Regularity Structures on Manifolds and Vector Bundles

Martin Hairer0000-0002-2141-6561
EPFL, Switzerland and Imperial College London, UK

Harprit Singh0000-0002-9991-8393
Imperial College London, UK

We develop a generalisation of the original theory of regularity structures, [28], which is able to treat SPDEs on manifolds with values in vector bundles. Assume $M$ is a Riemannian manifold and $E\to M$ and $F^i\to M$ are vector bundles (with a metric and connection), this theory allows to solve subcritical equations of the form

\begin{equation*}
\partial_t u + \mathcal{L}u = \sum_{i=0}^m G_i(u, \nabla u,\ldots, \nabla^n u)\xi_i\ ,
\end{equation*}

where $u$ is a (generalised) section of $E$, $\mathcal{L}$ is a uniformly elliptic operator on $E$ of order strictly greater than $n$, the $\xi_i$ are $F^i$-valued random distributions (e.g. $F^i$-valued white noises), and the $G_i:E\times TM^*\otimes E \times\ldots\times (TM^*)^{\otimes n} \otimes E \to L(F^i ,E)$ are local functions.

We apply our framework to three example equations which illustrate
that when $\mathcal{L}$ is a Laplacian it is possible in most cases to renormalise such equations by adding spatially homogeneous
counterterms and we discuss in which cases more sophisticated renormalisation procedures (involving
the curvature of the underlying manifold) are required.

§ INTRODUCTION

The purpose of this work is to extend the theory of regularity structures as developed in [28] for solving singular partial differential equations on $\mathbb{R}^d$ or $\mathbb{T}^d$ to a theory able to treat equations on vector bundles over a Riemannian manifold.
That is, assume $M$ is a Riemannian manifold and $E\to M$ and $F^i\to M$ are vector bundles (with a metric and connection), we develop a theory for solving subcritical equations of the form

\begin{equation}\label{equation to solve}
\partial_t u + \mathcal{L}u = \sum_{i=0}^m G_i(u, \nabla u,\ldots, \nabla^n u)\xi_i\ ,
\end{equation}

where $u$ is a (generalized) section of the vector bundle $E \to M$ and $\mathcal{L}$ is a uniformly elliptic operator on $E$ of order grater than $n$, $\xi_i$ are $F^i$ valued random distributions (e.g. $F^i$-valued white noise) and $G_i:E\times TM^*\otimes E \times\ldots\times (TM^*)^{\otimes n} \otimes E \to L(F^i ,E)$ are local functions.

As in [28] and subsequent works [11], <cit.>, [7] the treatment of such equations can naturally be divided into the following steps:

* Introducing regularity structures and models as general tool and developing an analytic framework allowing to reformulate and solve such equations as abstract equations “driven” by a model.

* Building a regularity structure for a specific equation (or system of equations) and a large enough renormalisation group, as well as identifying its action on the equation.

* Identifying the correct renormalisation procedure and showing convergence of renormalised models.

In this work we treat the first two steps in full generality and execute the last step for the following equations.

* Let $M$ be a compact Riemannian $2$-manifold, $f\in \cC^{\infty}(\RR)$ and $A(\cdot, \cdot)$ a smooth bilinear form on the tangent bundle $TM$. The g-PAM equation reads

\begin{equation}\tag{g-PAM}
\label{gpam_intro}
\partial_t u + \Delta u = A( \nabla u, \nabla u) +f(u)\xi,
\end{equation}

where $\Delta$ is the Laplace–Beltrami operator[We use the sign convention more commonly used in geometry. Thus, Laplacians are positive operators.] and $\xi$ is spatial white noise constant in time.

* Let $M$ be a compact Riemannian 2 or 3-manifold and $E\to M$ a vector bundle with a metric $h$ and compatible connection $\nabla^{E}$. The natural geometric generalisation of the $\Phi^4$ equation reads
∂_t u + Δ^E u =  -u·|u|^2_h +ξ,

where $\xi$ is an $E$-valued spacetime white noise, $|\cdot|_h$ the norm on $E$ induced by $h$ and $\Delta^E$
a generalised Laplacian acting on sections of $E$, an example being
the connection Laplacian induced by the connection $\nabla^E$ and the Levi-Civita connection, c.f. (<ref>).

The invariant measure of this dynamic is of interest in constructive quantum field theory and can formally be written as

$$\exp\Big(-\int_M |\nabla^E\phi|^2_{g\times h} +V_{C,\lambda}(\phi)  \Big)d\mathfrak{L}(\phi)\ , $$

$|\cdot |_{g\times h}$ is the induced norm on $T^*M\otimes E$,
$V_{C,\lambda}(u)= C|u|_{h}^4- \lambda |u|_{h}^2 \ $ is a double well potential and $\mathfrak{L}(\phi)$ denotes the (non-existent) Lebesgue measure on sections of $E$.

* While it turns out that in many cases, such as the two above, the renormalisation procedure needed to solve singular SPDEs is insensitive to the curvature of the underlying manifold, this is not always the case. The equation
∂_t u + Δu = u^2  +ξ
on a compact Riemannian $4$ manifold is an example where an additional counter-term proportional to the scalar curvature of $M$ has to be introduced.

At this point we mention the work [19] which builds a minimal framework to solve Equation (<ref>) in the special case $f=0$ and $g=\id$ within regularity structures. Of course flat compact manifolds, such as the torus or the Klein bottle could already be treated in [28], since all flat compact manifolds are quotients of Euclidean space by a free crystallographic group.
The theory of para-controlled distributions <cit.>, an alternative theory to deal with SPDEs, has been extended to the manifolds setting in [6] to be able to deal with the analytic step of solving Equation (<ref>), see also <cit.>.
Furthermore, Equation <ref> was recently independently studied in <cit.>, <cit.> based on a transform introduced in <cit.>.

§.§ Outline of the first half

Section <ref> introduces the geometric setting for this work, in particular a notion of scaling $\fraks$ similar to [28].

In Section <ref> we first recall known results about jets, the analogue of polynomials on manifolds. Then, the notion of an admissible realisation introduced.

Since in the setting here, i.e. in the presence of a connection and metric, there exists a canonical grading and norm on jet bundles $J^k$ which we use to construct a graded space $J$.

The space $J$ will play the role of abstract polynomials and comes with a canonical inclusion

$$J^k\hookrightarrow J \ $$

for each $k$.

Section <ref> introduces the notion of a regularity structure and model. As in [19], we only impose a condition informally expressible as

Π_p Γ_p,q∼Π_q

instead of the rigid relation $\Pi_p \Gamma_{p,q}= \Pi_q$ in [28] and further introduce a bigrading $T= \bigoplus_{\alpha,\delta} T_{(\alpha,\delta)}$ where the parameter $\delta$ keeps track of a quantitative bound on the error implicit in (<ref>).  While this allows for more flexibility in the definition of jet models, it has the drawback that in order to formulate a general Schauder estimate in Section <ref> it forces us to abandon the lower triangular structure of the maps $\Gamma$.
To be able to treat equations with components taking values in different vector bundles such as $u$ and $\nabla u$ in Equation (<ref>), we work with regularity structure ensembles. In contrast to [19] our maps $\Gamma$ and $\Pi$ are defined globally. We introduce vector bundle valued jet regularity structures and models as examples.

Up to modifications of some proofs the ideas in Section <ref>, where modelled distributions are introduced and a reconstruction theorem in our setting is provided, are the same as in <cit.>.

In Section <ref> we show how to implement local operations on regularity structures. The notion of products is similar to the one in [28] with the caveat of having to use regularity structure ensembles in the bundle valued setting and the need to keep track of the bi-grading. Compositions with smooth local functions relies on interpreting a local function $G: E\to F$, where $\pi_E: E\to M$, $\pi_F:F\to M$ are vector bundles, as a section of the bundle $E\ltimes F:= \pi_E^* (F)$ and needs additional considerations not present in the flat setting. Lastly, we explain in this section, how to lift differential operators relying on the fact that every $k$th order differential operator $\partial: \mathcal{C}^\infty(E)\to \mathcal{C}^\infty(F)$ corresponds to a bundle morphism $T_\partial\in L(J^k(E), F)$ together with considerations from Section <ref>.

In Section <ref> we follow <cit.> to define integration against singular kernels between vector bundles. Due to (<ref>)
which has as consequence that

\begin{equation}\label{locloc}
(I+J)\circ \Gamma\neq \Gamma\circ (I+J)
\end{equation}

(as opposed to the setting in [28]), we have to abandon the lower triangular structure of the maps $\Gamma$ and introduce a new operator to compensate (<ref>). While the presence of this operator technically complicates the analysis, it has a natural interpretation in terms of the “elementary” modelled distributions introduced in [3], see Remark <ref>.

Section <ref> and Section <ref> are except for technical changes close to <cit.>. Since throughout the main part of the text we only work with the integral form of the equations of interest, Section <ref> contains a short discussion about the existence of fundamental solutions for differential operators of the form $\partial_t + \mathfrak{L}$.

This completes the analytic part of the theory.

For readers mostly interested in the case of $M=\mathbb{R}^d$ or $\mathbb{T}^d$ the jet bundle can canonically be identified with polynomials, c.f. Sections <ref> and Remark <ref>. From this point of view one can interpret many results as recovering known statements of [28] under weaker assumptions, see also Remark <ref>.

§.§ Outline of the second half

In Section <ref> we recall the notion of symmetric sets developed in [8], and introduce vector bundle assignments, a direct modification of vector space assignments. We also recall the analogue of the functor $\Func_W$ therein.

In Section <ref> sets of (combinatorial decorated) trees which form the backbone of the constructions that follow are introduced. These, in contrast to previous works, are built from three sets of edge types $\mathcal{E}_+$, $\mathcal{E}_-$ and $\mathcal{E}_0$. The first encodes kernels, the second noises and the last serves as a place holder for jets. From these, similarly to [11], sets of trees $\mathfrak{T}, \mathfrak{T}_+$ and $\mathfrak{T}_-$ as well as coloured versions thereof are built from a rule.
The trees in $\mathfrak{T}_-$ are used later for negative renormalisation, contain no $\mathcal{E}_0$ type edges and are used to index spaces of differential operators $\mathfrak{Dif}$, while the trees in $\mathfrak{T}$ resp. $\mathfrak{T}_+$ are used in the construction of the regularity structure ensemble, resp. for positive renormalisation utilising on the machinery of symmetric sets.
The coloured versions of these sets of trees such as $\mathfrak{T}^{(-)}$ and $\mathfrak{T}_-^{(-)}$ are crucially used since both renormalisations are encoded by colouring operations instead of extractions.

In Section <ref> , after fixing a cutoff $\delta_0>0$, we construct vector bundles $\mathcal{T}$ and $\mathcal{T}_+$ from a subsets of $\mathfrak{T}$ and $\mathfrak{T}_+$. This is done by first turning trees into symmetric sets and then using the functor $\Func_W$,  informally this corresponds to “attaching"  jets to edges of type  $\mathcal{E}_0$. Finally, $\mathcal{T}$, is equipped with a bi-grading and thus a regularity structure ensemble is constructed.
Then, we construct pre-models which turn elements of $\mathcal{T}$ into concrete distributions as well as the renormalisation group $\mathfrak{G}_-$. After investigating the action of $\mathfrak{G}_-$ on a pre-model $\pmb{\Pi}$, we show how to construct a model $(\Pi, \Gamma)$ from it. To do so, positive renormalisation is still described by “admissible cuts", though in this setting, as a consequence of (<ref>) one has to cut trees of negative homogeneity as well.
Then, we study the action of $\mathfrak{G}_-$ on models. Since we do not restrict ourselves to constant coefficient differential operators, this action is not described by extraction operations but by colouring operations.
For both renormalisations, due to the fact that all bundles are constructed from symmetric sets, significant care has to be taken to ensure that everything is well defined.
Lastly, explicit formulas for the renormalisation of canonical smooth models are obtained in this section.

Section <ref> starts with a subcritical system of SPDEs and constructs the required bundle assignments, edge types and rule. Then the machinery of the previous sections yields a regularity structure as well as a renormalisation group. First, for the canonical model for smooth noises we check that the corresponding (reconstructed) abstract solution agrees the classical solution to the system of equations.
Then, in the central Proposition <ref> of this section, we prove a formula for the renormalised equation, which is solved (in the classical sense) by the reconstructed abstract solution with respect to a renormalised model.

In the context of Remark <ref> one can interpret the results of this second part of the article as giving an alternative non-algebraic point of view on the results in [11], [15] and [7]. In particular, our framework allows for a rather compact proof of
Proposition <ref>, a slight generalisation of the main result of [7] even in the case of the torus.

§.§ Outline of concrete applications and stochastic estimates

When working with non-translation invariant equations, one is faced with the problem that the strategy of regularity structures, when applied naively might yield an infinite-dimensional solution family. The renormalisation group constructed in Section <ref> is indeed infinite dimensional. In Section <ref>, we outline a general strategy to remedy this issue.

In particular, we explain why for most singular SPDEs where the linear term is given by a generalised Laplacian, renormalisation constants are sufficient. More precisely, one expects the convergence results for such singular SPDEs on trivial vector bundles over the torus to translate directly to any (possibly non-trivial) vector bundle over a compact manifold.

In Section <ref> we turn to the treatment of specific examples of singular SPDEs, each of which illustrates certain aspects of the theory developed in this article.

* While (<ref>) is a scalar valued equation, the right-hand side involves the gradient of the solution, i.e. a section of $TM$ and the equation involves a rather general non-linearity. Since it requires only Wick renormalisation, it serves as an example illustrative of the general workings of the theory which does not many stochastic estimates.
In Theorem <ref> we exhibit an (at most) two dimensional canonical solution family.

* The $E$-valued <ref>-equation illustrates how the theory developed in this article
can be used to solve singular SPDEs taking values in general vector bundles. While the scalar valued analogue
on a compact manifold can essentially be treated as in [28],
when the vector bundle $E$ is non-trivial, a technical novelty arises when establishing the convergence of

$$\left(\hat{\Pi}^\epsilon_q  \hat{\Gamma}^\epsilon_{q,p}- \hat{\Pi}^\epsilon_p\right) \left(\<3'2>_{_{(e_p)}}\right)\ , $$

see Lemma <ref>.
As in [28], we exhibit an (at most) $2$-dimensional solution family in Theorem <ref>.

* The last example, the <ref> equation, is meant to illustrate an SPDE, where a geometric counterterm is needed to renormalise the model. Here we obtain a (at most) $5$-dimensional solution family in Theorem <ref>.

Typically, in the framework of this article, when showing that a family of models $Z^\epsilon=(\Pi^\epsilon, \Gamma^\epsilon)$ converges, one has to not only obtain appropriate stochastic estimates on terms of the form
${\Pi}^\epsilon_p\tau (\phi^\lambda_p)$, but also additionally for

$$\left(\Pi^\epsilon_q\Gamma_{q,p}^\epsilon \tau_p-\Pi^\epsilon_p\tau_p \right)(\phi^\lambda_q)\ .$$

While this can significantly increase the number of estimates needed, their general form is close to the Feynman diagrams usually encountered in regularity structures. Thus, for example for the equations treated here, the straightforward adaptations of the kernel estimates in [28] and <cit.> in Section <ref>, are mostly sufficient.

§.§ Acknowledgements

HS gratefully acknowledges funding by a President’s PhD Scholarship from Imperial College London
and would like to thank S. Paycha for valuable discussions.
MH gratefully acknowledges funding from the Royal Society through
a research professorship, grant RP\R1\191065.

§ GEOMETRIC SETTING: ORTHOGONAL FOLIATIONS OF MANIFOLDS

In this section we motivate a natural geometric setup for incorporating a scaling $\mathfrak{s}$ in the sense of [28].
Let $(M,g)$ be a (complete) $d$-dimensional Riemannian manifold and suppose one is given $n\leq d$ mutually orthogonal distributions[Here “distribution” is meant in the geometric sense, i.e. a smooth section of the Grassmann bundle over $M$.] $\mathcal{D}^1,\ldots, \mathcal{D}^n\subset TM$, such that $T_p M= \bigoplus_{i=1}^n \mathcal{D}^i_p$ together with an element $\mathfrak{s}\in \mathbb{N}^n$. Then one defines the scaled length of a tangent vector
$$X_p= X_p^1+\ldots+X_p^n\in T_pM \ ,$$
where $X_p^i\in \mathcal{D}_p^i$ as

\begin{equation}\label{scaled norm}
|X_p|_\mathfrak{s}=\sum_{i=1}^n g_p(X_p^i,X_p^i)^{\frac{1}{2\mathfrak{s}_i}} \ .
\end{equation}

On geometric grounds it is natural to assume that these distributions are integrable and parallel.

