be found
e.\,g.\ in~\cite[Section~9]{KolvarMichorSlovak}.

\begin{definition}[Ehresmann connection]
    Let $\bundleproj\colon\,\totalfold\longmapsin\basefold$ be a (smooth) vector bundle.
    A (smooth) subbundle $\horbundle{\totalfold}\leq\tanbundle{\totalfold}$
    is called~\emph{Ehresmann connection} or \emph{horizontal (tangent) bundle} if
    \begin{displaymath}
        \tanbundle{\totalfold}{v}
        =
        \vertbundle{\totalfold}{v}
        \oplus
        \horbundle{\totalfold}{v}
        \qquad\text{for all }
        v\in\totalfold
        .
    \end{displaymath}
    For sake of readability we just write
    $\tanbundle{\totalfold} = \vertbundle{\totalfold} \oplus \horbundle{\totalfold}$
    in the sense of a Whitney sum.

    The \emph{horizontal lift at $v$} of $w\in \totalfold$ is the unique vector $\horlift_v(w)\in\horbundle{\totalfold}{v}$ such that
    \begin{displaymath}
        w = \dualpair{\horlift_v(w)}{d\bundleproj}
        .
    \end{displaymath}
    Finally, the projection of tangent vectors to their horizontal parts is denoted by~\horproj.
\end{definition}

Compare this usage of the terms `vertical' and `horizontal'
to the usage in stochastic analysis as e.\,g.\ in~\cite[Section~V.4]{IkedaWatanabe} or~\cite[Chapter~2]{Hsu}.
Luckily, we escaped the frame bundle via the McKean-Gangolli injection scheme
as described by E.~J\o{}ergensen.

Henceforth, we consider $\basefold = \posfold$ with the properties~\ref{cond:M},
since this is the sole instance of interest for this paper.
Furthermore, we just take the tangent space $\totalfold = \tanbundle\posfold$,
but note that we are interested in $\unittanbundle\posfold\subseteq\tanbundle\posfold$
in case of the fibre lay-down model.
Moreover, let us specify the one Ehresmann connection
we always will consider without further mentioning:
the Riemannian horizontal bundle.

\begin{definition}[connector map and Riemannian horizontal bundle]
    Let $U\subseteq\posfold$ be a neighbourhood of $o\in\posfold$
    with preimage $V\vdef \tanproj^{-1}(U)\subseteq\tanbundle{\posfold}$ such that
    the exponential $\exp_o\colon\,\tanbundle{\posfold}{o}\to\posfold$
    maps a 0\=/neighbourhood to~$V$ diffeomorphically.
    Let $\tau\colon\,V\to\tanbundle{\posfold}{o}$ denote parallel transport of~$v\in V$ along the unique geodesic arc connecting~$\pi(v)$ and~$o$.
    Let $r_{-u}\colon\,\tanbundle{\posfold}{o}\to\tanbundle{\posfold}{o}$
    be the translation $w\mapsto w-u$ by the vector $u\in\tanbundle{\posfold}{o}$.
    Now, consider the mapping
    \begin{displaymath}
        \kappa\colon\ V\longmapsin\posfold,\
        v \longmapsto \parentheses*{\exp_{o} \bincirc r_{-u} \bincirc \tau}(v)
    .
    \end{displaymath}
    The dependency on the chart vanishes when passing to the differential
    \begin{displaymath}
        d_u\kappa\colon\ \tantanbundle\posfold{u}\longmapsin\tanbundle\posfold{\tanproj(u)},\
        a \longmapsto \dualpair{a}{d(\exp_{\tanproj(u)} \bincirc r_{-u} \bincirc \tau)}
    \end{displaymath}
    which is called the \emph{connector map},
    cf.~\cite[Section~2]{Dombrowski}.
    Also see~\cite[Section~II.4]{Sakai} for explanation in terms of local coordinates.
    Via the assignment
    $\horbundle{\totalfold} \vdef \nullspace{d\kappa}$
    we gain an Ehresmann connection
    which we call the \emph{Riemannian horizontal bundle},
    see e.\,g\ \cite[Appendix~(ii)]{Dombrowski}.
    As the exponential map depends on the given Riemannian metric,
    so does this horizontal bundle.
\end{definition}

Some authors call $d\kappa$ the vertical projection
and $d\tanproj$ the horizontal projection.
E.\,g.\ in~\hyperref[not:func-lift]{Notation~\ref{not:func-lift}}
we use the mappings in a way that would justify such a naming.
However, we do not recommend this terminology
and introduced the notions~`$\vertproj$' and~`$\horproj$' to avoid confusion.
Concerning the vertical or horizontal lift of functions
there is a well-established consent what it should be.

\begin{notation}[vertical/horizontal lift of functions]
\label{not:func-lift}
    Let~$f_0$ be a real\-/valued function with domain in~\posfold.
    We call the pullback of~$f_0$  wrt.\ the tangent bundle projection~\tanproj
    the \emph{vertical lift of~$f_0$}.
    For sake of brev'ty we define $\vlfunc{f_0}\vdef\tanproj^\ast f_0$
    whenever it's defined.