Under these assumptions the following theorem of de Rham, c.f.  <cit.> holds.

In addition to the above setup, assume that $M$ is simply connected. We denote by $\mathfrak{F}^1,\ldots,\mathfrak{F}^n$ the foliations of $M$ corresponding to $\mathfrak{D}^1,\ldots,\mathfrak{D}^n$ respectively. Then, for every $p\in M$, there exists a foliation preserving isometry from $(M,g)$ to the Riemannian product manifold
$(L^1, g^1)\times\ldots\times (L^n,g^n)$, mapping each leaf $\mathfrak{F}^i$ to $L^i$.

The above discussion motivates the following definition.

A scaling $\mathfrak{s}$ on a manifold $(M,g)$ is a decomposition $M=(M^1,g^1)\times\ldots\times (M^n,g^n)$ together with an element $\mathfrak{s}\in \mathbb{N}^n$. To a scaling $\mathfrak{s}$ on $(M,g)$ we associate the scaled “norm” on $TM$ defined in (<ref>), as well as the distance[Sometimes it will be more convenient to work with the (equivalent) scaled distance $d_\mathfrak{s}(p,q):=\sup_{i\in \{1,\ldots,n\}} d^i(p_i,q_i)^{1/\mathfrak{s}_i}$, which we shall do without elaborate explanation.] defined for $p,q\in M$ by

$$d_\mathfrak{s}(p,q)=\sum_{i=1}^n d^i(p_i,q_i)^{\frac{1}{\mathfrak{s}_i}}$$

where $d^i$ denotes the usual distance induced on $M^i$ by $g^i$ and where $p=(p_1,\ldots,p_n)$, $q=(q_1,\ldots,q_n) \in (M^1,g^1)\times\ldots\times (M^n,g^n)$ .

The following simple lemma illustrates that the norm $|\cdot |_\mathfrak{s}$ and distance $d_\mathfrak{s}(\cdot, \ \cdot)$ fit together well.

In the above setting for every point $p\in M$ and $X_p\in T_p M$ small enough the identity

$$d_\mathfrak{s}(p,\exp_p{X_p}) = |X_p|_\mathfrak{s}\ ,$$

We call a chart $\phi : M\supset U \to V\subset \mathbb{R}^d$ an $\mathfrak{s}$-chart, if there exist charts
$\phi^{i}: M^i\supset U^i \to V^i\subset \mathbb{R}^{d_i}$ for $i\in \{1,\ldots,n\}$ such that
$U=U^1\times\ldots\times U^n$ and $\phi=(\phi^1,\ldots,\phi^n)$. For a multi-index $k=(k^1,\ldots,k^n)$, where $k^i\in \mathbb{N}^{d_i}$, we write
$$|k|_\mathfrak{s}=\sum_{i=1}^n \fraks_i\sum_{j=1}^{d_i} k^i_j \ .$$

In the sequel it will be useful to single out the vector fields $V$ on $M$, which take values in $TM^i\subset TM$. We call such a vector field an $\mathfrak{s}_i$-vector field. We say $W$ is a $\fraks$-vector field, if it is a $\mathfrak{s}_i$-vector field for some $i$ and declare the degree of a $\fraks$-vector field $V$ to be given by

\begin{equation}\label{deg of vec}
\text{deg}_\fraks V =\begin{cases}  0 & \text{if } V=0 \ ,\\
\fraks_i & \text{if } V \text{ is an }\fraks_i\text{-vector field .}
\end{cases}
\end{equation}

We extend this to tuples of $\fraks$-vector fields $ (V_1,\ldots,V_k)$ by

$$ \text{deg}_\fraks (V_1,\ldots,V_k) = \sum_j \deg_\fraks(V_j) \ .$$

One could also work with real-valued scalings $\fraks\in \mathbb{R}_+^n$, but the restriction $\fraks\in \mathbb{N}^n$ simplifies notation later on. Furthermore, since in the theory of regularity structures one always has “an epsilon of room”, this is not usually much of a restriction.

In the setting of Theorem <ref> the Frobenious theorem, c.f. <cit.>, characterises integrable distributions.
The somewhat complementary setting where $\mathcal{D}^1$ is spanned by Hörmander vector fields, the simplest examples being Carnot groups,  was considered in <cit.>.

§.§ Vector bundles and metric connections

Usually, when working with a vector bundle $\pi: E\to M$, we shall assume that it is equipped with a metric $\langle\cdot , \cdot\rangle_{E}$ and a connection $\nabla^E$ which is compatible with the metric, i.e. for any two sections $V,W$ of $E$ and $X_p\in T_p M$ one has

$$X_p(\langle V, W \rangle_E) =\langle \nabla_{X_p}^E V, W \rangle_E + \langle V, \nabla_{X_p}^E W \rangle_E \ , $$

where the left-hand side uses the usual identification of tangent vectors with derivations.
We shall write $(E,\nabla^E, \langle\cdot, \cdot\rangle_E)$ in this case. We shall denote by $\cC^\infty(E)$ the space of smooth sections of $E$ and by $\cC_c^\infty(E)$ those which are furthermore compactly supported.

Let us recall some important constructions and examples of vector bundles.

§.§.§ Trivial bundle

Any real-valued function on $M$ can naturally be identified with a section of the trivial bundle $M\times \RR$. (Recall that there exist a canonical metric and connection on this bundle, the latter being independent of the Riemannian structure of $M$.)

§.§.§ Tangent bundle

We shall always equip the tangent bundle $TM$ with the Levi-Civita connection $\nabla^{TM}$ .

§.§.§ Dual bundle

Given a vector bundle $(E,\nabla^E, \langle\cdot, \cdot\rangle_E)$ as above, denote by $E^*$ the dual bundle and $(\cdot, \cdot): E^*|_p \otimes E|_p\to \RR$ the natural pairing above any point $p\in M$. We shall always equip $E^*$ with the canonically induced metric $\langle\cdot , \cdot\rangle_{E^*}$ and connection  $\nabla^{E^*}$, which are characterised by the following properties:

* If $\{e_i\}$ is an orthonormal basis of $E|_p$, then the unique basis $\{e_i^*\}$ of $E^*|_p$ satisfying
$$(e_k^*,e_i)= \delta_{i,k} \ $$

is an orthonormal basis of $E^*|_p$.

* If $V, V^*$ are sections of $E$, resp. $E^*$ and $X_p\in T_p M$, then

$$\nabla_{X_p} (V^*,V) = (\nabla_{X_p}^{E^*} V^*, V) + (V^*,\nabla_{X_p}^{E}  V)\ .$$

One can observe that $\nabla^{E^*}$ is a metric connection with respect to $\langle\cdot , \cdot\rangle_{E^*}$.

§.§.§ Product bundle

Given two vector bundles $E\to M, \ F\to N$, let $E\hotimes F$ be the vector bundle over $M\times N$ with fibres $E\hotimes F|_{(p,q)}=  E|_p \otimes F|_q$ for $(p,q)\in M\times N$ and smooth structure characterised by the fact that if $V\in \cC^\infty(E),\ W\in \cC^\infty(F)$, then $(p,q)\mapsto V(p)\otimes W(q)$, for which we write $V(p)\hotimes W(q)$ from now, is an element of $\cC^\infty (E\hotimes F)$.

Furthermore $(E,\nabla^E, \langle\cdot, \cdot\rangle_E)$ and $(F,\nabla^F, \langle\cdot, \cdot\rangle_F)$ as above, then we shall always equip $E\hotimes F$ with the canonically induced metric $\langle\cdot , \cdot\rangle_{E\otimes F}$ and connection $\nabla^{E\otimes F}$, which are characterised by the following properties:

* If $\{e_i\}$, $\{f_j\}$ is an orthonormal basis of $E|_p$ respectively $F|_q$, then $\{e_i\hotimes f_j\}_{i,j}$ is an orthonormal basis of $(E\otimes F)|_{(p,q)}$ .

* If $V, W$ are local sections of $E$ resp. $F$, $X_p\in T_p M$ and $Y_q\in T_q N$, then
$$\nabla_{(X_p,Y_q)}^{E\hotimes F} (V\hotimes W) = (\nabla_{X_p}^{E} V)\otimes W(q) +  V(p)\otimes \nabla_{Y_q}^{F} W \ ,$$

where we implicitly identified $T(M\times N)\simeq TM\hotimes TN$ under the canonical isomorphism.

One can observe that $\nabla^{E\hotimes F}$ is compatible with $\langle\cdot , \cdot\rangle_{E\hotimes F}$.

§.§.§ Tensor product bundle

Given two vector bundles $(E,\nabla^E, \langle\cdot, \cdot\rangle_E)$ and $(F,\nabla^F, \langle\cdot, \cdot\rangle_F)$ with the same manifold $M$ as base space, let $E\otimes F:=\diag^*(E\hotimes F)$ where $\diag:M\to M\times M$ is the diagonal map and $\diag ^*$ the induced pullback of vector bundles. We equip $E\otimes F$ with the canonically induced metric $\langle\cdot , \cdot\rangle_{E\otimes F}$ and connection $\nabla^{E\otimes F}$, which are characterised by the following properties:

* If $\{e_i\}$ and $\{f_j\}$ are orthonormal bases of $E|_p$ and $F|_p$, then $\{e_i\otimes f_j\}_{i,j}$ is an orthonormal basis of $(E\otimes F)|_p$.

* If $V, W$ are local sections of $E$, resp. $F$ and $X_p\in T_p M$, then
$$\nabla_{X_p}^{E\otimes F} (V\otimes W) = (\nabla_{X_p}^{E} V)\otimes W +  V\otimes \nabla_{X_p}^{F} W \ .$$

One can observe that $\nabla^{E\otimes F}$ is compatible with $\langle\cdot , \cdot\rangle_{E\otimes F}$. Furthermore they agree with the pulled back metric and connection under $\diag$ .

§.§.§ Symmetric tensor product bundle

Working with one vector bundle $(E,\nabla^E, \langle\cdot, \cdot\rangle_E)$, we write $E\otimes_s E:=\diag^*(E\hotimes_s E)$ for the subbundle of $E\otimes E$ consisting of symmetric tensors equipped with the metric and connection obtained by restriction of the ones on $E\otimes E$.

§.§.§ Higher order derivatives

The constructions above give a meaning to higher order derivatives of sections of $ E$ given inductively as
$$(\nabla^E)^{n+1}:= \nabla^{T^*M^{\otimes n}\otimes E}\circ (\nabla^E)^n \ , $$

where $(\nabla^E)^1:=\nabla^E$ .

We shall usually simply write $\langle\cdot , \cdot \rangle$ and $\nabla$ omitting reference to the underlying vector bundle.

§.§ Bundle valued distributions

Given a vector bundle $E\to M$, we write $\mathcal{D}(E)$ for the space of smooth compactly supported sections. We equip
$$\mathcal{D}(E)= \bigcup_{K\subset M\ {\text{compact}}} \mathcal{C}^\infty_{c,K}(E)\ $$
with the inductive limit topology, where each $\mathcal{C}^\infty_{c,K}(E):=\{ \phi\in \mathcal{C}^\infty_{c}(E) \ : \supp (\phi)\subset K\}$ is equipped with the usual Fréchet topology induced be the semi-norms
$$\|\phi \|_k:= \sup\{ |\nabla^l \phi (x)|\ : \ x\in K, \ l\leq k\}\ .$$

The space of generalised sections (or $E$-valued distributions) $\mathcal{D}'(E)$ is defined as the dual space of $\mathcal{D}(E)$.
We shall usually identify $u\in \mathcal{C}^\infty (E)$ with an element of $\mathcal{D}'(E)$ via the canonical inclusion

$$\mathcal{C}^\infty (E)\hookrightarrow \mathcal{D}'(E),\qquad u\mapsto T_u \ , $$

where $T_u(\phi):= \int_M \langle u(p), \phi(p)\rangle_E  \, \dVol_p$ for $\phi\in \mathcal{D}(E)$ and
$\dVol_p$ stands for integration in $p$ with respect to the usual Riemannian density.

§ JET BUNDLES

In this section we collect the definitions and basic properties of the jet bundle of a vector bundle $\pi:E\to M$ where $M$ is equipped with a scaling $\fraks$ and at the same time we fix notation used in the sequel. For a beautiful exposition on jet bundles we refer to [5].

§.§ Definition and basic properties

Throughout this section we do not assume any structure other than $M$ being a smooth manifold and $\pi:E\to M$ being a smooth vector bundle.

For $k\in \mathbb{N},p\in M$ we define an equivalence relation on $\cC^\infty(M)$ by postulating that

$$f\sim_{k,p} g$$
if in some (and therefore any) $\fraks$-chart $(U,\phi)$ around $p=\phi (0) \in M$ and for all multiindices
$l$ with $|l|_{\fraks}\leq k$ one has $D^l(f\circ \phi)(0) =D^l(g\circ \phi)(0)$. We define

$$J_p^k M := \cC^\infty(M)/\sim_{k,p}$$
and denote by $j^k_p f$ the equivalence class of $f\in \mathcal{C}^\infty(M)$.

For a vector bundle $E\to M$ we define the space

$$J_p^k E:= \cC^\infty(E)/\sim_{E,k,p}\ ,$$

where two section $f,g\in \cC^\infty(E)$ are equivalent with respect to $\sim_{E,k,p}$ if in some (and therefore any) smooth local frame $\{e_i\}_i$ of $E$ around $p$ the component functions of $f=\sum_i f^i e_i$, $g=\sum_i g^i e_i$ satisfy $f^i\sim_{k,p} g^i$ for each $i$ .

We define $J^k E:= \bigcup_{p} J_p^k E$, which can be endowed with the structure of a vector bundle (c.f. <cit.> or <cit.>) and is called the $k$-jet bundle of $E$. To any smooth section $f\in \mathcal{C}^\infty (E)$ one can associate the following section of $J^k E$:

$$j^k f :M \to J^k E,\qquad  p\mapsto  j_p^k f\ .$$

The map $j^k$ is called the prolongation map.

To make notation consistent one should write $J^k (M\times\RR)$, instead of  $J^k M$ but we shall usually opt for the latter for notational convenience.

Classically jet bundles are defined in the absence of a non-trivial scaling. But since the scaling only causes minor modifications of the classical concepts, committing slight abuse of nomenclature, we use the classical terminology in this setting as well.

The notion of jet bundles extends to fibre bundles, but the jet bundle of a fibre bundle is only an affine space in general, c.f. [46].

In the absence of a scaling $\mathfrak{s}$,  an equivalent condition
for $f\sim_{k,p} g$ for two functions $f,g\in \cC^\infty(M)$ is to ask that for any smooth curve $\gamma \colon (-\epsilon, \epsilon) \to M$
such that $\gamma(0)=p$, one has
$$\partial_s^n (f\circ \gamma)(s)|_{s=0}=\partial_s^n (g\circ \gamma)(s)|_{s=0}$$
for all $n\leq k\ .$
This is not true in the presence of a non-trivial scaling $\mathfrak{s}$!

Let us mention an alternative characterisation of the space $J_p^k M$, resp. $J_p^k E$, for a proof in the absence of a scaling we refer to the discussion in <cit.>.

For $p=(p_1,\ldots,p_n)\in M$ let $I^i_p:=\{f\in  \cC^\infty(M^i)\,:\, f(p)=0\}$ be the ideal of $\mathcal{C}^\infty(M^i)$ of functions vanishing at $p$.
Identify $I^i_p\subset \mathcal{C}^\infty (M)$ in the canonical way and define $\bar{I}_p^k$ as the smallest ideal containing
$$\bigcup_{|l|_\fraks > k} (I_p^1)^{l^1} \cdots (I_p^n)^{l^n}\;.$$

Then one can define $J^k_p(M)= \mathcal{C}^\infty(M)/\bar{I}^k_p(M)$, which shows in particular, that $J^k(M)$ has a natural algebra structure.