    A direct analogy would be that
    the horizontal lift of~$f_0$ is the pullback wrt.\ the connector map
    and we declare $\hlfunc{f_0}\vdef\kappa^\ast f_0$
    whenever it's defined.
    However, this is not straight forward,
    since~$\kappa$ depends on the choice of base point~$o$ of the exponential and the translation vector~$u$.
    At first, the horizontal lift of a smooth function~$f_0$ is a function
    \begin{displaymath}
        (u,v)\longmapsto \parentheses*{f_0\bincirc\exp_{\tanproj(u)}\bincirc r_{-u}\bincirc \tau}(v)
    \end{displaymath}
    or a bevy of functions indexed by~$u$;
    we should exclude the case that~$u$ and~$v$ are in the same fibre
    as this would yield the vertical lift again.
    We do not worry too much about well-definedness and differentiability:
    Due to assumption~\ref{cond:M:complete} and the Hopf-Rinow Theorem
    the exponential mapping at~$\tanproj(u)$ is defined everywhere on the tangent space,
    furthermore it is almost everywhere a diffeomorphism by~\cite[Lemma~III.4.4]{Sakai}.
    But as aforementioned earlier
    we can project from the double tangent space into the tangent space via~$d\tanproj$ and~$d\kappa$
    and get
    $df_0 (d\tanproj + d\kappa) = d\vlfunc{f_0} + df_0\bincirc d\kappa$.
    The \emph{horizontal lift of~$f_0$} is a function~\hlfunc{f_0} such that
    $
        \dualpair{a}{d\hlfunc{f_0}}
        =
        \dualpair{a}{df_0\bincirc d\kappa}
    $
    for all $a\in\tantanbundle\posfold$ --
    algebraically this makes perfectly sense and is unique up to constant offsets.
    A more analytic intuition would be
    $
        \hlfunc{f_0}(v)
        =
        \int_{\tantanbundle\posfold{v}} df_0\bincirc d\kappa\,\diffd\Leb
    $,
    where we consider the Lebesgue measure on the standard fibre $F=\Rnum^{2\posdim}$ of the double tangent bundle.
\end{notation}

\begin{lemma}
\label{lem:vert-hor-acting}
    Let $f_0\in\smoothfunc{\posfold}$
    and $\Xfield\in\smoothsec{\tanbundle\posfold}$.
    Then, the following relations hold
    \begin{align*}
        \vertlift(\Xfield)\vlfunc{f_0} = 0
        \quad&\text{and}\quad
        \horlift(\Xfield)\vlfunc{f_0} = \vlfunc*{\Xfield f_0},
        \\
        \vertlift(\Xfield)\hlfunc{f_0} = \vlfunc*{\Xfield f_0}
        \quad&\text{and}\quad
        \horlift(\Xfield)\hlfunc{f_0} = 0
    .
    \end{align*}
\end{lemma}
\begin{proof}
    Elementary.
\end{proof}

Next, we define the Sasaki metric.
It's original definition was given in the famous paper~\cite{Sasaki58}
containing many other results of general interest.
Sasaki refered just to the Riemannian horizontal bundle and so do we,
but it's clear that the following definition can easily be adapted to a situation without connector map.

\begin{definition}[Sasaki metric]
\label{def:Sasaki}
    The \emph{Sasaki metric \sasakimetric* on \tanbundle{\posfold}}
    is uniquely characterised as
    the Riemannian metric respecting inner products in the given Ehresmann connection, i.\,e.\
    the metric \sasakimetric* is \emph{natural} in the sense of~\cite{GudmundssonKappos} meaning that
    \begin{displaymath}
        \sasakimetric{\horlift(\Xfield)}{\horlift(\Yfield)}
        =
        \riemannmetric{\Xfield}{\Yfield} \bincirc\tanproj
        \qquad\text{and}\qquad
        \sasakimetric{\horlift(\Xfield)}{\vertlift(\Yfield)}
        =
        0
    \end{displaymath}
    for all $\Xfield,\Yfield\in\smoothsec{\tanbundle{\posfold}}$,
    and additionally the metric respects the inner product under vertical lifting in the sense that
    \begin{displaymath}
        \sasakimetric{\vertlift(\Xfield)}{\vertlift(\Yfield)}
        =
        \riemannmetric{\Xfield}{\Yfield} \bincirc\tanproj
    \end{displaymath}
    for all $\Xfield,\Yfield\in\smoothsec{\tanbundle{\posfold}}$.
    This metric is explicitly given as the sum of pullbacks of the metric tensor of the base manifold:
    \begin{align*}
        \sasakimetric{a}{b}
        &{}\vdef
        \kappa^\ast\riemannmetric{a}{b}
        +
        \tanproj^\ast\riemannmetric{a}{b}
        \\
        &=
        \riemannmetric{\dualpair{a}{d\kappa}}{\dualpair{b}{d\kappa}}
        +
        \riemannmetric{\dualpair{a}{d\tanproj}}{\dualpair{b}{d\tanproj}}
    \end{align*}
    for all $a,b\in\tantanbundle{\posfold}$.
    We call the component $\hormetric*\vdef\tanproj^\ast\riemannmetric*$ the \emph{horizontal (Sasaki) metric}.
    Mutatis mutandis, the \emph{vertical (Sasaki) metric} \vertmetric* is $\vertmetric*\vdef\kappa^\ast\riemannmetric*$.
    This yields $\sasakimetric* = \vertmetric* + \hormetric*$.
\end{definition}

\begin{notation}
    If we consider the $\rho_{\posfold}$\=/weighted manifold $(\posfold,\wriemannmetric*)$,
    then the weighting procedure is naturally reflected
    in the horizontal bundle via the vertical lift.
    Explicitly, we think of $(\tanbundle\posfold,\whormetric*)$
    as a $\vlfunc{\rho_{\posfold}}$\=/weighted Riemannian manifold,
    where~\whormetric* is the weighted version of the horizontal Sasaki metric~\hormetric*.
    In the first place, the vertical bundle is not affected.
    If we weight the fibre $F=\Rnum^{\posdim}$ by~$\rho_F$,
    then the vertical metric changes and the horizontal one does not.
    The weighted vertical Sasaki metric is denoted by~\wvertmetric*.
    If and only if a bundle weighting is specified and no confusion possible,
    we denote the corresponding weighted Sasaki metric by~\wsasakimetric*
    and the weighted tangent bundle just as $(\tanbundle\posfold,\wsasakimetric*)$.
\end{notation}

It turns out that the volume form corresponding to the Sasaki metric --
    it exists independently of orientability of~\posfold
    as tangent spaces always are orientable --
induces a volume measure
with very neat loc-product structure.