Similarly for a vector bundle $E$, if one defines $Z^k_p= \bar{I}_p^k \cdot \cC^\infty(E)\subset \cC^\infty(E)$ , one can set
$$J^k_p(E)= \mathcal{C}^\infty(E)/Z^k_p\;,$$
which implies that $J^k_p(E)$ is a $J^k_p(M)$-module.

§.§.§ Pull-back of jets

Let $\phi:M\to N$ be a smooth map, the notion of pullback
$$\phi^*: C^{\infty}(N)\to C^{\infty}(M)\ ,\qquad  g\mapsto g\circ \phi\ ,$$
induces a notion of pullback on jets. We define a map $\phi^*:J^k N \to J^k M$ determined for $p\in U$, $q=\phi(p)$ by

\begin{equation}\label{eq pullback of scalar jets}
\phi^*|_q: J^k_q V \to J^k_p U \ , \qquad  j^k_q f\mapsto j^k_{p} (\phi^* f) =j^k_{p} (f\circ \phi) \ .
\end{equation}

Working on vector bundles $E\to M$, $F\to N$, let $\Phi: E\to F$ be a bundle morphism which restricts to isomorphisms on fibres.
Write $\Phi=(\phi, G)$ where $\phi:M\to N$ is a smooth map and $G(p):E|_p\to F|_{\phi(p)}$ is an invertible linear map for each $p\in M$,  such that one can write $\Phi(p,v)= (\phi(p), G(p)(v))$ for every $(p,v)\in E$. Such a bundle morphism induces a pullback $\Phi^*: \cC^\infty(F)\to \cC^\infty(E)$ by

\begin{equation}\label{eq pullback of general jets}
\big(\Phi^* g\big) (p)= G^{-1}(p) \big(g\circ \phi(p)\big)\ .
\end{equation}

One can thus define a map $\Phi^*:J^k F \to J^k E$ determined by the fact that for any $g\in\cC^\infty(F)$ and $p\in M$, $q=\phi(p)$

$$\Phi^*|_q: J^k_q F \to J^k_p E \ , \qquad  j^k_q g\mapsto j^k_{p} (\Phi^*g) \ .$$

One easily checks that the maps introduced in (<ref>) and (<ref>) are well defined.

§.§.§ Differentiation of jets

For $V$ a vector field in a neighbourhood of $p$ the map
$$\nabla_V: J^k_p(E) \to J_p^{{\lfloor k-\max_{i}\fraks_i\rfloor}} (E),\quad
j^k_p f \mapsto j^{{\lfloor k-\max_{i}\fraks_i\rfloor}}_p \nabla_Vf $$
is well defined. If $V$ is an $\fraks_i$-vector field, this can improved and the  map

\begin{equation}\label{differentiation of jets}
\nabla_V: J^k_p(M) \to J_p^{{\lfloor k-\fraks_i\rfloor}}(M) ,\quad
j^k_p f \mapsto j^{{\lfloor k-\fraks_i\rfloor}}_p \nabla_Vf \
\end{equation}

is well defined. This follows from a direct computation in charts, or alternatively from the fact that in the setting of Remark <ref> if $r\in Z_p^k$, then $\nabla_{V}r \in Z_p^{\lfloor k-\fraks_i\rfloor}$.
Given a global $\fraks_i$-vector field $V$, this induces a smooth map $J^k E\to J^{\lfloor k-\fraks_i\rfloor} E$.

§.§.§ Scalar valued jets in local coordinates

Let $\mathbb{R}^d$ be equipped with a scaling $\mathfrak{s}$ in the sense of <cit.> and denote by $\Poly$ polynomials in indeterminates $X_1,\ldots,X_d$. If $M=U$ is an open subset of $\mathbb{R}^d$, we can and will identify $J^k M$ with $U\times \Poly^k$, where $\Poly^k\subset \Poly$ denotes the space of abstract polynomials of (scaled) degree up to $k$
and we identify for $f\in \mathcal{C}^\infty(M)$

$$j^k_p f= \sum_{|l|_\fraks\leq k} \frac{X^l}{l!} D^{l}f(p) \ .$$

Note that $\Poly^k$ comes with a natural algebra structure, where the product is given by the usual product on $\Poly$ composed with the projection onto $\Poly^k$ and, for any $p \in M$, this agrees with the algebra structure on $J^k_p$ previously mentioned.
We now investigate the notion of pullback of jets under this identification. Let $U,V \subset \mathbb{R}^m$ and $\phi: U\to V$ be a smooth map. For $p\in U$, $q=\phi(p)$ one can express the map

$$\phi^*|_q: J^k_q V \to J^k_p U \ , \qquad  j^k_q f\mapsto j^k_{p} (\phi^* f) =j^k_{p} (f\circ \phi) \ ,$$

under the above identification of jets with polynomials as follows: Denoting by $P^k_{q}\phi$ the polynomial of order $k$ associated to the $\mathbb{R}^d$-valued jet $j_q^k \phi$, the map $\phi^*$ is given by the algebra (bundle) morphism $\{q\}\times \Poly^k \to \{p\} \times \Poly^k$ determined by $X_i \mapsto \big( P_{q}\phi - p \big)_i$. Note that this map is not graded in general, but upper triangular with respect to the natural grading on polynomials.

The fact that the map $\phi^*|_q: J^k_q V \to J^k_p U$ does not preserve the grading on polynomials is consistent with the fact that in the setting of smooth manifolds, the jet bundle is only a filtered bundle as will be discussed in Section <ref>.

If $(U,\phi)$ is a chart of $M$ at $p\in U \subset M$, then this induces a chart

\begin{equation}\label{chart for jets}
\theta^k_{(U,\phi)}:J^k U \to \phi(U)\times \Poly^k
\end{equation}

determined by $\theta^k_{(U,\phi)}:  j^k_q f\mapsto j^k_{\phi(q)} (\phi^{-1})^* f$, where we can identify $ j^k_{\phi(q)} (\phi^{-1})^* f$ with a polynomial. The smooth structure of $JM$ agrees with the one given by such charts.

§.§.§ Bundle valued jets in local coordinates

We extend the previous discussion to bundle valued jets. Let $M=U$ be an open subset of $\mathbb{R}^d$ as before, let $E=U\times\mathbb{R}^n$ be the trivial bundle over $U$, and denote by $\{e_i\}_{i=1,..,n}$ the canonical basis of $\mathbb{R}^n$. Then we can and will identify $J^k E$  with $U\times \Poly^k(\RR^n)$, where $\Poly^k(\RR^n)=\Poly^k \otimes \mathbb{R}^n$ denotes the space of abstract $\RR^n$-valued polynomials of (scaled) degree up to $k$. Note that $\Poly^k(\RR^n)$ is naturally a $\Poly^k$-module. We thus identify for $f=\sum_{i=1}^n f^i\otimes e_i$

$$j^k_p f= \sum_{|l|_\fraks\leq k} \sum_{i=1}^n \frac{D^{l}f^i(p)}{l!} X^l\otimes e_i \ .$$

Let $U,V \subset \mathbb{R}^d$ and $E=U\times \RR^n$, $F=V\times \mathbb{R}^n$ be trivial bundles. Let $\Phi: E\to F$ be a bundle morphism restricting to an isomorphism on fibres. Write $\Phi=(\phi, G)$ where $\phi:U\to V$ and $G:U\mapsto \text{GL}(\mathbb{R}^n)$ are smooth maps such that $\Phi(p,v)= (\phi(p), G(p)(v))$ for each $(p,v)\in E$. Then the pull-back map $\Phi^*: \cC^\infty(F)\to \cC^\infty(E)$ is given on $g= \sum_i g^i e_i \in\cC^\infty(F)$ by

$$\big(\Phi^* g\big) (p)= G^{-1}(p) \big(g\circ \phi(p)\big)= \sum_i (g^i\circ\phi)(p) \, G^{-1}(p)(e_i) \ .$$

To describe the map
$$\Phi^*|_q: J^k_q F \to J^k_p E \ , \qquad  j^k_q g\mapsto j^k_{p} (\Phi^*g) \ $$
for $p\in U$, $q=\phi(p)$ explicitly, we use the algebra structure of $\Poly^k$ and that $\Poly^k(\RR^n)$ is a $\Poly^k$-module. Then, making the above identification of jets with polynomials and denoting for $\Phi= (\phi, G)$ by $\{g^{i,j}\}_{i,j}$ the components of $G^{-1}$, the map $\Phi^*|_q: J^k_q F\to J^k_p E$ is the unique linear map such that for every multi index $l$

\begin{equation}\label{equationn}
X^l  e_j\mapsto (P^k_{q}\phi - p)^l \cdot (P^k_p G^{-1})e_j =\sum_{i} (P_{q}\phi - p)^l \cdot (P^k_p g^{i,j}) e_i
\end{equation}

where $P^k_{q}\phi$ is the polynomial of order $k$ associated to the $\mathbb{R}^d$ valued jet $j_q \phi$ and similarly for $P^k_p (G^{-1})$ and $P^k_p g^{i,j}$.

Now suppose $(\pi^{-1} U,\Phi)$ is a local trivialisation of $E$, then this induces a chart
$\theta^k_{(\pi^{-1} U,\Phi)}: J^k(\pi^{-1} U) \to J^k(\phi(U)\times \mathbb{R}^n)=\phi(U)\times \Poly^k(\RR^n)$ determined by

\begin{equation}\label{bundle jet chart}
\theta^k_{(\pi^{-1} U,\Phi)}:  j^k_q f \mapsto j^k_{\phi(q)} \big((\Phi^{-1})^* f\big) \ ,
\end{equation}

where we identify $j^k_{\phi(q)} \big((\Phi^{-1})^* f\big)$ with a polynomial.

As in the scalar case one sees from (<ref>) the jet bundle is only a filtered bundle and the maps in (<ref>) are charts for the jet bundle.

§.§.§ Differential operators

Given vector bundles $E, F$ over a manifold $(M,g)$ with a scaling $\fraks$, we call $\mathcal{A}: \mathcal{D}(E) \to \mathcal{D}(F)$ a (smooth) $k$th order differential operator if there exists $T_\mathcal{A} \in \mathcal{C}^\infty (L(J^kE , F))$ such that
$ \mathcal{A}= T_\mathcal{A}\circ j^k\ .$
We denote by $\mathfrak{Dif}_k(E,F)$ the space of such differential operators.

Observe that if the scaling is trivial, this notion of $k$-th order differential operators agrees with the standard notion, c.f. <cit.>.

§.§ Grading on jets

In this section we observe that there is a canonical grading on $J^k E$, if the underlying manifold $M$ is equipped with a Riemannian metric and $E$ with a (metric) connection.

Note that for each $l<k$ the linear map $p_{k,l}: J^k E\to J^l E$, $j^k f \mapsto j^l f$ is well defined, which
turns $(J^kE)_{k\in \mathbb{N}}$ into  a filtered bundle.
One then has a highly non-canonical isomorphism of vector bundles

$$J^k E \simeq \ker p_{k,k-1} \oplus \Im p_{k,k-1} \simeq \ker p_{k,k-1} \oplus J^{k-1} E\;,$$

so that one can write

$$J^k E \simeq \bigoplus_{l=0}^k \ker p_{l,l-1}\ ,$$

where we identify $J^{-1} E=M\times \{0\}$ and $ p_{0,-1}: J^{0} E\to J^{-1} E$ is the unique bundle morphism preserving the base space.
Thus, by choosing for each $k$ an isomorphism $ \ker p_{k,k-1} \oplus J^{k-1} E\to J^k E $, one constructs a grading on $J^k M$. It is natural to restrict our attention to maps which act on $\ker p_{k,k-1}$ as the identity, and thus it suffices to choose an injection

$$i_{k-1,k}: J^{k-1} E\to J^k E \ ,$$

for which it is natural to further impose that $p_{k,k-1}\circ i_{k-1,k}= \id_{J^{k-1}(E)}$.

We first discuss the case when $E=M\times \RR$ is the trivial bundle and later turn to the more general case.
For $p\in M$ we define

\begin{equation}\label{def of i}
i_{k-1,k}: J^{k-1}_p M \to J^{k}_p M
\end{equation}

as follows. Let $\phi$ be an exponential $\fraks$-chart at $p$, i.e. an exponential chart at $p$ which is also an $\fraks$-chart, then for
$f\in \cC^\infty(M)$ set

$$i_{k-1,k}(j^{k-1}_p f) = j_p^k h \ ,$$

where $h$ is a function satisfying

* $j^{k-1} h = j^{k-1} f$

* For any multi-index $l$ with $|l|_{\mathfrak{s}}=k$, $D^l h (p)=0$.

For each $k$ the map  $i_{k-1,k}$ is a well defined bundle morphism and furthermore it is a right inverse of the map $p_{k,k-1}$, i.e. $p_{k,k-1}\circ i_{k-1,k}= \id_{J^{k-1}M}$.

Before proceeding to the proof, let us point out that in the vector bundle case, these two conditions (respectively their naive adaptation) are in general not compatible as can be seen from the following example. Let $E=TM$, then in the same exponential chart $(\nabla_{\partial_1}\nabla_{\partial_2}-\nabla_{\partial_2}\nabla_{\partial_1} )f= R(\partial_1,\partial_2)f$ where $R$ denotes the Riemann curvature tensor.

To show that the map $i_{k-1,k}$ is well defined, it suffices to take $h$, in an exponential $\fraks$-chart, to be the polynomial of scaled degree $k$ satisfying these conditions.
To see that it is a right inverse of $p_{k,k-1}$, note that

$$p_{k,k-1}\circ i_{k-1,k} j_p^{k-1} f= p_{k,k-1} j^k_p h = j_p^{k-1} h = j_p^{k-1} f\ . $$

To observe that $i_{k-1, k}$ is indeed a bundle morphism, it remains to show that it is smooth. Fix some point $p\in M$, and let $U\subset M$ be a neighbourhood of $p$, such that there exists an orthonormal frame of $TU$ consisting of $\fraks$-vector fields. Fix such a frame $(V_1,\ldots,V_d)$ and denote by $r_{\tilde{p}}\in (0,\infty)$ the injectivity radius of the exponential map at ${\tilde{p}}\in M$. Denote by

$$\phi_{\tilde{p}}: B_{r_{\tilde{p}}}(0)\to M\ ,\qquad (\lambda_1,\ldots,\lambda_d)\mapsto \exp_{\tilde{p}} (\lambda_1V_1({\tilde{p}}),\ldots,\lambda_d V_d({\tilde{p}}))$$
the induced exponential chart at ${\tilde{p}}\in U$. Write $\neigh=\{({\tilde{p}},q)\in M\times M\ | \  d({\tilde{p}},q)<r_{\tilde{p}}\}$. It is well known that $\neigh\ni ({\tilde{p}},q)\mapsto (\phi_{\tilde{p}})^{-1}(q)$ is smooth.

Let $O:=\{({\tilde{p}},q,v_q)\in U\times JU \ | \ d({\tilde{p}},q)< r_{\tilde{p}} \}$ and define
$$\psi^k: U\times JU \supset O\to U\times \Poly^k, \qquad ({\tilde{p}},v_q)\mapsto \theta^k_{\tilde{p}}(v_q) \ ,$$

where $\theta^k_{\tilde{p}}=(\phi^{-1}_{\tilde{p}})^*$ is defined as in (<ref>). Observe that $\psi^k$ is smooth, which follows from the fact that the exponential chart map is smooth in both entries and the explicit transformation rule for jets in charts from Section <ref>.
We define
$$\theta^k: JU \to U\times \Poly^k, \qquad ({\tilde{p}},v_{\tilde{p}})\mapsto \psi^k({\tilde{p}},v_{\tilde{p}})\;.$$

Then $i_{k-1,k}$ can locally be factored as

$$J^{k-1}M\supset J^{k-1}U \overset{\theta^{k-1}}\longrightarrow U\times \Poly^{k-1}\hookrightarrow U\times \Poly^{k} \overset{(\theta^{k})^{-1}}\longrightarrow  J^{k}U\subset  J^{k}M \ ,$$

and since all maps involved are smooth, the claim follows.