\begin{lemma}[Sasakian volume measure]
    The volume measure~\Leb{\sasakimetric*} wrt.\ the Sasaki metric
    coincides with the bundle measure on~\tanbundle{\posfold}
    supplemented with Lebesgue fibre measure,
    i.\,e.\ it holds
    $\Leb{\sasakimetric*} = \Leb{\riemannmetric*} \locprod \Leb$,
    where we abbreviated the $\posdim$\=/fold Lebesgue measure
    $\Leb^{\otimes\posdim}$ just by~\Leb.
\end{lemma}
\begin{proof}
    Form~\hyperref[def:Sasaki]{Definition~\ref{def:Sasaki}}
    we know that the Sasaki metric can be written as the sum
    $\sasakimetric* = \vertmetric* + \hormetric*$
    of vertical and horizontal metric.
    We encounter this situation when considering a product of Riemannian manifolds and
    thus, we know that the Sasakian volume measure is a product measure basically.
    Indeed, let a chart $(v^j)_{j=1}^{2\posdim}$ with domain $V\subseteq\tanbundle{\posfold}$
    respecting the Ehresmann connection
    in the sense that $(\partial v^j)_{j=1}^{\posdim}$ provides local basis for
    either the vertical or the horizontal vector fields and
    $(\partial v^j)_{j=\posdim+1}^{2\posdim}$ provides a basis for the complementary type of vector fields;
    the matrix representation of the Sasaki metric is of block diagonal form
    with zero matrix at south west and north east position.
    Therefore, the Sasakian volume form~$\diffd\Leb{\sasakimetric*}$ reads in those coordinates as
    \begin{displaymath}
        \diffd\Leb{\sasakimetric*}
        =
        \sqrt{\abs*{\tanproj^\ast\riemannmetric*}}
        \sqrt{\abs*{\kappa^\ast\riemannmetric*}}
        \bigwedge_{j=1}^{2\posdim} \diffd v^j
        .
    \end{displaymath}
    Wlog.\ $V$ is preimage of a chart domain in~\posfold, i.\,e.\ $V=\tanproj^{-1}(U)$,
    such that there is a diffeomorphism~$\varphi$
    rendering the diagram in~\autoref{fig:local-trivialisation} commutative.
    Hence the pushforward measure $\varphi_\ast\Leb{\sasakimetric*}$ has to coincide
    with product volume measure on $U\times F$,
    where the fibre $F=\Rnum^{\posdim}$ is naturally endowed with the Lebesgue measure.
    But~\hyperref[lem:loc-prod-measure]{Lemma~\ref{lem:loc-prod-measure}}
    uniquely determines the bundle measure on~\tanbundle{\posfold}
    supplemented the fibre measure $\nu_F=\Leb$,
    so the Sasakian volume measure has to be this bundle measure.
\end{proof}

Note that by~\hyperref[lem:vert-hor-acting]{Lemma~\ref{lem:vert-hor-acting}}
we already know that
the gradients~\grad{\vertmetric*} and~\grad{\hormetric*} wrt.\ vertical and horizontal Sasaki metric respectively
satisfy $\grad{\vertmetric*}\vlfunc{f_0} = 0$ and $\grad{\hormetric*}\hlfunc{f_0} = 0$.
In the next lemma we give more insight into the three basic operators
induced by the metrics from the Ehresmann connection.

\begin{lemma}[Sasakian gradient, divergence and Laplacian]
\label{lem:Sasaki-operators}
    The gradient, divergence and Laplace-Beltrami operators
    corresponding to the vertical Sasaki metric are characterised by
    \begin{displaymath}
        \grad{\vertmetric*}\hlfunc{f_0}
        =
        \vertlift\parentheses*{\grad{\riemannmetric*}f_0\bincirc\tanproj}
        ,\
        \divergence{\vertmetric*}\parentheses*{\vertlift{\Xfield}}
        =
        \parentheses*{\divergence{\riemannmetric*}\Xfield}\bincirc\tanproj
        \text{ and }
        \laplace{\vertmetric*}\hlfunc{f_0}
        =
        \parentheses*{\laplace{\riemannmetric*}f_0}\bincirc\tanproj
    \end{displaymath}
    for all $f_0\in\smoothfunc{\posfold}$
    and $\Xfield\in\smoothsec{\tanbundle{\posfold}}$.
    Similarly, for the case of the horizontal Sasaki metric we have that
    \begin{displaymath}
        \grad{\hormetric*}\vlfunc{f_0}
        =
        \horlift\parentheses*{\grad{\riemannmetric*}f_0\bincirc\tanproj}
        ,\
        \divergence{\hormetric*}\parentheses*{\horlift{\Xfield}}
        =
        \parentheses*{\divergence{\riemannmetric*}\Xfield}\bincirc\tanproj
        \text{ and }
        \laplace{\hormetric*}\vlfunc{f_0}
        =
        \parentheses*{\laplace{\riemannmetric*}f_0}\bincirc\tanproj
    \end{displaymath}
    for all $f_0\in\smoothfunc{\posfold}$
    and $\Xfield\in\smoothsec{\tanbundle{\posfold}}$.