By defining for $l<k-1$ the maps $i_{l,k}:= i_{l, l+1}\circ \ldots\circ i_{k-1,k}$, we have fixed an injection

$$J^l M\supset \ker p_{{l},{(l-1)}}\to J^k M\ .$$

We set $(J^k M)_l:= i_{l,k}(\ker p_{{l},{(l-1)}})$ and obtain a (canonical) grading

\begin{equation}
J^k M = \bigoplus_{l\leq k} (J^k M)_l \ .
\end{equation}

For $k\geq l$ we shall write $Q^k_l$ for the canonical projection

$$J^k M\to (J^k M)_l\subset J^k M\ ,$$

which can explicitly be written as

\begin{equation}
Q^k_l=  i_{l,k}\circ p_{k,l} -  i_{l-1,k}\circ p_{k,l-1} \ .
\end{equation}

Before we construct a (canonical) grading on the jet bundle of a genuine vector bundle $E$, we turn to a short discussion about higher order covariant derivatives.

§.§.§ Higher order derivatives

Denote by $T^*M$ the dual space of the tangent space of $M$. This space naturally decomposes as

$$T^*M= T^*M^1\oplus \ldots\oplus T^*M^n \ .$$
For a multiindex $l=(l^1,\ldots,l^n)\in \mathbb{N}^n$ define

$$ |l|_\fraks:=\sum_{i=1}^n \fraks_i l^i \qquad \text{ and }\qquad  |l|:=\sum_{i=1}^n  l^i\ .$$

and define

$$(T^*M)^{\otimes_s l }:= \bigotimes_{i=1}^n (T^*M^i)^{\otimes_s l^i} \ ,$$
$\otimes_s$ denotes the symmetric tensor product. For $m\in \mathbb{N}$ set

$$(T^*M)^{m}= \bigoplus_{|l|_\mathfrak{s} =m} (T^*M)^{\otimes_s l } \ .$$

Define for a multi-index  $l\in \mathbb{N}^n$

\begin{equation}\label{canonical differential op}
D^l: \mathcal{C}^\infty(M) \to \cC^\infty\big( (T^*M)^{\otimes_s l }\big) \ \text{ respectively }\  D^l: \mathcal{C}^\infty(E) \to \cC^\infty\big((T^*M)^{\otimes_s l }\otimes E \big)
\end{equation}

to be given as follows:
For $f\in \mathcal{C}^\infty(M) $ , respectively  $f\in \mathcal{C}^\infty(E)$

$$D^l f:= \nabla^{|l|} f\big|_{(TM)^{\otimes_s l }}\ ,$$

where we implicitly use the canonical identification $\big((TM)^{\otimes_s l }\big)^*= (T^*M)^{\otimes_s l }$ .
We define the map

\begin{equation}\label{differential operator}
\bar D_{\fraks}^k: \mathcal{C}^\infty(E)\to\cC^\infty\big( \bigoplus_{m=0}^k (T^*M)^m\otimes E \big)\ ,\qquad f\mapsto \sum_{|l|_\mathfrak{s} \leq k} D^l f \ .
\end{equation}

Note that in the absence of a non-trivial scaling $\bar{D}^k f= \bigoplus_{m=0}^k \nabla^m f$.

§.§.§ Grading on bundle valued jets

Once we have (<ref>), we have a natural injection $i_{k-1,k}: J^{k-1} E\to J^k E$
by setting, for any section $f\in \cC^\infty(E)$,
$$ i_{k-1,k}(j^{k-1}_p f)= j_p^k h \ ,$$

where $h$ is a section of $E$ satisfying

* $j_p^{k-1} h = j_p^{k-1} f$

* for any multi-index $l\in \mathbb{N}^n$ with $|l|_{\mathfrak{s}}=k$, $\big(D^l h\big) (p)=0$ in $(T^*M)_p^{\otimes_s l}\otimes E_p$ .

Note that in the case of the trivial bundle $E=M\times \RR$ this definition of $i_{k-1,k}$ agrees with the one in (<ref>) since $M\times \RR$ is a flat bundle and thus $D^l= \nabla^{|l|}$ in this case.

For each $k$ the map  $i_{k-1,k}$ is a well defined bundle morphism, and furthermore it is a right inverse of the map $p_{k,k-1}$, i.e. $p_{k,k-1}\circ i_{k-1,k}= \id_{J^{k-1}E}$.

The proof proceeds similarly to the scalar valued case, but with some modifications.

For $p\in M$, we again use $r_p$ to denote the injectivity radius of the exponential map. Given an orthonormal basis $\{e_{i,p}\}$ of $E_p$, we construct the following “special” orthonormal frame on $B_{r_p}(p)$ where for $q\in B_{r_p}(p)$

\begin{equation}\label{paralell}
e_{i,p}(q)\in E_q \text{ is obtained by by parallel translation of } e_{i,p}
\end{equation}

along the shortest geodesic connecting $p$ to $q$. We denote by $\Phi^k_p$ the trivialisation of $\pi^{-1}(B_{r_p}(p))\subset E$ induced by the exponential chart and the frame $e_{i,p}(\cdot)$.

To show that the map $i_{k-1,k}$ is well defined, it suffices to take $h$ in the trivialisation $\Phi^k_p$ to be the (unique)
appropriate polynomial of scaled degree $k$ and to note that in this trivialisation

$$j_p^{k-1} h = j_p^{k-1} f$$

is a condition on the part of the polynomial of scaled degree up to $k-1$,  while

$$\big(D^l h\big) (p)=0  \ \text{for each } l\in \mathbb{N}^n \text{ such that}\ |l|_{\mathfrak{s}}=k$$

is a condition on the part of the polynomial of (scaled) homogeneity exactly $k$.

The map $i_{k-1,k}$ is a right inverse of $p_{k,k-1}$ since

$$p_{k,k-1}\circ i_{k-1,k} j_p^{k-1} f= p_{k,k-1} j^k_p h = j_p^{k-1} h = j_p^{k-1} f\ . $$

To observe that $i_{k-1, k}$ is indeed a bundle morphism, it remains to show that it is smooth. We proceed by a slightly more elaborate version of the construction for the $J^k M$ case.
Fix some point $p_0\in M$, and let $U\subset M$ be a neighbourhood of $p_0$, such that there exists an orthonormal frame of $TU$ consisting of $\fraks$ vector fields and an orthonormal frame of $\pi^{-1}(U)\subset E$. Fix such frames $\{V_1(p),\ldots,V_d(p)\}$ of $T_p M$ and $\{e_{p,1},\ldots,e_{p,m}\}$ of $E|_p$ and denote by $r_p\in (0,\infty)$ the injectivity radius of the exponential map at $p$. Denote by

$$\phi_p: B_{r_p}(0)\to M\ ,\quad (\lambda_1,\ldots,\lambda_d)\mapsto \exp_p (\lambda_1V_1(p),\ldots,\lambda_d V_d(p)) $$

the induced exponential chart at $p\in U$. Write $\neigh=\{(p,q)\in M\times M\ | \  d(p,q)<r_p\}$ and recall that $\neigh \ni (p,q)\mapsto (\phi_p)^{-1}(q)$ is smooth.

Denote for $(p,q)\in \neigh$ by $e_{p,i}(q)\in E_q$ the vector obtained by parallel translating $e_{p,i}$ to $q$ as in (<ref>) and note that the maps $ e_i: \neigh\to E,\ (p,q)\mapsto e_{p,i}(q)$ are smooth by construction. Write $\Phi_p:  \pi^{-1}(B_{r_p}(p))\to B_{r_p}(0)\times \mathbb{R}^n$ for the induced trivialisation.

Let $O:=\{(p,q,v_q)\in U\times\pi^{-1}(U) \ | \ d(p,q)< r_p \}$ and define

$$\Psi^k: U\times JU \supset O\to U\times \Poly^k(\mathbb{R}^n),\quad \ (p,v_q)\mapsto \theta^k_p(v_q) \ ,$$

where $\theta^k_p:=(\Psi_p^{-1})^*$ is defined as in (<ref>). Observe that the map $\Psi^k$ is smooth, which follows from the fact that the exponential chart map as well as the maps $e_i$ are smooth (in both entries) and the explicit transformation rule for jets  under change of bundle trivialisations (<ref>) .
We define
$$\theta^k: JU \to U\times \Poly^k, \quad (p,v_p)\mapsto \Psi^k(p,v_p) \ .$$

Then $i_{k-1,k}$ can locally be factored as

$$J^{k-1}E\supset J^{k-1}\pi^{-1}(U) \overset{\theta^{k-1}}\longrightarrow U\times \Poly^{k-1}(\mathbb{R}^n)\hookrightarrow U\times \Poly^{k}(\mathbb{R}^n) \overset{(\theta^{k})^{-1}}\longrightarrow  J^{k}\pi^{-1}(U)\subset  J^{k}E \ $$

and since all involved maps are smooth, the claim follows.

Exactly as in the scalar valued case, we set $i_{l,k}:= i_{l, l+1}\circ \ldots\circ i_{k-1,k}$ and thus single out a “canonical” injection

$$J^l E\supset \ker p_{{l},{(l-1)}}\to J^k E\ .$$

Set $(J^k E)_l:= i_{l,k}(\ker p_{{l},{(l-1)}})$ to obtain a grading

\begin{equation}\label{grading of J^k}
J^k E = \bigoplus_{l\leq k} (J^k E)_l \ .
\end{equation}

For $k\geq l$ we shall write $Q^k_l$ for the associated canonical projection

$$J^k E\to (J^k M)_l\subset J^k E\ ,$$

which can explicitly be written as

\begin{equation}\label{explicit formula for jet projection}
Q^k_l=  i_{l,k}\circ p_{k,l} -  i_{l-1,k}\circ p_{k,l-1} \ .
\end{equation}

We define a map
$${\pmb{D}}_\fraks^k: JE^k\to \bigoplus_{m=0}^k (T^*M)^m\otimes E$$
as the unique bundle morphism such that the following diagram commutes

\[\begin{tikzcd}
C^{\infty}(E) \arrow{r}{j^k_{\cdot}} \arrow[swap]{dr}{\bar{D}_\fraks^k}  & J^k E \arrow{d}{{\pmb{D}_\fraks^k}}\\
&\bigoplus_{m=0}^k (T^*M)^m\otimes E \ ,
\end{tikzcd}
\]

where $\bar{D}_\fraks^k$ was introduced in (<ref>).
We recover the following (in the non-scaled setting classical) result, c.f. the Corollary after Theorem 7 in <cit.>. The result also substantiates the claim that the grading constructed above is “canonical”.

The map $\pmb{D}^k$ is well defined and an isomorphism of graded vector bundles.

We first construct and show uniqueness of the map $\pmb{D}^k_p$ which makes the following diagram commute.

\[\begin{tikzcd}
C^{\infty}(E) \arrow{r}{j_p^k} \arrow[swap]{dr}{\bar{D}^k|_p}  & J_p^k E \arrow{d}{\pmb{D}_p^k}\\
& \bigoplus_{m=0}^k (T_p^*M)^k\otimes E|_p \ .
\end{tikzcd}
\]

Indeed, existence, uniqueness and the fact that $\pmb{D}^k_p:   \bigoplus_{m=0}^k J_p^k E\to  (T_p^*M)^k\otimes E|_p$ is an isomorphism follow from the fact that $\ker j_p^k= \ker \bar{D}^k|_p$ as can, for example, be seen by going to an exponential chart at $p$.

It follows directly from the construction of the grading that $\pmb{D}^k_p$ is graded.

To see that the thus defined map $\pmb{D}^k$ with the property
$\pmb{D}^k|_p=\pmb{D}^k_p$ is indeed a bundle morphism, it suffices to check smoothness. This can for example be seen similarly as in the smoothness part of the proof of Lemma <ref> by using a map $\theta$ defined in a neighbourhood of $p$. Alternatively it follows directly from <cit.>.

We fix the scalar product on $J^kE$ making $\pmb{D}^k$ an isomorphism of inner product bundles.

Let $(U,\phi)$ be an exponential $\fraks$-chart at $p$ and denote by $\partial_i$ the coordinate vector fields. Then one sees from the above construction, that

\begin{align*}
|j^k_p f|_n:= |Q^k_n j^k_p f|&\lesssim \sup_{\deg(\partial_{i_1},\ldots,\partial_{i_m})=n} |(\nabla_{\partial_{i_1}}\circ\ldots\circ \nabla_{\partial_{i_m}} f)(p)|\\
&=\sup_{\deg(\partial_{i_1},\ldots,\partial_{i_m})=n} |(\nabla^m_{(\partial_{i_1},\ldots,\partial_{i_m})} f)(p)|  \ ,
\end{align*}

for every $f\in \mathcal{C}^\infty (E)$ .

The following is a direct consequence of the preceding discussion.

For a point $p$ denote by $\phi_p$ $\fraks$-normal coordinates in a neighbourhood $U$ of $p$. Then, the map $\theta_p:=(\phi_p^{-1})^*|_{J^k_p M}$ is graded with respect to the grading introduced in (<ref>) .
Similarly, in the vector bundle valued case, denote by $\Phi_p$ a trivialisation of $\pi^{-1}(U)\subset E$ constructed as in (<ref>) from an orthonormal frame $\{e_{i,p}\}_i$ of $E|_p$ and an underlying $\fraks$-exponential chart. Then the map
$\theta_p:=(\Phi^{-1}_p)^*|_{J^k_p M}$ is graded.

Note that the search for a canonical map $i_{k-1,k}$ in this section is equivalent to the search for a canonical splitting of the jet bundle exact sequence

\[ \begin{tikzcd}
0  \arrow{r} & (T^*M)^k\otimes E \arrow{r}  & J^k(E) \arrow{r}{p_{k,k-1}} & J^{k-1}(E)   \arrow{r} & 0 \ ,
\end{tikzcd}
\]

c.f. <cit.>.
A differential operator $\mathcal{A}\in \mathfrak{Dif}_k(E,T^*M)^k\otimes E)$ such that $T_\mathcal{A}$ splits the jet bundle exact sequence is called a $k$th total differential.[It turns out that in the setting of a trivial scaling, $k$th total differentials can be characterised as exactly those differential operators $\mathcal{A}$ whose symbol, c.f. Section <ref>, is given by $\sigma_k(\mathcal{A})(v_p)(e_p)= v_p^{\otimes k}\otimes e_p$, see <cit.>. ] The operator

$$\mathcal{A}=\big(\sum_{l\in \mathbb{N}^n : |l|_\fraks=k } D^l \big)\, : \mathcal{C}^\infty(E)\to \mathcal{C}^\infty((T^*M)_p^k\otimes E )\ ,$$

where $D^l$ was defined in (<ref>), has this property.

§.§ Infinite jet bundle

At this point it is natural to introduce the infinite jet bundle $J^\infty E$ as the inverse limit of the inverse system of vector bundles $(J^k E, p_{k,l})_{l,k\in \mathbb{N}: \ l\leq k}$.

It is straightforward to see that $(J^k E, p_{k,l})_{l,k\in \mathbb{N}: \ l\leq k}$ is a perfect inverse system and that there is a canonical isomorphism $J_p^\infty E\simeq \mathcal{C}^\infty(E)/\sim_{\infty,p}$
where the equivalence relation $\sim_{\infty,p}$ is defined analogously to the relation $\sim_{k,p}$ introduced in Section <ref>. This in particular implies that for each $k\in \mathbb{N}$ there is a unique projection

$$p_{\infty,k}: J^\infty E\to J^kE$$

such that the family of maps $\{p_{\infty, k}\}_{k\in \mathbb{N}}$ satisfies the identity $p_{\infty,l}=p_{\infty,k}\circ  p_{k,l}$ for all $l\leq k$. The analogous statement also holds for the maps $i_{k,\infty}$.

In particular, it follows that the maps $\{Q^k_l\}_{k,l\in \mathbb{N}}$ introduced in (<ref>) extend to a map $Q_l:J^\infty E\to J^\infty E$ given by

\begin{equation}\label{def of proj}
Q_l=i_{l,\infty}\circ p_{\infty,l} -  i_{l-1,\infty}\circ p_{\infty,l-1} \ .
\end{equation}

We define the spaces $(JE)_l= Q_l J^\infty E$, which come with a natural scalar scalar product induced from the one on $J^kE$ .
We define the graded space

$$JE:= \bigoplus_{m=0}^\infty (JE)_m \ ,$$

which in view of Proposition <ref> is canonically isomorphic to $\bigoplus_{m=0}^\infty (T^*M)^m\otimes E$.
Note the strict inclusion
$JE\subsetneq J^\infty E$.