    Eventually, we have
    \begin{displaymath}
        \divergence{\sasakimetric*}(\Yfield)
        =
        \divergence{\vertmetric*}(\vertproj(\Yfield))
        +
        \divergence{\hormetric*}(\horproj(\Yfield))
        \qquad\text{for all }\Yfield\in\smoothsec{\tanbundle\posfold}{\tantanbundle\posfold}
    .
    \end{displaymath}
\end{lemma}
\begin{proof}
    We restrict ourselves to the vertical case, since the other statements follow analogously.
    Let $f_0\in\smoothfunc{\posfold}$
    and $\Xfield\in\smoothsec{\tanbundle{\posfold}}$ arbitrary be fixed.
    Then, by definition and from~\hyperref[lem:vert-hor-acting]{Lemma~\ref{lem:vert-hor-acting}}
    we know that the following two equations hold simultaneously
    which characterises the vertical gradient:
    \begin{align*}
        \vertmetric{\vertlift\Xfield}{\grad{\vertmetric*}\hlfunc{f_0}}
        &=
        (\Xfield f_0)\bincirc\tanproj
        \\
        \text{and}\quad
        \vertmetric{\vertlift\Xfield}{\vertlift\parentheses*{\grad{\riemannmetric*}f_0}}
        &=
        \riemannmetric{\Xfield}{\grad{\riemannmetric*}f_0}
        =
        (\Xfield f_0)\bincirc\tanproj
    .
    \end{align*}
    Regarding the divergence,
    we use the definition in terms of the Lie derivative,
    which is mostly preferred in manifold theory,
    and apply Cartan's magical formula.
    We have for the Riemannian metrics \riemannmetric* and~\vertmetric* the defining equations of divergence:
    \begin{align*}
        \Liederivative{\Xfield} \Leb{\riemannmetric*}
        &=
        \divergence{\riemannmetric*}(\Xfield) \pdot \Leb{\riemannmetric*}
        \\
        \text{and}\quad
        \Liederivative{\vertlift\Xfield} \Leb{\vertmetric*}
        &=
        \divergence{\vertmetric*}(\vertlift\Xfield) \pdot \Leb{\vertmetric*}
    ,
    \end{align*}
    where `\Liederivative' denotes the Lie derivative.
    Again, the symbol~\Leb{\riemannmetric*} is to be interpreted
    as either volume form or 1\=/density.
    Now, in case of a volume form Cartan's formula yields the equation
    \begin{align*}
        \Liederivative{\vertlift\Xfield} \Leb{\vertmetric*}
        &=
        d\parentheses*{\vertlift(\Xfield)\insertmorphism \kappa^\ast\Leb{\riemannmetric*}}
        +
        \vertlift(\Xfield)\insertmorphism d\kappa^\ast\Leb{\riemannmetric*}
        =
        \parentheses*{d \Xfield\insertmorphism\Leb{\riemannmetric*}}\bincirc\tanproj
        +
        \vertlift(\Xfield)\insertmorphism d\kappa^\ast\Leb{\riemannmetric*}
        \\
        &=
        \parentheses*{d \Xfield\insertmorphism\Leb{\riemannmetric*}}\bincirc\tanproj
        +
        \parentheses*{\Xfield\insertmorphism d\Leb{\riemannmetric*}}\bincirc\tanproj
        =
        \parentheses*{\Liederivative{\Xfield} \Leb{\riemannmetric*}}\bincirc\tanproj
    .
    \end{align*}
    Thus, the statement follows;
    this can locally be reused for the case of a 1\=/density,
    cf.\ the proof of~\cite[Proposition~3.4.3]{Nicolaescu}.
    Combining the results for gradient and divergence
    we get the equation for the Laplace-Beltrami operator.

    Finally, let $\Yfield\in\smoothsec{\tanbundle\posfold}{\tantanbundle\posfold}$.
    Recall that
    the Sasakian volume measure is realised by a volume form~$\diffd\Leb{\sasakimetric*}$
    which could be written as wedge product of a vertical and horizontal volume form:
    $
        \diffd\Leb{\sasakimetric*}
        =
        \diffd\Leb{\vertmetric*}
        \wedge
        \diffd\Leb{\hormetric*}
    $.
    Then, the divergence wrt.\ Sasaki metric is characterised by
    \begin{align}
        \Liederivative{\Yfield} \diffd\Leb{\sasakimetric*}
        &=
        \Liederivative{\Yfield}
        \parentheses*{
            \diffd\Leb{\vertmetric*} \wedge \diffd\Leb{\hormetric*}
        }
        =
        \Liederivative{\Yfield}
            \diffd\Leb{\vertmetric*}
        \wedge
        \diffd\Leb{\hormetric*}
        +
        \diffd\Leb{\vertmetric*}
        \wedge
        \Liederivative{\Yfield}
             \diffd\Leb{\hormetric*}
        \notag
        \\
        &\begin{aligned}
        =
        \Liederivative{\vertproj(\Yfield)}
            \diffd\Leb{\vertmetric*}
        \wedge
        \diffd\Leb{\hormetric*}
        +
        \diffd\Leb{\vertmetric*}
        \wedge
        \Liederivative{\horproj(\Yfield)}
             \diffd\Leb{\hormetric*}
        \end{aligned}
        \label{eq:vertical-horizontal-Lie-derivative}
        \\
        &=
        \parentheses*{
            \divergence{\vertmetric*}(\vertproj(\Yfield))
            +
            \divergence{\hormetric*}(\horproj(\Yfield))
        }
        \pdot
        \diffd\Leb{\vertmetric*}
        \wedge
        \diffd\Leb{\hormetric*}
        =
        \divergence{\sasakimetric*}(\Yfield) \pdot \diffd\Leb{\sasakimetric*}
        \notag
    ,
    \end{align}
    where we used the local coordinate description of Lie derivatives in line~\eqref{eq:vertical-horizontal-Lie-derivative}.
\end{proof}