The reason for introducing the space $JE\subset J^\infty E$ is that working with $J^\infty E$ would introduce non-trivial convergence considerations in subsequent sections, while this can be circumvented on $JE$. In principle it would have been possible to work with $J^k M$ for some big enough $k\in \mathbb{N}$ throughout this article, but our decision to work with $JE$ seems more natural from the point of view of [28], as $J_pE$ is the canonical analogue of the space of polynomials of arbitrary but finite degree at $p$, while $J_p^\infty$ allows for infinite power series.

From now on we shall use the notation

$$j_p f:= j^\infty_p f \ ,$$

since this will make many expressions less cluttered. Note that $j_p f\notin J_p M$ in general, but this will cause no confusion, since the space $J^\infty M$ will play no role from Section <ref> onwards.

One can see that $JE$ and $JM\otimes E$ are canonically isometrically isomorphic, since it follows from Proposition <ref> that for each $k$ the vector bundles $J^k E$ and $J^kM \otimes E$ are isometrically isomorphic.

§.§ Jets and analytic bounds

We make some remarks on the growth of a function $f$ depending on its jet $j^k_p f$.

Suppose $W=(W_1,\ldots,W_n)$ is a tuple of $\fraks$-vector fields defined on $B_1(p)$. Then there exists a constant (depending on $W$), such that the following bound holds uniformly over $f, g\in \mathcal{C}^\infty(M)$ satisfying $j^k_p f= j^k_p g$

\begin{align*}
|(\nabla_{W_1}\cdots \nabla_{W_n} f)(q)&-(\nabla_{W_1}\cdots \nabla_{W_n} g)(q)|\\
& \lesssim_{W}  d_\fraks(p,q)^{\lfloor k+1-\deg_\fraks(W)\rfloor\vee 0}\ \big\| |j_\cdot^{\lfloor k+\max_i \fraks_i\rfloor\vee \deg(W)} (f-g)|\big\|_{L^\infty(B_1)} ,
\end{align*}

where $\deg_\fraks(W)$ was defined in (<ref>) .

We show the claim for $|j_p^k f|=|j_p^k f|=1$, the general case follows from linearity. We begin with the case $n=0$:
Let $h= f-g$, since $j^k_p h=0$ the discussion in this Section <ref> implies that in any $\fraks$-coordinate chart $(U,\phi)$ around $p$ the Taylor polynomial $j^k_{\phi(p)} (h\circ \phi^{-1})$ of order $k$ vanishes. Thus it follows from <cit.>, that one has a bound of the form $|h\circ \phi^{-1}(x)|\lesssim |x-\phi(p)|_\fraks^{k+1}$, which in turn implies the bound.
In the bundle valued case the same argument applies replacing the $\fraks$-chart $(U,\phi)$ by a trivialisation.

If $n>0$, note that $j^{\lfloor k-\text{deg}_\fraks(W)\rfloor}_p(\nabla_{W_1}\cdots \nabla_{W_n} h)=0$ by (<ref>) and thus the claim follows as above.

§.§ Admissible realisations

In this section we define a notion of “admissible realisation” of $J E$ for a smooth vector bundle $E$ which will lay the groundwork for defining the analogue of the polynomial regularity structure. In short, an admissible realisation of $J E$ associates to each element $v\in J E$ a concrete function. We shall first define admissible realisations on $J^kE$ and obtain admissible realisations of $JE$ by a (soft) limiting argument. The only place where more than the smooth structure on $E$ is needed is when formulating quantitative analytic bounds on admissible realisations (Lemma <ref>, Lemma <ref> and Lemma <ref>).

We call a map $R^k: J^k E \to \mathcal{C}^\infty(E)$ an admissible realisation of $J^k E$ if the following properties are satisfied:

* For each $p\in M$, the map $R^k|_{J^k_pE}$ is a right inverse of $j_p^k$, i.e.  $j_p^k \circ R^k|_{J^k_pE}= \id|_{J^k_p E}$.

* The map $R^k$ is linear on each fiber of $J^k E$.

* The map $J^k E \times M \to E, ((p, v_p),q) \mapsto R^k( (p, v_p))(q)$ is smooth.

We shall often just write $R^k(v_p)$ instead of $R^k(p,v_p)$ or $R^k_{E|_p}(v_p)$.

Let $F\mapsto N$ be a vector bundle and $\Phi:E\to F$ bundle isomorphism. Then the notion of pullback of sections $\Phi^*: \mathcal{C}^\infty (F)\to \mathcal{C}^\infty (E)$ induces the notion of pullback $\Phi^*: JF\to JE$ discussed in Section <ref>. This in turn induces a
notion of pullback of admissible realisations.

Let $\Phi:E\to F$ be a bundle isomorphism  and
$R^k$ be an admissible realisation of $JF$, then $\Phi^* R^k$ is defined as the unique map $JE\to \mathcal{C}^\infty(E)$ such that the following diagram commutes

\[ \begin{tikzcd}
J^kF  \arrow{r}{\Phi^*} \arrow[swap]{d}{R^k} &J^kE \arrow{d}{\Phi^* R^k} \\%
\mathcal{C}^\infty (F) \arrow{r}{\Phi^*}& \mathcal{C}^\infty (E)
\end{tikzcd}
\]

Note that this diagram defines the map $\Phi^* E^k$ uniquely and that the smoothness and linearity conditions in Definition <ref> are trivially satisfied. To conclude that $\Phi^* R^k$ is an admissible realisation, it suffices to observe that for $p\in M$, $v_p\in J_pE$ one has $j_p^k (\Phi^* R^k(v_p))=v_p$ . Indeed, writing $\bar{p}= \phi(p)$, one has

$$j_p^k ((\Phi^* R^k)(v_p)) = j_p^k (\Phi^* (R^k ((\Phi^{-1})^* v_p)))= \Phi^*j_{\bar{p}}^k R^k((\Phi^{-1})^* v_p)=\Phi^* (\Phi^{-1})^* v_p=v_p\ .$$

One can see that the notion of pullback of admissible realisations also makes sense if $\Phi=(\phi, G)$ is a bundle morphism restricting to isomorphisms on fibres and such that $\phi$ is an immersion.

For any vector bundle $E\to M$ and any $k\in \mathbb{N}$ there exists an admissible realisation of $J^k E$.

Let us fix some function $\psi: \mathcal{C}^\infty_c (B_1(0))$, such that $\psi|_{B_{1/2}(0)} = 1$. For $r>0$ we denote by $\psi_r$ the map given by $\psi_r(x)= \psi(\frac{x}{r})$ and extended to be vanish outside of $B_r(0)$.

First assume that $V \subset W\subset \mathbb{R}^d$ are such that $\text{dist}(V,W^c)>r$ for some $r>0$.
Then we define the map,

$$ R_{V,W,r}: V\times \Poly^k(\mathbb{R}^m)\to \mathcal{C}^\infty(W)\otimes\mathbb{R}^n,\quad (p, v_p)\mapsto \psi_r (\cdot - p)(\Pi_p v_p)(\cdot)\ ,$$

where $\Pi_p$ is the map that maps an abstract polynomial in $\Poly^k(\mathbb{R}^n)$ to the corresponding concrete $\mathbb{R}^n$ valued polynomial function centred at $p$.
Note that

* $j_p^k R_{V,W,r}(v_p) = v_p$ for all $p\in V$,

*  the map $R_{V,W,r}$ is $\mathcal{C}^\infty(V)$-linear,

*  the map $J^k(V\times \mathbb{R}^m)\times W \to \mathbb{R}^m,\ ((p,v_p), q)\mapsto R_{V,W,r}(v_p)(q)$ is smooth.

Now we construct an admissible realisation on an arbitrary vector bundle $E\to M$. Let $\{(U_i,\phi_i)\}_{i\in I}$ be a locally finite atlas of $M$ such that there exist $V_i\subset \phi (U_i)$,  satisfying $\text{dist}(V_i,\phi(U_i)^c):=r_i>0$ and $M=\bigcup_i \phi_i^{-1}(V_i)$ and such that there exists a partition of unity $\mathbb{1}_i: M\to [0,1]$ subordinate to the cover $\{V_i\}$ as well as trivialisations $\Phi_i= (\phi_i, G_i)$ of $\pi^{-1}(U_i)$. Then we define $R^k: JE \to C_c^\infty(E)$ by the following formula:

\begin{equation}\label{formula for admissible realisation}
R^k(p,v_p)(q):= \sum_i  \mathbb{1}_i(p)\, \Phi_i^*\big(R_{V_i,\phi_i(U_i),r_i}(\theta^k_i v_p )\big)(q) \,
\end{equation}

where $\theta^k_i:= (\Phi_i^{-1})^* : J^k(\pi^{-1}(V))\to V\times \Poly^k(\mathbb{R}^m)$  .

Next we check that this is indeed an admissible realisation. The second and third condition in Definition <ref> follow directly from (<ref>) and (<ref>) above.
For the first condition note that one has

$$j^k_p\big(\Phi_i^*R_{V_i,\phi_i(U_i),r_i}((\Phi_i^{-1})^* v_p )\big) = v_p ,$$

by definition.
Thus it suffices to show that, if $f , f_1,\ldots,f_n$ are smooth
sections of a vector bundle defined in a neighbourhood of $p\in M$ satisfying $j^k_p f = j^k_p f_1 =\ldots= j^k_p f_n$ and $\eta_1,\ldots,\eta_n$ are smooth functions satisfying $\sum_{i=1}^n \eta=1$ in a neighbourhood of
$p$, then
$$j_p^k \big(\sum_{i=1}^n \eta_i f_i\big) = j^k_p f\ .$$
But since this holds on the trivial bundle $\mathbb{R}^d\times \mathbb{R}^n$ it follows in the general case.

The next lemma contains bounds which are used for the construction of the analogue of the polynomial model. Assume that $M$ is Riemannian and $(E, \nabla, \langle \cdot , \cdot\rangle)$ is as in Section <ref> and recall for $k>l$ the map $Q^k_l:  J^k E\to (J^k E)_l$ from (<ref>).

Let $R^k$ be an admissible realisation of $J^k E$. For every compact set $K\subset M$ the bound

$$|Q^k_m j_q^k R(v_p)|_m\lesssim_K d_\fraks(p,q)^{(n-m)\vee 0}|v_p|_n$$

holds uniformly over $p,q\in K$,  $m,n\leq k$ and $v_p\in (J^k_pE)_n$.

Let $(U,\phi)$ be a chart around $p$ and $\Phi$ a compatible trivialisation of $\pi^{-1}(U)$. Note that it suffices to show the claim for $q\in U$, by continuity and compactness if follows for $q$ further away from $p$. Since $(\Phi^{-1})^*R(v_p) $ is a smooth function on $\phi(U)$ vanishing of order $n$ at $\phi(p)$, it follows that the coefficients in front of monomials of homogeneity $m$ of the abstract polynomial $j_{\phi(q)}^k (\Phi^{-1})^*R(v_p)$ vanish of order $n-m$. Since the map $\Phi^*$ is upper triangular, the same is true for $ j^k_q R ( v_p)$.

We call a family of admissible realisations $\{R^k\}_{k\in\mathbb{N}}$ for ${J^k E}$ consistent, if for each $k\geq l$ and each map $p_{k,l}:J^k E \to J^lE$ introduced at the beginning of Section <ref> one has the identity

$$R^l= R^k \circ p_{k,l} \ .$$

We call a map $R: JE\to \mathcal{C}^\infty(E)$ an admissible realisation of $JE$ if there exists a family of admissible realisations  $\{R^k\}_{k\in\mathbb{N}}$ of $J^k E$ such that for each $k\geq l$ one has the identity

\begin{equation}\label{admissible re}
R|_{(JE)_l} = R^k\circ p_{\infty,k}|_{(JE)_l}\ .
\end{equation}

Note that there is an obvious one to one correspondence between consistent families of  admissible realisations of $\{J^kE\}_k$ and admissible realisations of $JE$. In particular every consistent family of  admissible realisations gives rise to an admissible realisation of $JE$.

For any vector bundle $E\to M$ there exists a family of admissible realisations $\{R^k\}_k$ which is consistent and therefore there exists an admissible realisation of $JE$.

In view of the previous remark it suffices to show that there exists a consistent family of admissible realisations.
Note that the admissible realisations constructed for each $k\in \mathbb{N}$ in the proof of Proposition <ref> would form a consistent admissible realisation, if we could replace each $\theta_i^k$ by $\theta_i^\infty$ in equation (<ref>).
To this end, define for $V,W$ as in the proof of Proposition <ref> and any sequence $\{r_n\}_{n\in \mathbb{N}}$ such that $d(V,W^c)>r_n$ for all $n$ the following partial map

$$R_{V,W,\{r^n\}_{n\in \mathbb{N}}}: J^\infty(V\times \mathbb{R}^m) \nrightarrow \mathcal{C}^\infty(W)\otimes \mathbb{R}^m$$

which on $J (V\times \mathbb{R}^m)_n\subset V\times P(\mathbb{R}^m)$ agrees with $R_{V,W,r_n}$.

Now the point is, that depending on the chart maps and trivialisation $(U_i,\Phi_i)$ one can choose sequences $\{r_{n,i}\}_{n\in \mathbb{N}}$ such that no convergence issues arise when setting $k=\infty$ and replacing $R_{V_i,\phi_i(U_i),r_i}$ by $R_{V_i,\phi_i(U_i),\{r_{n,i}\}_{n\in \mathbb{N}}}$ in  (<ref>).

Note that the bounds obtained for admissible realisations of $J^k E$ in Lemma <ref> translate directly into bounds for admissible realisations $R$ of $JE$.

Let $R$ be an admissible realisation of $JE$.
For every $k\in \mathbb{N}$ and $K\subset M$ compact, there exists a constant $C_{k,K}$ such that for all $n,m\leq k$ the bound

\begin{equation}\label{norm on admissible }
|j_q R(v_p)|_m \lesssim C_{k,K} d_\fraks (p,q)^{(n-m)\vee 0} |v_p|
\end{equation}

holds uniformly over $p,q\in K$ and
$v_p\in (T_p E)_n$ .

We point out the following direct consequence of Lemma <ref> .

Let $R, \tilde R$ be two admissible realisations of $J E$, then for every compact set $K$ and $n\in \mathbb{N}$

$$|R(\tau_p)(q)-\tilde R(\tau_p) (q)|\lesssim_{R, \tilde R, K} |\tau | d_{\fraks}(p,q)^{k+n}$$

uniformly over $p,q \in K$, $\tau_p\in (J_p E)_{<k}$ .

§.§.§ Further conditions on admissible realisations

We point out that the notion of admissible realisation is rather general and depending on the context one might want to impose further conditions such as the following.

*  Recall, that $JM$ is has a natural algebra-bundle structure. Thus one might want impose that $R$ is an algebra-bundle morphism. Indeed, the existence of such admissible realisation is clear: since given any admissible realisation $\tilde{R}$ of $JM$ one can simply define $R$ fibre wise as the unique algebra morphism satisfying $R(\mathbf{1}_p)=1$ where $\mathbf{1}_p:=j_p 1 \in (JM_p)_0$ and $R|_{(JM)_1}= \tilde{R}_{(JM)_1}$.

*  Recall that $JE$ is naturally a $JM$ module, i.e there is a fibre wise multiplication. Suppose one is given an admissible realisation $R$ of $JM$ which is a bundle algebra morphism. Then, one could restrict attention to admissible realisations $\bar{R}$ of $JE$ which are compatible with this multiplicative structure, i.e. such that the following diagram commutes

\[ \begin{tikzcd}
JM\otimes JE  \arrow{r}{m} \arrow[swap]{d}{R\otimes \bar{R}} & J E \arrow{d}{\bar{R}} \\%
\mathcal{C}^\infty (M)\otimes \mathcal{C}^\infty(E) \arrow{r}{m} & \mathcal{C}^\infty (E)\ .
\end{tikzcd}
\]

*  More generally, given two vector bundles $E$ and $F$, recall that in general there is a canonical product

$$m: JE\otimes JF\to J(E\otimes F) \ ,$$

specified by the fact that for $e\in \mathcal{C}^\infty(E)$, $f\in \mathcal{C}^\infty(F)$ one has
$$Q_k j(e\otimes g) =\sum_{k=m+n}  m(Q_{n} je, Q_m jf)\ .$$
Similarly to above one might restrict attention to admissible realisations $R, R'$ and $R''$ such that the following diagram commutes

\[ \begin{tikzcd}
JE\otimes JF  \arrow{r}{m} \arrow[swap]{d}{R\otimes {R'}} & J (E\otimes F) \arrow{d}{{R''}} \\%
\mathcal{C}^\infty (E)\otimes \mathcal{C}^\infty(F) \arrow{r}{} & \mathcal{C}^\infty (E\otimes F)\ ,
\end{tikzcd}
\]

where the lower arrow denotes the pointwise tensor product.