\begin{remark}[Sasaki gradient of loc-density]
\label{rem:grad-loc-density}
    Basically, the Sasakian gradient of
    a tangent bundle weighting~$\rho = \rho_{\posfold}\locprod\rho_F$
    decomposes as
    \begin{displaymath}
        \grad{\sasakimetric*}\rho
        =
        \vlfunc{\rho_{\posfold}} \pdot \grad{\vertmetric*}\rho_F
        \oplus
        \rho_F \pdot \grad{\hormetric*}\vlfunc{\rho_{\posfold}}
    \end{displaymath}
    reminiscent of the Leibniz rule.
    Here, we think of~$\rho_F$ as an independent copy defined on the pointwise tangent spaces.
    Actually, the symbol
    $\grad{\vertmetric*}\rho_F\vdef\vertproj(\grad{\sasakimetric*}\rho)$
    is an intuitive short hand for the vertical component of the local vector field
    $
        \varphi_2^{-1}(\ID,\grad{\mathrm{euc}}\rho_F\bincirc\proj_2\bincirc\varphi_1)
    $,
    where the diffeomorphisms $\varphi_1$, $\varphi_2$ render the diagram
    in~\autoref{fig:trivialisation-for-grad} commutative.
    \begin{figure}[ht]\large
    \begin{tikzcd}
	    \arrow{r}{\varphi_2}
        \arrow[swap]{d}{\tanproj}
        \arrow[swap, bend right=55]{dd}{\tantanproj}
        \tantanproj^{-1}(U)
        &
        \tanproj^{-1}\parentheses*{ U\times\Rnum^{\posdim} }
        \arrow{d}{\tanproj}
        \arrow{r}{\simeq}
        &
        \tanproj^{-1}(U)\times\Rnum^{\posdim}
        \arrow{dl}{(\tanproj,\ID)}
        \\
	    \arrow{r}{\varphi_1}
        \arrow[swap]{d}{\tanproj}
        \tanproj^{-1}(U)
        &
        U\times\Rnum^{\posdim} \arrow{dl}{\proj_{1}}
        &
        \\
	    U & &
    \end{tikzcd}
    \caption{Local trivialisation reflected in the double tangent bundle}\label{fig:trivialisation-for-grad}
    \end{figure}
\end{remark}

We define the basic concept of semisprays
which arises in the abstract study of second order ordinary differential equations --
particularly, of Newton's (second) law on manifolds, see~\cite[Section~4]{Gliklikh} --,
and also encodes geometric extra structure in several interesting ways.
As a consequence of the discussion,
the direct sum $\canonfield\oplus\spray$ of canonical vector field and semispray~\spray
is understood as the diagonal mapping
\begin{displaymath}
    \tanbundle\posfold\to\tantanbundle\posfold,\
    v\longmapsto \vertlift(v)\oplus\horlift(v)
.
\end{displaymath}

\begin{definition}[Semispray]
\label{def:semispray}
    A section $\spray\in\smoothsec{\tanbundle{\posfold}}{\tantanbundle{\posfold}}$
    is a \emph{semispray} if it satisfies
    $\dualpair{\spray}{d\tanproj} = \ID_{\tanbundle{\posfold}}$
    or equivalently if
    any integral curve $s\colon\,\indexset\to\tanbundle{\posfold}$
    takes the form $s = (\tanproj\bincirc s)'$.
    A curve $c\colon\,\indexset\to\posfold$ is called \emph{geodesic of the semispray \spray} if
    there is an integral curve $s\colon\,\indexset\to\tanbundle{\posfold}$
    such that $c = \tanproj\bincirc s$.
    Equivalently, $c$ is geodesic if $\spray\bincirc c' = c''$.
    For a local coordinate form of semisprays
    see the~\autoref{app:loc-coord}.
\end{definition}

\begin{example}[Semispray associated to an Ehresmann connection]
\label{ex:Riemann-spray}
    Consider an Ehresmann connection
    $\tantanbundle{\posfold} = \vertbundle{\tanbundle{\posfold}} \oplus \horbundle{\tanbundle{\posfold}}$.
    Then, given $v\in\tanbundle{\posfold}$ there is a unique horizontal vector $a$ --
    namely the horizontal lift of $v$ --
    such that $v = \dualpair{a}{d\tanproj}$.
    Furthermore, $a$~depends on~$v$ smoothly.
    Hence, there is $\spray\in\smoothsec{\tanbundle{\posfold}}{\horbundle{\tanbundle\posfold}}$
    such that $\ID_{\tanbundle{\posfold}} = \dualpair{\spray}{d\tanproj}$.
    Then, \spray~is a semispray
    and we say that it is the \emph{semispray associated to the Ehresmann connection}.
    In particular, there is the semispray~$\spray_{\riemannmetric*}$
    associated to the Riemann metric~\riemannmetric* via the corresponding Ehresmann connection.
    We call it \emph{Riemannian semispray}.

    Another common name is \emph{geodesic spray},
    since it can be constructed just in terms of geodesics corresponding to the Riemannian metric,
    see~\cite[Lemma~III.2.3]{doCarmo}.
    So, the geodesics of this semispray are just classical geodesics.
    In fact, this semispray satisfies a homogeneity condition making it a (full) spray
    whence the name --
    we will use the spray structure a few times indeed.
\end{example}

\begin{remark}
\label{rem:Ehresmann-connection-semispray-relation}
    In a nutshell, choosing an Ehresmann connection adds the same geometric information
    as choosing a covariant derivative or a semispray.
    See~\cite{habil-Bucataru} and various references therein.
    We think that this observation might be a good starting point for
    generalisations even to rough geometries,
    as it translates well to Lagrange and Finsler spaces.
\end{remark}