Observe that for any vector bundles $E, F$, the canonical trace map $\tr: E\otimes E^* \otimes F\to F$ extends to a trace map
$\trr  : J(E\otimes E^* \otimes F)\to JF$ on jets, where the latter is characterised by the fact that for $g\in \mathcal{C}^\infty(E\otimes E^* \otimes F)$ one has $ \trr j g = j \tr g $ .

For a fixed manifold $M$, we give a universal construction of an admissible realisation for any vector bundle (with a connection) such that the above properties are always satisfied.
First, for a vector bundle $E\to M$, recall the isomorphism $JE\simeq JM\otimes E$.

* Fix a multiplicative admissible realisation $\tilde{R}$ as in Point <ref> of $JM$ with the property that $\supp(v_p)\subset B_{r_p}$ for all $v_p$ in $J_pM$ where $r_p$ denotes the exponential radius at $p$.

* For $e_p\in E|_p$ denote by $e_p(\cdot)$ the vector field obtained by parallel transport as in (<ref>).

* Define $R: JE\simeq  JM\otimes E \to \mathcal{C}^\infty (E)$ by

$$JM\times E\to \mathcal{C}^\infty (E), \qquad  (v_p, e_p)\mapsto R(v_p)(\cdot)e_p(\cdot)\ .$$

It is straightforward to check that the above properties are satisfied.

§ REGULARITY STRUCTURES AND MODELS ON VECTOR BUNDLES

As already seen in the introduction, when solving equations such as (<ref>), we need to work with several vector bundles simultaneously. In order to do so we first introduce some convenient notion.

§.§ Maps between families of spaces

For a family of vector spaces or vector bundles $V=\{V_i\}_{i\in I}$ we shall write $v\in V$ to mean $v\in \bigcup_i V_i$.
Given a second family $W= \{W_j\}_{j\in J}$ and a map $\sigma: I\to J$ we define
L_σ(V,W):=  ∏_i ∈I L(V_i,W_σ(i)) = ∏_i ∈I  W_σ(i)⊗V_i^*,   L̂_σ(V,W):= ∏_i ∈I  W_σ(i)V_i^* .

Given $A\in L_\sigma(V,W)$ and $v \in V$, we write $Av$ to denote $A_{i}v$, where $i\in I$ is the index such
that $v\in V_i$. (In particular $v \in V_i$ implies that $Av \in W_{\sigma(i)}$.)
For a third family $U=\{U_k\}_{k\in K}$ and $\sigma': K\to I$ and $B\in L_{\sigma'}(U,V)$ the composition
$AB\in L_{\sigma\circ \sigma'}(V,W)$ is defined by component-wise composition, i.e. $(AB)_{k}= A_{\sigma'(k)} \circ B_{k}$.

Furthermore, if the index sets agree, i.e. $I=J$ and $\sigma=\id$, we simply write $L(V,W)$ and $\hat{L} (V,W)$ instead of $L_{\id}(V,W)$ and $\hat{L}_{\id} (V,W)$.
We use a similar notation for spaces of sections, such as $\mathcal{D}(V)= \{ \mathcal{D} (V_i) \}_{i\in I}$ and $\mathcal{D}'(V)= \{ \mathcal{D}' (V_i) \}_{i\in I}$ for vector bundle valued test functions and distributions.

We shall sometimes also use the notation $L(V,W)$ where $V$ is a family of vector bundles, while $W$ is only a family of vector spaces. In this case we implicitly identify $W_j$ with the trivial bundle over one point and mean $L(V,W):= \hat{L}(V,W)$.

In this article, when working with the notions introduced above, we shall often work with finite families of finite dimensional vector bundles.
In this case one has the canonical isomorphisms

$$L_\sigma (V,W) \simeq \bigoplus_{i \in I}  W_{\sigma(i)}\otimes V_i^*, \quad \hat{L}_\sigma (V,W)\simeq \bigoplus_{i \in I}  W_{\sigma(i)}\hotimes V_i^*\ . $$

§.§ Regularity structure ensemble

We henceforth assume that the manifold $M$ is equipped with a Riemannian structure and any vector bundle $E\to M$ is equipped with a metric and compatible connection as in Section <ref>.

The appearance of several vector bundles in equations motivates the definition of a regularity structure ensemble.

Let $M$ be a smooth manifold, a regularity structure ensemble consists of an indexing set $\mathfrak{L}$ and a triple $\mathcal{T}=(\{{E}^\mathfrak{t}\}_{\mathfrak{t}\in \mathfrak{L}},\{{T}^\mathfrak{t}\}_{\mathfrak{t}\in \mathfrak{L}}, L)$ consisting of the following elements:

* For each $\frakt\in \mathfrak{L}$, ${E}^\mathfrak{t}$ is a finite dimensional vector bundle over $M$ equipped with a metric and a compatible connection.

*  For each $\frakt\in \mathfrak{L}$, $T^{\frakt}= \bigoplus_{(\alpha,\delta)\in A\times \triangle} T^{\frakt}_{\alpha,\delta}$ is a finite dimensional vector bundle over $M$ bi-graded by $A\subset \RR$ and $\triangle\subset \RR\cup \{+\infty\}$ which are bounded from below, discrete sets.

*  The bundle $T^{\frakt}_{0,:}:= \bigoplus_{\delta\in \triangle} T^{\frakt}_{0,\delta}$ comes with a canonical isomorphism $T^{\frakt}_0\simeq E^{\frakt}$.

* For every $\alpha\in A$ the bundle $T^\frakt_{\alpha,\infty}$ is trivial.

*  $L$ is a sub-bundle of $\hat{L}(T,T)$ such that for every $p,q,r\in M$ composition maps $L_{p,q}\times L_{q, r}$ to $L_{p,r}$.

Let us elaborate on the definition of a regularity structure ensemble:

* The spaces $T^\frakt$ and the $A$-grading play the same role as in [28]. A model, c.f. Definition <ref>, shall turn elements of $T^\frakt$ into $E^\frakt$-valued distributions. The second grading $\triangle$ plays the role of “precision”, c.f. again Definition <ref> and the definition of a model in [19]. In contrast to [19] we allow different values of $\delta$ for different elements of $T^\frakt$, this is natural when introducing products on regularity structures, c.f. Section <ref>, and when renormalising models, c.f. Section <ref>.

* The isomorphism Point (<ref>) is crucial when extending non-linear functions on the vector bundle $E^\frakt$ to functions on $T^\frakt$, c.f. Section <ref>. (Using the vocabulary introduced below, $T^\frakt_0$ will always be part of a “jet sector”.)

* The reason that the action of $L$ on $T^\frakt$ is in general not lower triangular in $A$ stems from the fact that when $\delta<+\infty$ an additional “higher order” term appears when lifting singular kernels, c.f. Section <ref> and in particular Remark <ref>.

In order to lighten notation, whenever we write $\tau_p\in T^\frakt_{\alpha,\delta}$ the subscript $p$ signifies that $\tau_p$ belongs to the fibre of $T^\frakt_{\alpha,\delta}$ above $p\in M$.

We set $T^{\frakt}_{\alpha,:}:=  \bigoplus_{\delta \in \triangle} T_{\alpha,\delta}^\frakt$ and $T_{:,\delta}^\frakt=\bigoplus_{\alpha\in A} T^\frakt_{\alpha,\delta}$.
For $\gamma\in \mathbb{R}$ we introduce
$T^{\frakt,\gamma}:=\bigoplus_{\alpha<\gamma< \delta} T^\frakt_{\alpha,\delta}\ .$
Lastly, for $\alpha\in A$ we define the canonical projection maps $Q_{\alpha}: T^\frakt\to T_{\alpha, :}^\frakt$.

Analogously to <cit.>, we can define what a sector of a regularity structure ensemble is.

Given a regularity structure ensemble $\mathcal{T}=(\{E^{\mathfrak{t}}\}_{t\in \mathfrak{L}},\{T^{\mathfrak{t}}\}_{t\in \mathfrak{L}}, L)$, a sector ensemble of regularity $\underline{\alpha}\in \mathbb{R}^{\mathfrak{L}}$ and precision $\delta\in \mathbb{R}\cup \{+\infty\}$ is a family $\{V^\frakt\}_{\frakt \in \mathfrak{L}}$ of bi-graded subbundles $V^\frakt=\bigoplus_{\alpha'\in A, \delta' \in \triangle} V^\frakt_{\alpha', \delta'}\subset T^\frakt$ with $V^\frakt_{\alpha', \delta'}\subset T^\frakt_{\alpha', \delta'}$ satisfying the following properties for each $\frakt \in \mathfrak{L}$

* $V^{\frakt}_{\alpha', \delta'} = M\times \{0\}\subset T^{\frakt}_{\alpha', \delta'}$ whenever $\alpha'<\underline{\alpha}_\mathfrak{t}$ or $\delta'<\delta$.

* For all $p,q\in M$ and all $\Gamma\in L_{p,q}$ one has $\Gamma_{p,q}V^{\frakt}_q\subset V^{\frakt}_p$ .

* For every $\alpha'\in A$, $\delta'\in \triangle$,

there exists a subbundle $\bar{V}^{\frakt}_{\alpha', \delta'}\subset T^{\frakt}_{\alpha', \delta'}$, such that

$$T^{\frakt}_{\alpha', \delta'} = V^{\frakt}_{\alpha', \delta'}\oplus  \bar{V}^{\frakt}_{\alpha', \delta'}\ .$$

As usual in the theory of regularity structures, we call the bundles $V^\frakt$ sectors of
regularity $\underline{\alpha}_\frakt$. If $\underline{\alpha}=0$, then the sector (or
sector ensemble) is called function-like.

We say that a regularity structure ensemble
$\mathcal{T}=(\{E^{\mathfrak{t}}\}_{t\in \mathfrak{L}},\{T^{\mathfrak{t}}\}_{t\in \mathfrak{L}}, L)$
is contained in a regularity structure ensemble
$\hat{\mathcal{T}}=(\{E^{\mathfrak{t}}\}_{t\in \hat{\mathfrak{L}}},\{\hat{T}^{\mathfrak{t}}\}_{t\in \hat{\mathfrak{L}}},\hat L)$

* one has $\mathfrak{L}\subset \hat{\mathfrak{L}}$ ,

* for each $\frakt\in \mathfrak{L}$ there is a graded injection $\iota: T^{\frakt} \to \hat T^{\frakt}$ ,

* the spaces $\iota T^\frakt\subset \hat{T}^\frakt$ are sectors and $\iota^{-1} \circ \hat L \circ \iota = L$.

Before we turn to the definition of a model, we introduce the following useful space of test functions. Let $V\to M$ be a vector bundle (equipped with the usual structure), for $r\in \mathbb{N}, \lambda\in (0,1), p\in M$ let

\begin{equation}\label{good textfunctions}
\mathcal{B}_p^{r,\lambda}(V) := \Big \{\phi \in \cC_c^\infty\big(V \big)\  \Big| \ \supp( \phi) \subset B_\lambda(p) \ , \
\| |j^r_\cdot \phi|_k \|_{L^\infty}<\frac{1}{\lambda^{|\fraks|+k}} \text{ for all } k\leq r\Big\}\ .
\end{equation}

We shall often write $\mathcal{B}_p^{r,\lambda}$ or $\mathcal{B}_p^{\lambda}$ instead of $\mathcal{B}_p^{r,\lambda}(V)$ if $r$ resp. $V$ is fixed.
Furthermore, instead of just writing $\phi\in\mathcal{B}_p^{r,\lambda}$, we shall sometimes use the notation $\phi_p^\lambda$ to emphasise $\lambda\in (0,1]$ and $p\in M$.
Lastly, we use the usual notational convention for families of vector bundles.

Instead of requiring bounds on the jet of a section, one could require bounds on the (covariant) derivatives in (<ref>). This is equivalent, as can be seen from the discussion around Proposition <ref>.

We point out the following useful property of the spaces $\mathcal{B}_p^{r,\lambda}$. For every $c\in (0,1)$ there exists $N\in \mathbb{N}$, such that the following property holds: For each $p$ there exist $N$ points $p_1,\ldots,p_N$, such that
each $\phi^\lambda_p \in \mathcal{B}_p^{r,\lambda}$ can be written as

$$\phi^\lambda_p= \sum_{i=1}^N \phi^{c\lambda}_{p_i}\ ,$$

where $\phi^{c\lambda}_{p_i}\in \mathcal{B}_{p_i}^{r,c\lambda}$. The points $p_i\in M$ can furthermore be chosen to satisfy $d_\fraks (p, p_i)\leq \lambda $.

Fix some constant $D>1$ and $K\subset M$ compact, observe that there exists a constant $C=C(D,K)$ such that uniformly over $\lambda <\frac{1}{D}$, $p\in K$ and $d(p,\bar{p})<D$ one has
$$\phi^\lambda_{\bar{p}} \in C\mathcal{B}_p^{r,D\lambda} \ .$$

Let $\{e_i\}_i$ be a frame of $\pi^{-1}(U)\subset V$. Then there exist $C, C'$ depending on this frame, such that

$$\phi^\lambda_p= \sum \phi^{\lambda,i}_p e_i  \in \mathcal{B}_p^{r,\lambda}(V)\quad \Rightarrow\quad  \phi^{\lambda,i}_p\in C\mathcal{B}_{p}^{r,\lambda}(M) \text{ for all } i$$

$$ \phi^{\lambda,i}_p\in \mathcal{B}_{p}^{r,\lambda}(M)\text{ for all } i\quad \Rightarrow\quad \phi^\lambda_p= \sum \phi^{\lambda,i}_p e_i  \in C'\mathcal{B}_p^{r,\lambda}(E) \ . $$

A model for a regularity structure ensemble $\mathcal{T}$ consists of a pair $Z = (\Pi, \Gamma)$, where

* $\Pi$ is an element of $L(T, \mathcal{D}'(E)\big)$ (in the notation of Remark <ref>),

* $\Gamma$ is a continuous section of $L$.

Furthermore, a model is required to satisfy the following analytic conditions for every $\frakt \in \mathfrak{L}$, compact subset $K\in M$ and for all $p,q\in K$, $\alpha \in A$, $\delta \in \triangle$ and $\tau_p\in (T_p)^\frakt_{\alpha,\delta}$:

* $|\Pi_p \tau_p (\phi^\lambda_p) | \lesssim_{K,\alpha} \lambda^\alpha |\tau_p|_\alpha$,

* $|\Gamma_{p,q} \tau_q |_{\beta} \lesssim_{K,\alpha} d_\fraks(p,q)^{(\alpha-\beta)\vee 0}|\tau_q|_\alpha$ for all $\beta\in A$ and $p,q\in K$,

* $|(\Pi_p \Gamma_{p,q} \tau_q - \Pi_q \tau_q) (\phi_p^\lambda) | \lesssim_{K,\alpha} \lambda^\delta |\tau_q|_\alpha$ for $p,q\in K$,

uniformly[Here it is understood that $\lambda^{+\infty}=0$ for all $\lambda\in (0,1 )$.] over $\lambda\in (0,1 )$ and $\phi_p^\lambda \in \mathcal{B}_p^{r,\lambda}(E^\frakt)$ for fixed $r>|{\min A}|$.
We denote by $\mathcal{M}_{\mathcal{T}}$ the space of all models for $\mathcal{T}$.

From now on, whenever we are working with a regularity structure or regularity structure ensemble we implicitly fix some integer $r>|{\min A}|$.

In practice models usually satisfy for $\tau_p \in T^\frakt_0$

$$\Pi_p \tau_p\in \mathcal{C}^\infty (E^\frakt), \qquad \Pi_p \tau_p (p)=\tau_p\ , $$
where the second equality makes use of the canonical isomorphism $ T^\frakt_0\simeq E^\frakt$ in Definition <ref>.