\begin{example}[Semisprays induced by Lagrangians]
\label{ex:spray-as-gradient}
    Let the Lagrangian $\Lagrange = \frac12\,\abs{\cdot}{\riemannmetric*}^2$.
    Then, the Lagrangian vector field~\spray{\Lagrange}
    is a semispray\footnote{In fact, it is the geodesic spray again.},
    see~\cite[Section~7.5]{MarsdenRatiu}.
    As outlined in~\cite[Section~VII.6]{SergeLang}
    this relation can be translated to general Riemannian manifolds modelled on some Hilbert space.
    Note that existence of such a vector field is due to
    the more general result~\cite[Proposition~VII.5.9]{SergeLang}:
    Let $\configfold\vdef\tanbundle\posfold$,
    then the cotangent space of~\configfold naturally\footnote{
        Up to sign conventions:
        Let~$\Theta$ denote the canonical 1\=/form,
        then we define the 2\=/form~$\Omega$ by $\Omega=-d\Theta$.
    } yields the structure of a symplectic manifold
    $(\tanbundle*\configfold,\Omega)$.
    Furthermore, we make~\configfold into a Riemannian manifold
    endowing it with a `natural metric' in the sense of Gudmundsson-Kappos.
    This specifies the musical isomorphisms, in particular~$\sharp$.
    Thus, there is a unique section
    $\spray{\Lagrange}\in\smoothsec{\tanbundle\configfold}$ such that
    \begin{displaymath}
        \parentheses*{\Omega^\sharp\bincirc\spray{\Lagrange}}_w(a)
        \vdef
        \Omega_w^\sharp(\spray{\Lagrange}(w),a)
        =
        d_w\Lagrange(a)
        \qquad\text{for all }w\in\configfold,a\in\tanbundle\configfold{w}
    ,
    \end{displaymath}
    where $\Omega^\sharp$ denotes the pullback of~$\Omega$ wrt.~$\sharp$.
    With that, we can say that
    $
        \spray{\Lagrange}
        =
        (d\,\Lagrange)^\sharp
        =
        \grad\Lagrange
    $,
    where the gradient is taken wrt.\ the `natural' metric chosen before.

    See~\cite[Section~3]{metric-nonlin-connections-Bucataru}
    for another short discussion of semisprays on general Lagrange spaces.
\end{example}

\begin{remark}[Revisiting {\hyperref[ex:Riemann-spray]{Example~\ref{ex:Riemann-spray}}}]
\label{rem:Riemann-spray-revisited}
    We shall characterise the Riemannian semispray using~\cite[Proposition~VII.5.9]{SergeLang}
    similar to the previous example.
    Again, let $\configfold\vdef\tanbundle\posfold$
    and the symplectic manifold $(\tanbundle*\configfold,\Omega)$.
    Let $\hormetric*=\tanproj^\ast\riemannmetric*$ be the horizontal metric,
    which is the simplest `natural' metric,
    and denote the pullback of the canonical 2\=/form~$\Omega$ wrt.\ this metric by~\hlfunc{\Omega}.
    Locally we can think of
    $a^\prime,b^\prime\in\tanbundle*\configfold{w}$, $w\in\configfold$,
    as tuples $a^\prime=(u_1,u_2^\prime)$ and $b^\prime=(v_1,v_2^\prime)$
    with $u_1,u_2,v_1,v_2\in\configfold$
    and
    $
        u_2^\prime=\riemannmetric{\cdot}{u_2}{x},
        v_2^\prime=\riemannmetric{\cdot}{v_2}{x}\in\configfold^\ast
    $,
    where $x\vdef\tanproj(w)$.
    Then, the form~{\hlfunc{\Omega}}\xspace reads as
    \begin{align*}
        \hlfunc{\Omega}_w((u_1,u_2),(v_1,v_2))
        &=
        \riemannmetric{u_1}{v_2}{x}
        -
        \riemannmetric{v_1}{u_2}{x}
        \\
        &=
        \dualpair{u_1}{v_2^\prime}
        -
        \dualpair{v_1}{u_2^\prime}
        =
        \Omega_w(a^\prime,b^\prime)
    .
    \end{align*}
    There is a 1\=/form~$\omega$
    which reads in such local regimes as
    $\omega_w(b)=\omega_w(v_1,v_2)=\riemannmetric{w}{v_1-v_2}{\tanproj(w)}$.
    The unique section $\spray\in\smoothsec{\tanbundle\configfold}$
    such that $\hlfunc{\Omega}\bincirc\spray=\omega$
    is the Riemannian semispray.
\end{remark}

\begin{example}[Euclidean case]
\label{ex:spray-acting}
    The Riemannian semispray~\spray{\riemannmetric*}
    acts on vertically lifted functions~$f=\vlfunc{f_0}$ with $f_0\in\smoothfunc{\posfold}$ as
    \begin{displaymath}
        \spray{\riemannmetric*}f
        =
        \hormetric
            {\spray{\riemannmetric*}}
            {\grad{\hormetric*}\vlfunc{f_0}}
        =
        \riemannmetric
            {\ID_{\tanbundle\posfold}}
            {\grad{\riemannmetric*}f_0 \bincirc\tanproj}
            {\tanproj}
    .
    \end{displaymath}
    In case of $\posfold = \Rnum^{\posdim}$ with standard Riemannian metric
    this action is written as
    $   \spray{\mathrm{euc}}f(x,v)
        =
        \scalarprod{v}{\nabla_x f_0(x)}{\mathrm{euc}}
    $
    for smooth functions
    $f\colon\,\Rnum^{\posdim}_x\times\Rnum^{\posdim}_v\to\Rnum,\,(x,v)\mapsto f_0(x)$
    and $x,v\in\Rnum^{\posdim}$.

    By analogy, the canonical vector field~\canonfield acts on
    horizontally lifted functions~$f=\hlfunc{f_0}$ with $f_0\in\smoothfunc{\posfold}$ as
    \begin{displaymath}
        \canonfield f
        =
        \vertmetric
            {\canonfield}
            {\grad{\vertmetric*}\hlfunc{f_0}}
        =
        \riemannmetric
            {\ID_{\tanbundle\posfold}}
            {\grad{\riemannmetric*}f_0 \bincirc\tanproj}
            {\tanproj}
    .
    \end{displaymath}
    In the Euclidean case,
    this action is written as
    $   \canonfield f(x,v)
        =
        \scalarprod{v}{\nabla_v f_0(v)}{\mathrm{euc}}
    $
    for smooth functions
    $f\colon\,\Rnum^{\posdim}_x\times\Rnum^{\posdim}_v\to\Rnum,\,(x,v)\mapsto f_0(v)$
    and $x,v\in\Rnum^{\posdim}$.
\end{example}

The following theorem is well-known,
but comes in a few quite different formulations e.\,g.:
The geodesic flow preserves the volume of~\tanbundle{\posfold}.
We just stick to the formulation below,
since it tells us that
the Riemannian semispray is an antisymmetric operator
wrt.~\bigL2{\tanbundle\posfold}{\sasakimetric*}\=/scalar product.

\begin{theorem}[Liouville's Theorem]
\label{thm:Liouville-thm}
\namedlabel{thm:named:Liouville-thm}{Liouville's Theorem}
    The semispray $\spray{\riemannmetric*}$ is solenoidal wrt.\ Sasaki metric~\sasakimetric*.
\end{theorem}
\begin{proof}
    A proof could be done via
    explicitly calculating the Sasakian volume form in normal coordinates.
    This approach is taken from the neat book by M.\ do~Carmo,
    cf.~\cite[Exercise~3.14]{doCarmo}.
\end{proof}

The final lemma of this section
characterises the test functions on the tangent space
exploiting again the `almost product' structure due to local trivialisation.
This result will be very important arguing
for a reasonable set of test functions being a core of the (Langevin/fibre lay-down) generator.

\begin{lemma}[Tensor product of lifted test functions]
\label{lem:D0isdense}
    Define the tensor product space of pulled back test functions
    \begin{align*}
        D_0
        &{}\vdef
        \tanproj^\ast\smoothcompact{\posfold}
        \tensorprod
        \kappa^\ast\smoothcompact{\posfold}
        \\
        &=
        \linhull
        \left\{
            \tanproj^\ast f_0
            \tensorprod
            \kappa^\ast g_0
            \vdef
            \tanproj^\ast f_0
            \pdot
            \kappa^\ast g_0
            \ \middle\vert\
            f_0, g_0\in\smoothcompact{\posfold}
        \right\}
        \\
        &{}=
        \linhull
        \left\{
            \vlfunc{f_0}
            \tensorprod
            \hlfunc{g_0}
            \ \middle\vert\
            f_0, g_0\in\smoothcompact{\posfold}
        \right\}
    .
    \end{align*}
    Then, $D_0$ is dense\footnote{
        Dense wrt.~the usual locally convex topology induced by appropriate seminorms,
        which implies uniform convergence of all derivatives on compacts;
        for the well-known Euclidean case see~\cite[Example~2.4.10]{Horvath}
        and for the generalised geometric case see~\cite[Section~3.1.3]{GKOS}.
        In common notation for test function spaces with this topology,
        the result reads as
        $\tanproj^\ast\mathcal{D}(\posfold)\tensorprod\kappa^\ast\mathcal{D}(\posfold)$
        is dense in
        $\mathcal{D}(\tanbundle\posfold)$.
        By the way,
        the authors of~\cite{GKOS} not only discuss generalised distribution spaces rigorously,
        they shed some light on integration of 1\=/densities from a very analytic perspective.
    }
    in $\smoothcompact{\tanbundle{\posfold}}$.
\end{lemma}
\begin{proof}
    Consider a local trivialisation in $U\vdef U_x\cap W$
    for a chart domain $W\subseteq\posfold$ at $x\in\posfold$
    with $V\vdef\tanproj^{-1}(U)$ and the diffeomorphism~$\varphi$
    rendering the diagram in~\autoref{fig:local-trivialisation} commutative.
    Then, we immediately conclude the relations
    \begin{displaymath}
        \tanproj^\ast\smoothcompact{U}
        =
        \varphi^\ast\parentheses*{ \smoothcompact{U}\tensorprod\{1\} }
        \qquad\text{and}\qquad
        \kappa^\ast\smoothcompact{U}
        =
        \varphi^\ast\parentheses*{ \{1\}\tensorprod\smoothcompact{\Rnum^\posdim} }
        .
    \end{displaymath}
    Thus, by~\cite[Proposition~4.8.1]{Horvath} the tensor product
    $
    \tanproj^\ast\smoothcompact{U}
    \tensorprod
    \kappa^\ast\smoothcompact{U}
    $
    is dense in~
    $
    \smoothcompact{V}
    =
    \varphi^\ast\smoothcompact{U\times\Rnum^{\posdim}}
    $
    and the proof is finished via a partition of unity argument.
\end{proof}


TITLE: Exponential growth with a constant
QUESTION [1 upvotes]: Some guy opens a bank account with an initial amount of $\$1,000$. Each month he deposits $\$200$ and the bank gives him a monthly interest of $6\%$. I want to find the closed formula.
Given this, we can derive the recurrence-relation: $y_n=1.06y_{n-1} + 200, y_0 = 1000, n \ge 1$. Now, I know that it can also be written as: $y_n=1.06^ny_0 + 200n = 1.06^n*1000 + 200n$. Or so I thought. When I try this formula for $n=2\ or\ n=3$, the formula is not equivalent to the recurrence-relation.
I think I know this has something to do with the fact that the coefficient $1.06$ only applies to the initial cash, and not the deposits he made after that, and so I tried the formula: $$y_n = 1.06^n*1000 + 1.06^{n-1}*200n$$ However, this formula is also wrong, maybe even more so. I'm in need of some help to get past this obstacle.
Own thoughts on how to progress (in a structural manner)
$y_n = 1.06y_{n-1}+200, y_0 = 1000, n \ge 1$
$y_1 = 1.06y_{0} + 200 = 1.06*1000 + 200$
$y_2 = 1.06y_{1} + 200 = 1.06(1.06*1000 + 200) + 200 = 1.06^2*1000 + 1.06*200 + 200$
$\begin{align}
y_3 = 1.06y_{2} + 200 &= 1.06(1.06^2*1000+1.06*200+200) + 200\\
&=1.06^3*1000 + 1.06^2*200 + 1.06*200 + 200 \\
&= 1.06^3*1000+1.06^2*200+1.06*200+1.06^0*200
\end{align}$
Don't know if this even helps me. Any help is appreciated, many thanks.