Given a regularity structure $\mathcal{T}=(E, T, L)$ such that the bundles
$T_{\alpha, \delta}$ are all trivial, one can reformulate the notions of a regularity structure and model in a way closer to the definition in [28].

* For each $\alpha\in A$, $\delta\in \triangle$, let $N_{\alpha,\delta}\in \mathbb{N}$ and $\{\tilde{\tau}^i_{\alpha,\delta}\}_{i\in\{1,\ldots,N_{_{\alpha,\delta}}\} }$ be a global frame of $T_{\alpha, \delta}$.
Define the vector spaces $\tilde{T}_{\alpha,\delta}=\text{span}\{  \tilde{\tau}^i_{\alpha,\delta} \ : i=1,\ldots, N_{\alpha,\delta}\ \}$ and

$$\tilde{T}= \bigoplus_{\alpha,\delta}\tilde{T}_{\alpha,\delta}\ .$$

* Then, let $\tilde{L}$ consist of all linear maps $\tilde{\Gamma}: \tilde{T}\to \tilde{T}$ such that there exist $p,q\in M$ and $\Gamma_{p,q}\in L$ such that for every $\tilde{\tau}\in \tilde{T}$ one has

\begin{equation}\label{eq:local rhs}
\big(\tilde{\Gamma} \tilde{\tau}\big)(p)= \Gamma_{p,q} (\tilde{\tau}(q)) \ .
\end{equation}

* Given a model $Z=(\Pi, \Gamma)$ for $\mathcal{T}$ this induces the maps

* $\tilde{\Pi}: M\to L(\tilde{T}, \mathcal{D}'(E))$ given by

\tilde{\Pi}_p \tilde{\tau}: = \Pi_p \left(\tilde{\tau}(p)\right)\ ,

* $\tilde{\Gamma}: M\times M\to \tilde{L}$ is the unique map such that each $p,q\in M$ and $\tilde{\tau}\in \tilde{T}$ the term $\left(\tilde{\Gamma}_{p,q}\tilde{\tau}\right)(p)$ agrees with the right-hand side of (<ref>).

The bundles $T^\frakt$ appearing when treating singular SPDEs will in general only be trivial if the jet bundles $JE^\frakt$ are trivial.

Whenever $M$ is a compact manifold, it is possible to artificially put oneself in the setting of Remark <ref> by enlarging $T_{\alpha,\delta}$ since every vector bundle over a compact manifold can be seen as a summand with respect to the direct sum of vector bundlesof a trivial vector bundle.

Next we introduce semi-norms on models.

Given a regularity structure ensemble  $\mathcal{T}$, we introduce the following “seminorms”
on models $Z = (\Pi, \Gamma)$:

* For $c>0$ and a compact set $K\subset M$, let

$$\| \Pi\|_{c,K}:= \sup_{\lambda<c}\sup_{\frakt\in \mathfrak{L}} \sup_{p\in K} \sup_{\alpha\in A}\sup_{\tau_p\in T^\frakt_{\alpha,:}}   \sup_{\phi\in \mathcal{B}^{r,\lambda}_p} \frac{|\Pi_p \tau_p (\phi) |}{\lambda^\alpha |\tau_p|} \ .$$

We also write $\| \Pi\|_{ K}: = \| \Pi\|_{ 1,K}$.

* Define
$$\|\Gamma\|_{K}:=\sup_{\frakt\in \mathfrak{L}}\sup_{p,q\in K: d_\fraks(p,q)<1} \sup_{\alpha,\beta\in A}\sup_{\tau_q\in T^\frakt_{\alpha,:}} \frac{|\Gamma_{p,q}\tau_q|_\beta}{d_\fraks(p,q)^{(\alpha-\beta)\vee 0}|\tau_q|} \ . $$

* Set
$$\llbracket Z \rrbracket_{ c,K}:= \sup_{\lambda<c}\sup_{\frakt\in \mathfrak{L}} \sup_{p,q\in K} \sup_{\alpha\in A} \sup_{\delta\in\triangle}\sup_{\tau_q\in T^\frakt_{\alpha,\delta}}  \sup_{\phi\in \mathcal{B}^{r,\lambda}_p} \frac{|(\Pi_p \Gamma_{p,q} \tau_q - \Pi_q \tau_q) (\phi) |}{\lambda^\delta |\tau_q|} $$

and, as before, $\llbracket Z  \rrbracket_{ K}:=\llbracket Z  \rrbracket_{ 1,K}$.

We  set $\|Z\|_{ c,K}:=\| \Pi\|_{ c,K}+\|\Gamma\|_{ K}+\llbracket Z \rrbracket_{ c,K}$.

Similarly, we define for a second model $\tilde{Z}=(\tilde{\Pi},\tilde{\Gamma})$

$$\llbracket Z ;\tilde{Z} \rrbracket_{ c,K}:= \sup_{\lambda<c}\sup_{p,q\in K} \sup_{\alpha\in A}\sup_{\delta \in \triangle}\sup_{\tau_q\in T_{\alpha,\delta}}  \sup_{\phi\in \mathcal{B}^{r,\lambda}_p} \frac{|(\Pi_p \Gamma_{p,q} \tau_q - \Pi_q \tau_q- \tilde{\Pi}_p \tilde{\Gamma}_{p,q} \tau_q + \tilde{\Pi}_q \tau_q) (\phi) |}{\lambda^\delta |\tau_q|} $$

and set $\|Z;\tilde{Z}\|_{ c,K}=\| \Pi-\tilde{\Pi}\|_{ c,K}+\|\Gamma-\tilde{\Gamma}\|_{ K}+\llbracket Z ;\tilde{Z} \rrbracket_{ c,K}$.

The following technical lemma will be useful in the sequel.

Let $\tilde Z= (\tilde{\Pi}, \tilde{\Gamma})$ be such that for each compact set $K\subset M$, there exists a constant $c_K$, such that properties (1), (2), and (3) of Definition <ref> are satisfied for $\lambda\leq c_K$, or equivalently, for each $K$ compact there exists $c_K>0$, such that $\|\tilde Z\|_{c_K,K}< \infty$. Then $\tilde Z$ is a model.
Furthermore for every $C>0$ the bounds

$$\|\tilde Z\|_{ 1,K}\lesssim_{C,c_{\bar{K}}} \|\tilde Z\|_{ c_{\bar{K}},\bar K}\;,\qquad
\|Z; \tilde Z\|_{ 1,K}\lesssim_{C,c_{\bar{K}}}  \|Z;\tilde Z\|_{ c_{\bar{K}},\bar K}\;,

hold uniformly over models $Z$, $\tilde Z$ satisfying $\|Z\|_{c_{\bar{K}},\bar K} \vee \|\tilde Z\|_{c_{\bar{K}},\bar K} < C$, where $\bar{K}$ is the 2-fattening of $K$.

We check the required bounds separately:

* Note that the bound on $\Gamma_{p,q}\tau$ holds by assumption.

* Next we check the bound on $\Pi$. We use Remark <ref> with $c=c_{\bar{K}}$ and decompose $\phi^\lambda_p \in \mathcal{B}^{r,\lambda}_{p}$ as $\phi^\lambda_p=\sum_{i=1}^N \phi^{c_{\bar{K}}\lambda}_{p_i}$. Thus we can write for $\tau_p \in T^\frakt_{\alpha,\delta}$, $p,q\in K$, $\lambda\in (0,1)$

\begin{align*}
|\Pi_p \tau_p (\phi_p^\lambda)| &\leq \sum_{i=1}^N |\Pi_p \tau_p (\phi^{c_{\bar{K}}\lambda}_{p_i})|\\
& \leq \sum_{i=1}^N |(\Pi_p \tau_p - \Pi_i \Gamma_{p_i, p}\tau_p)  (\phi^{c_{\bar{K}}\lambda}_{p_i}) |+| \Pi_{p_i} \Gamma_{p_i, p}\tau_p (\phi^{c_{\bar{K}}\lambda} _{p_i}) | \\
&\leq \llbracket Z \rrbracket_{ c_{\bar{K}},\bar K} |\tau_p| N \cdot (c_k\lambda)^\delta + \|\Pi\|_{ c_{\bar{K}},\bar K} \|\Gamma\|_{ \bar K} |\tau_p|\sum_i \sum_{\beta\in A} d_\fraks (p_i,p)^{(\alpha-\beta)\vee 0}(c_{\bar{K}}\lambda)^\beta\\
&\lesssim_{c_{\bar{K}}} (\llbracket Z \rrbracket_{ c_{\bar{K}},\bar K} + \|\Pi\|_{ c_{\bar{K}},\bar K} \|\Gamma\|_{ \bar K})|\tau_p|\lambda^{\alpha}
\end{align*}

where we used the fact that $d_\fraks (p_i,p) \leq \lambda$. Observe that the obtained bound is uniform over $p\in K$, $\alpha\in A $, $\tau_p\in T^\frakt_{\alpha,\delta}$ and $\lambda\in (0,1)$, $\phi_p^\lambda\in \mathcal{B}^{r,\lambda}_p$ .
$$\|\Pi\|_{K}\lesssim_{c_{\bar{K}}} \llbracket Z \rrbracket_{ c_{\bar{K}},\bar K} + \|\Pi\|_{ c_{\bar{K}},\bar K} \|\Gamma\|_{ \bar K} \ .$$

Similarly one finds for a second model $\tilde Z=(\tilde{\Pi},\tilde{\Gamma})$

$$\|\Pi-\tilde{\Pi}\|_{K}\lesssim\llbracket Z ;\tilde{Z} \rrbracket_{ c_{\bar{K}},\bar K} + \|\Pi-\tilde{\Pi}\|_{ c_{\bar{K}},\bar K} \|\Gamma\|_{ \bar K}+\|\tilde{\Pi}\|_{ c_{\bar{K}},\bar K} \|\Gamma-\tilde{\Gamma}\|_{ \bar K} \ .$$

* We proceed similarly to obtain the desired bound on $\Pi_p\tau_p-\Pi_q\Gamma_{q,p}\tau_p$. We treat only the case $\lambda\geq c_{\bar{K}}$, since for $\lambda< c_{\bar{K}}$ the desired bound holds by assumption. W.l.o.g assume $\tau_p\in T_{\alpha,\delta}$ for $\delta<+\infty$, then

\begin{align*}
(\Pi_q \tau_q -\Pi_p\Gamma_{p,q}\tau_q)(\phi_p^\lambda)& = \sum_{i=1}^N  (\Pi_q \tau_q -\Pi_p\Gamma_{p,q}\tau_q)(\phi_{p_i}^{c_{\bar{K}}\lambda})\\
&= \sum_{i=1}^N (\Pi_q \tau_q - \Pi_{p_i}\Gamma_{p_i, q}\tau_q)(\phi_{p_i}^{c_{\bar{K}}\lambda})\\
&+ \sum_{i=1}^N (\Pi_{p_i}\Gamma_{p_i, p}\Gamma_{p,q}\tau_q -\Pi_p\Gamma_{p,q}\tau_q )(\phi_{p_i}^{c_{\bar{K}}\lambda})\\
&+\sum_{i=1}^N (\Pi_{p_i}\Gamma_{p_i, q} \tau_q- \Pi_{p_i}\Gamma_{p_i, p}\Gamma_{p,q}\tau_q)(\phi_{p_i}^{c_{\bar{K}}\lambda})
\end{align*}

* The summands of the first sum can bounded as follows:

\begin{align*}
|(\Pi_q \tau_q - \Pi_{p_i}\Gamma_{p_i, q}\tau_q)(\phi_{p_i}^{c_{\bar{K}}\lambda})|&\leq \llbracket Z \rrbracket_{ c_{\bar{K}},\bar K} |\tau| (c_{\bar{K}}\lambda)^ \delta
\end{align*}

* For the summands in the second sum one has

\begin{align*}
|(\Pi_{p_i}\Gamma_{p_i, p}\Gamma_{p,q}\tau_q -\Pi_p\Gamma_{p,q}\tau_q )(\phi_{p_i}^{c_{\bar{K}}\lambda})|
&\leq \sum_{\beta\in A}
|(\Pi_{p_i}\Gamma_{p_i, p} Q_{\beta} \Gamma_{p,q}\tau_q -\Pi_p Q_{\beta} \Gamma_{p,q}\tau_q )(\phi_{p_i}^{c_{\bar{K}}\lambda})|\\
&\leq \llbracket Z  \rrbracket_{ c_{\bar{K}},\bar K} \sum_\beta |Q_{\beta} \Gamma_{p,q}\tau_q| (c_{\bar{K}}\lambda)^\delta\\
&\lesssim_K \llbracket Z  \rrbracket_{ c_{\bar{K}},\bar K} \|\Gamma\|_{  K} |\tau_q| (c_{\bar{K}}\lambda)^ \delta
\end{align*}

* For the summands in the last sum we use the following very crude bounds:

\begin{align*}
|\Pi_{p_i}\Gamma_{p_i, q} \tau_q (\phi_{p_i}^{c_{\bar{K}}\lambda})| &\leq \|\Pi\|_{ 1,K}  \sum_{\beta}|Q_\beta \Gamma_{p_i, q}\tau_q| (c_{\bar{K}}\lambda )^\beta\\
&\lesssim_K  \|\Pi\|_{ c_{\bar{K}},\bar K} \|\Gamma\|_{  K}|\tau_q| \lambda^{\min A}\\
&\lesssim_K \|\Pi\|_{ c_{\bar{K}},\bar K} \|\Gamma\|_{  K}|\tau_q| c_{\bar{K}}^{\min A -\delta} \lambda^\delta\\
&\lesssim_{K,{c_{\bar{K}}}}\|\Pi\|_{ c_{\bar{K}},\bar K} \|\Gamma\|_{  K}|\tau_q|\lambda^\delta
\end{align*}

where in the last line we used that $\lambda^{\min A}\leq c_{\bar{K}}^{\min A -\delta} \lambda^\delta$ for $\lambda\geq c_{\bar{K}}$.
Very similarly one obtains

\begin{align*}
|\Pi_{p_i}\Gamma_{p_i, p}\Gamma_{p,q}\tau_q (\phi_{p_i}^{c_{\bar{K}}\lambda}) |&\lesssim_{K,{c_{\bar{K}}}} \|\Pi\|_{ c_{\bar{K}},\bar K} \|\Gamma\|^2_{  \bar K}|\tau_q|\lambda^\delta
\end{align*}

for $\lambda\geq c_{\bar{K}}$.

Putting this together,

$$|(\Pi_q \tau_q -\Pi_p\Gamma_{p,q}\tau_q)(\phi_p^\lambda)|\lesssim_{ c_{\bar{K}},\bar K} (1+\|\Gamma\|_{  \bar K}) (\llbracket Z  \rrbracket_{ c_{\bar{K}},\bar K}  +\|\Pi\|_{ c_{\bar{K}},\bar K} \|\Gamma\|_{  \bar K} )    |\tau_q| \lambda^ \delta $$

holds uniformly over $p,q\in K$, $\tau_q\in (T_q)_{\alpha,\delta}^\frakt,$ and $\lambda\in (0,1)$, $\phi_p^\lambda\in \mathcal{B}^{r,\lambda}_p$ .

For the last claim, if $\tilde Z=(\tilde{\Pi},\tilde{\Gamma})$ is a second model, one similarly finds

\begin{align*}
\llbracket
Z;\tilde{Z}  \rrbracket_{{ c_{\bar{K}},\bar K}} \lesssim & \|\Gamma-\tilde{\Gamma}\|_{  \bar K} \left(\llbracket Z\rrbracket_{ c_{\bar{K}},\bar K}  +\|\Pi\|_{ c_{\bar{K}},\bar K} \|\Gamma\|_{  \bar K} \right)  \\
&+(1+\|\tilde\Gamma\|_{  \bar K}) \left(\llbracket Z ;\tilde{Z} \rrbracket_{ c_{\bar{K}},\bar K}  +\|\Pi-\tilde{\Pi}\|_{ c_{\bar{K}},\bar K} \|\Gamma\|_{  \bar K} +\|\tilde \Pi\|_{ c_{\bar{K}},\bar K} \|\Gamma-\tilde{\Gamma}\|_{  \bar K} \right)  \ .
\end{align*}

Next, we make the following useful observation, which holds in the flat setting as well, though is less important there.