REPLY [1 votes]: let:
$M = \mbox{monthly deposit}$
$R = \mbox{interest rate}$
$S = \mbox{starting capital}$
the recurrence-relation is:
$$
\mbox{given: } n \in \mathbb{N}\\
\begin{cases}
y_0 = S& \mbox{if } n=0\\
y_n = Ry_{n-1}+M &\mbox{otherwise}
\end{cases}
$$
by calculating manually the first few terms, and performing some algebra one can find that:
$$
y_n=S \cdot R^n + M \cdot \sum_{i=0}^{n-1} R^i
$$
now let's remember the Geometric Series:
$$
\sum_{i=0}^{n} x^i =\frac{1-x^{n+1}}{1-x}
$$
we will use this to explicitly calculate $\sum_{i=0}^{n-1} R^i$, substituting and correcting the upper bound gives:
$$
y_n=S \cdot R^n + M \cdot \sum_{i=0}^{n-1} R^i \iff \boxed{y_n=S \cdot R^n + M \cdot \frac{1-R^{n}}{1-R}}
$$


TITLE: Prove that $<$ defined above on $\hat f$ is a well-ordering of order type $\epsilon_{0}$
QUESTION [3 upvotes]: If $f,g:\Bbb R \to \Bbb R$ define $f<g \iff \exists M:\forall x>M$, $f(x)<g(x)$
[Does this mean that $g$ dominates $f$].
Now define $\hat f$ to be the smallest set of functions containing $\Bbb N[x]$ and closed under the operation $f\mapsto x^{f}$ and $(f,g)\mapsto f+g$
[e,g $x^{x^{3x+2}+5x^{x}}+2x+4\in \hat f$, but $(x+1)^{x} \notin \hat f$]
I want to prove that $<$ defined above on $\hat f$ is a well-ordering of order type $\epsilon_0$.
My attempt:
Recall that  a set  $x$ is said to be well oredered by $<$ if $x$ is linearly ordered by $<$ and has the property if $A\subseteq X$ non- empty then $A$ contains a least element i,e $\exists x \in A : x\le a ,\forall a \in A$.
I need to show that $(\hat f,<)\cong (\epsilon_{0},<)$ to ensure that it is a well ordering of type $\epsilon_0$ that is i need to find a Bijection?

REPLY [0 votes]: If $\hat{f}$ is also stable under sum, then you could try to prove that the map recursively defined on $\varepsilon_0$ by $\varphi(0) = 0$ and $\varphi(\sum \limits_{k=0}^p \omega^{\alpha_k}.n_k) := \sum \limits_{k=0}^p x^{\varphi(\alpha_k)}.n_k$ is an isomorphism.
The best way to do this is to prove it by induction on the least integer $n_{\lambda}$ such that $\lambda \in x_n:= \omega^{\omega^{... \omega}}$ where $\omega$ appears $(n+1)$-times: prove that $\varphi$ is well-defined on $x_n$, strictly increasing, and that $\hat{f} = \bigcup \limits_{n \in \mathbb{N}} \varphi(x_n)$.


TITLE: integral of $\tan^2(x/2)$
QUESTION [0 upvotes]: Whats the integral of $$\tan^2(x/2)dx$$ so basically what i tried is integration by parts which gave me results like $$2\tan(x/2)\log|sec(x/2)|+\frac{1}{2}\int \sec^2(x/2).\log|sec(x/2)|dx$$ but after this i tried $sec(x/2)=u$ also IVP but dint help. Help is appreciated.

REPLY [1 votes]: Set $\nu=x/2$ and $d\nu=\frac 1 2 dx$
$$=2\int \tan^2 \nu d\nu$$
$$\color{blue}{\tan^2(\nu)=\sec^2(\nu)-1}$$
$$=2\int \left(\sec^2(\nu)-1\right)d\nu=2\tan(\nu)-2\nu+\mathcal C=\color{red}{2\tan\left(\frac x 2\right)-x+\mathcal C}$$


TITLE: Using the Casorati-Weierstrass theorem.
QUESTION [2 upvotes]: Show that there is a complex number $z$ such that:$$\left|\cos{\left(\frac{1}{2z^4+3z^2+1}\right)}+100\tan^2{z}+e^{-z^2}\right|<1$$
It's easy to see that $z=i$ is a simple pole of $\frac{1}{2z^4+3z^2+1}$, but I want to know how to conclude that $z=i$ is an essential singularity of $\cos{\left(\frac{1}{2z^4+3z^2+1}\right)}$ so that I can use the Casorati-Weierstrass theorem.

REPLY [0 votes]: Denote $$f(z) = \cos\left(\frac{1}{2z^4+3z^2+1}\right)$$
To prove $z=i$ is an essential singularity of $f(z)$, just find two complex sequences of $\{z_n\}$ and $\{w_n\}$ so that $z_n, w_n\rightarrow i$ but $f(z_n) = 1$, $f(w_n)=-1$.
This means both $\lim_{z->i} f(z)$ and $\lim_{z->i} 1/f(z)$ do not exist. By definition, $z=i$ is an essential singularity.


TITLE: Intersection between 2 planes
QUESTION [2 upvotes]: I need help understanding the formula for intersection between 2 planes

Where is $\alpha$ actually, its not clear in the diagram. It appears that I could just interpret it as the below?

Where $\beta = \theta$?
UPDATE
Now given the question:

What I did

Whats wrong? Correct answer $60.7^{\circ}$. Did I "expanded" the equation for plane whose equation was in Cartesian form correctly?

REPLY [2 votes]: The norm $\vec{n}=\langle 0,3,5 \rangle$ is $\sqrt{34}$ and not $\sqrt{24}$. Then, the computation adds up and the angle amounts to 60.7$^\circ$.