Fix a regularity structure and a model. For $D>0$, denote by $K^D$, the $D$-enlargement of any set $K$. Then, for every compact set $K\subset M$ uniformly in
$p\in K$ and $\tilde{p}\in K$ satisfying $d(p,\tilde{p})<D\lambda$ as well as $\tau_p\in T_{\alpha,:}$ and $\phi^\lambda_{\tilde{p}}\in B^{r,\lambda}_{\tilde{p}}$

$$|\Pi_p \tau_p (\phi^\lambda_{\tilde{p}})| \lesssim_{D}\big(\|\Pi\|_{K^D}\|\Gamma\|_{K^D}+\llbracket Z  \rrbracket_{K^D } \big) |\tau_p| \lambda^{\alpha}$$

for $\lambda \in (0,1)$.
For any second model $\tilde{Z}=(\tilde{\Pi},\tilde{\Gamma})$ the analogous bound

$$|\big(\Pi_p-\tilde{\Pi}_p\big) \tau_p (\phi^\lambda_{\tilde{p}})| \lesssim_{D}\big(\|\Pi-\tilde\Pi\|_{K_{D+1}}\|\tilde\Gamma\|_{K_{D+1}}+\|\Pi\|_{K^D}\|\Gamma-\tilde{\Gamma}\|_{K^D }+\llbracket Z  ;\tilde Z  \rrbracket_{K^D } \big) |\tau_p| \lambda^{\alpha} \ ,$$

First assume $\lambda< \frac{1}{D}$, then

\begin{align*}
\Pi_p\tau_p(\phi^\lambda_{\tilde{p}})&=(\Pi_p\tau_p-\Pi_{\tilde{p}}\Gamma_{\tilde{p},p}\tau_p)(\phi^\lambda_{\tilde{p}}) + (\Pi_{\tilde{p}}\Gamma_{{\tilde{p}},p}\tau_p)(\phi_{\tilde{p}}^\lambda)\\
&=(\Pi_p\tau_p-\Pi_{\tilde{p}}\Gamma_{\tilde{p},p}\tau_p)(\phi^\lambda_{\tilde{p}}) + \sum_\beta(\Pi_{\tilde{p}}Q_\beta \Gamma_{{\tilde{p}},p}\tau_p)(\phi_{\tilde{p}}^\lambda)\\
&\lesssim \llbracket Z  \rrbracket_{K^D }|\tau_p|\lambda^ \delta + \|\Pi\|_{K^D }\sum_{\beta }|Q_\beta \Gamma_{{\tilde{p}},p}\tau_p|\lambda^{\beta}\\
&\lesssim \llbracket Z  \rrbracket_{K^D }|\tau_p|\lambda^ \delta + \|\Pi\|_{K^D }\|\Gamma\|_{K^D }|\tau_p|\sum_{\beta } (D\lambda)^{(\alpha-\beta)\vee 0}\lambda^{\beta} \ .
\end{align*}

A similar calculation works the second part of the claim for $\lambda< \frac{1}{D}$.
The general case follows along the lines of the proof of Lemma <ref>.

§.§.§ Jet regularity structures

We define jet regularity structures in the case $|\mathfrak{L}|=1$ and use the obvious adaptation for cases $|\mathfrak{L}|>1$.

For a vector bundle $E\to M$, an $E$-valued jet regularity structure $\bar{\mathcal{T}}:=(E,\bar T, L)$ of precision $\delta_0$ consists of a bi-graded finite dimensional subbundle $\bar{T}= \bigoplus_{\alpha\in A, \delta \in \triangle} \bar{T}_{\alpha,\delta} \subset JE$ where

* $A\subset \mathbb{N}$ and for every $\alpha\in A$ one has $\bar{T}_{\alpha,:}=\bigoplus_{\delta \in \triangle} \bar{T}_{\alpha,\delta}=(JE)_\alpha$, where the grading on $JE$ was introduced in (<ref>).

* The isomorphism $\bar{T}_{0}\simeq E$ is the canonical isomorphism $\bar{T}_{0}=(JE)_0 \simeq E$.

* $L$ consists of all maps satisfying the properties in Definition <ref> Point (<ref>).

An $E$-valued jet regularity structure is said to be of precision $\delta_0$ if it satisfies
$A= \mathbb{N}\cap [0, \delta_0)$ and $\triangle\subset [ \delta_0, + \infty ]$. It is called the standard $E$-valued jet regularity structure if $\triangle=\{\delta_0\} .$

Now we define the analogue of the polynomial model.

Let $\bar{\mathcal{T}}:=(E,\bar T, L)$ be an $E$-valued jet regularity structure. An admissible realisation $R$ of $JE$ is said to induce a model for $\bar{\mathcal{T}}$ if the maps $Z=(\Pi, \Gamma)$ defined as

* $\Pi_p v_p = R (v_p)$ for $v_p\in \bar{T}_{\alpha,\delta}$,

* $\Gamma_{q,p}v_p= {Q}_{<\delta} j_q R (v_p)$ for $ v_p \in \bar{T}_{\alpha,\delta}$,
where ${Q}_{<\delta}: JE\to (JE)_{<\delta}$ denotes the canonical projection defined in (<ref>) ,

satisfy the properties of a model. Then $Z$ is called a jet model for $\bar{\mathcal{T}}$ .

For any $\delta_0>0$, every admissible realisation $R$ of $JE$ induces a model for the standard jet regularity structure of precision $\delta_0$.

Set $k= \max\{n\in \mathbb{N}\ : \ n<\delta_0\}$ and define $(\Pi, \Gamma)$ as in Definition <ref>.
Recall the canonical identification $(JE)_0\simeq E$ induced by $Q_0 j_pf= f(p)$ for all $f\in \mathcal{C}^\infty (E)$ and
that $\Pi_p v_p (p)=0$ for $v_p\in (T_p)_{>0}$ since for all $\tau_p\in \bar T_p$

$$j_p \Pi_p \tau_p = j_p R(\tau_p) = \tau_p \ .$$

\begin{equation}\label{good for bounds bellow}
|\Pi_p\tau_p (q)|= |R (Q_{<k} j_q \Pi_p\tau_p)(q)| = |R (Q_{<k} j_q R \tau_p)(q)| =|Q_0 j_q R \tau_p | \
\end{equation}

\begin{equation}\label{good for bound below 2}
|\Gamma_{p,q} \tau_q |_m = |j_p R\tau_q |_m
\end{equation}

We check that the properties of a model are satisfied.

* Note that (<ref>) combined with Lemma <ref> implies that for $v_p\in (T_p)_n $

$$ |\Pi_p\tau_p (q)|\lesssim d_\fraks (p,q)^n \ .$$

* The same Lemma <ref>  combined with (<ref>)  implies for $v_p\in (T_p)_n $

\begin{equation}
|\Gamma_{p,q} \tau_q |_m = |Q_{\leq k}j_p R\tau_q |_m \lesssim d_\fraks (p,q)^{(n-m)\vee 0} \ .
\end{equation}

* To check the last condition, denote by $\bar K:= \{p\in M \ : \ d(p,K)\leq 1\}$ the $1$-fattening of $K$. Then, since

$$\Pi_p \Gamma_{p,q} \tau_q - \Pi_q \tau_q =  R Q_{\leq k} j_{p} R\tau_q - R \tau_q $$

it follows by taking  $f= R\tau_q$ and $g= R (Q_{\leq k} j_{p} f)$ in Lemma <ref> that

$$|(\Pi_p \Gamma_{p,q} \tau_q - \Pi_q \tau_q) (r) | \lesssim_{R,\delta} d_\fraks (p,r)^{\delta_0} |\tau_q|_\alpha \ .$$

When $E=\mathbb{R}^d \times \RR^n$ is the trivial bundle over $M=\RR^d$ one can identify jets with polynomials as in Section <ref>. Then the polynomial model in the sense of [28] induces a model for the any $E$ valued-jet regularity structure with $\triangle=\{+\infty\}$ (up to an appropriate identification of the different fibres of $\bar{T}$, c.f. Remark <ref>, and a truncation to make the space finite dimensional).

Note that the definition of “non-standard” jet regularity structures allows some flexibility on the analytic conditions one imposes on the corresponding model. These conditions are particularly interesting in the case that $\infty\in \triangle$ and $\bar{T}_\infty\neq \{0\}$.

* For example, when $E=M \times \RR$ is the trivial bundle one can always set $\bar{T}_{0,\infty}= (JM)_0$ and obtain a model for it by working with admissible realisations $R$ mapping $j_p 1\mapsto 1$ for every $p\in M$.

* More generally, if there exists a global parallel section $e$ of $E$, it it can be useful to restrict attention to admissible realisations satisfying
$Ej_p e = e$. An important example is the metric on some bundle $F$, which is a parallel section of $E=F^*\oplus F^*$.

Given an arbitrary jet regularity structure, in general it is a non-trivial geometric question whether there exists an admissible realisation inducing a model for it.

Note that in this section there was no mention of the algebra structure of jet bundles, c.f. Remark <ref>, which will be used again in Section <ref>. There it will be crucial that flexible notion of admissible realisations allows one to impose further conditions with respect to this structure, c.f. Section <ref>.

§.§ Operations on models

Let $E\to M$, $F\to N$ be vector bundles and $\Phi=(\phi, G): E\to F$ a bundle isomorphism. There is a natural notion of push-forward of a regularity structure  $\mathcal{T}_E=(E,T,L)$ on $M$ to a regularity structure $\Phi_*\mathcal{T}_E = (F,\phi_*T,\phi_*L)$ on $N$, where $\phi_*T,\phi_*L$ are the pullback-vector bundles under $\phi^{-1}$, resp. $\phi^{-1}\times \phi^{-1}$ and the isomorphism $(\phi_*T)_0\simeq F$ is given by composing the isomorphism $T_0\simeq E$ with $\Phi$.

The corresponding concept of push-forward of models

$$\mathcal{M}_{\mathcal{T}_E}\to\mathcal{M}_{\Phi_*\mathcal{T}_M}, \qquad Z \mapsto \Phi_*Z \ ,$$

c.f. <cit.> for the scalar case, is given as follows: Let $Z=(\Pi,\Gamma)$ be a model for $\mathcal{T}_E$, then for $\bar{p}, \bar{q}\in N$

(Φ_*Γ)_p̅,q̅   :=Γ_ϕ^-1(p̅),ϕ^-1(p̅)

(Φ_*Π)_q̅   := Φ_* ∘Π_ϕ^-1p̅  ,

where the last $\Phi_*$ denotes the push-forward of distributions $\Phi_*: \mathcal{D}'(E)\to \mathcal{D}'(F)$.

In the special case of the jet regularity structure, this simply corresponds to the pushforward of admissible realisations, see Remark <ref>.

Note that in the setting of Definition <ref>, for every compact set $K\subset N$, there exist $C=C(K,\Phi)>0$, $c=c(K,\Phi)\in (0,\infty)$ such that uniformly over $\bar p \in K$ and $\lambda\in (0,1)$

$$\mathcal{C}^\infty_c (F)\supset\Phi^*( \mathcal{B}^\lambda_{\bar{p}}(F) )\subset C\mathcal{B}^{c\lambda}_{p}(E) \subset \mathcal{C}^\infty_c (E)\ ,$$

where we write $p=\phi^{-1}(\bar{p})$.

Since our definitions differ slightly from the ones in [19], we state and prove the following lemma, which is close to <cit.>.

The pushforward map of Definition <ref> is continuous. For every compact set $K\subset N$ there exists a constant $c_K$, such that

\begin{align}
&\|\Phi_* \Pi \|_{K,c_K}\lesssim_{\phi} \| \Pi \|_{\phi^{-1}( K)}\label{first}\\
&\llbracket \Phi_* Z  \rrbracket _{K} \lesssim_{\phi} \llbracket Z  \rrbracket_{\phi^{-1}( K)}\label{second}\\
&\|\Phi_*\Gamma \|_{c_K,K} \lesssim_{\phi} \|\Gamma \|_{\phi^{-1}( K)} \label{third}
\end{align}

and the analogous bound holds for the difference of two models.
Therefore, by Lemma <ref>, the bound

$$\|\Phi_* Z \|_{K}\lesssim_{C,K} \|Z \|_{\phi^{-1}(\bar K)}$$

holds uniformly over models $Z$ satisfying $\|Z \|_{\phi^{-1}(\bar K)}<C$, as well as the similar bound for two models $Z, \bar{Z}$

$$\|\Phi_* Z;\Phi_* \bar{Z} \|_{K, \delta}\lesssim_{C,K} \|Z;\bar{Z} \|_{\phi^{-1}(\bar K)} \ .$$

Recall that by Remark <ref> there exist $C=C(K,\phi)>0$, $c=c(K,\phi)\in (0,\infty)$ such that uniformly over $\bar p \in K$ and $\lambda\in (0,1)$

$$\mathcal{C}^\infty_c (F)\supset\Phi^*( \mathcal{B}^\lambda_{\bar{p}}(F) )\subset C\mathcal{B}^{c\lambda}_{p}(E) \subset \mathcal{C}^\infty_c (E)\ ,$$

where we write $p=\phi^{-1}(\bar{p})$.
For $\tau_{\bar{p}}\in (\phi_*T)_{\bar p}$, denote by $\tau_p$ the corresponding element of $T_p$. We have

$$(\Phi_* \Pi)_{\bar p} \tau_{\bar p} \rho^\lambda_{\bar p} =\Pi_p \tau_p ( \Phi^*\rho_{\bar p}^\lambda) \ .$$

Similarly writing $q=\phi^{-1}(\bar{q})$, we have

$$\big((\Phi_* \Pi)_{\bar q} (\phi_*\Gamma_{\bar q, \bar p})\tau_{\bar{q}} - (\Phi_* \Pi)_{\bar p} \tau_{\bar{q}} \big)\rho^\lambda_{\bar q}=
(\Pi_{ q} \Gamma_{q, p}\tau_{q} -  \Pi_{ p} \tau_{q}) (\Phi^*\rho_{\bar q}^\lambda) \ .$$

Thus, if we set $c_K:= \frac{1}{c}\wedge 1$ the bounds (<ref>) and (<ref>) follow.  The bound (<ref>) follows directly from the Lipschitz continuity of $\phi^{-1}$.

In the sequel we will want to pushforward a regularity structure and model, using chart mapstrivialisations. To do so, we need to give a meaning to a model $Z$ on $M$ restricted to an open set $V$. Let $U\subset M$ be such that $V\Subset U\subset M$ and $\mathbb{1}_V\in C_c^\infty (U)$ such that $ \mathbb{1}_V|_V=1$.
Define $\mathcal{T}_M|_U:= (A, T|_U, L|_{U\times U})$. Then, we define

$$Z|_{V}:=(\tilde{\Pi},\tilde{\Gamma}) \ ,$$

$$\tilde{\Pi}_p\tau (\cdot):= \mathbb{1}_V(\cdot)\Pi_p \tau (\cdot)$$

$$\tilde \Gamma_{p,q}=  \Gamma_{p,q} \ .$$

We say that $\bar Z$ is a restriction of $Z$ to $V$ subordinate to $U$. Note that this definition does depend on the choice of $\mathbb{1}_V\in C_c^\infty (U)$.  It is clear how to pushforward such restricted models.
In particular it will be useful to pushforward restricted models using bundle trivialisations.

It follows from the definitions that restriction is a continuous operation, i.e. one has for each compact set $K\subset U$

$$\|\tilde Z\|_{K}\lesssim \|Z\|_{K} \ ,$$

where the implicit constant depends on the function $\mathbb{1}_V$ .

Given a regularity structure on $M$, assume $U\subset M$ is open. We call a model on $Z=(\Pi, \Gamma)$ on $U$ compactly supported, if for all $p\in U$ and $\tau_p\in T_p$, $\Pi_p\tau_p$ is compactly supported in $U$.
If $U\subset \RR^d$ is convex, then there exists  extension of the regularity structure and model $Z$ to a model on all of $\RR^d$. The first claim follows from the fact that any bundle over a convex base space is trivial and a corresponding model is easily constructed as well.

§ MODELLED DISTRIBUTIONS AND RECONSTRUCTION

Now we define modelled distributions analogously to [28].

Given a regularity structure ensemble $\mathcal{T}=(\{E^{\mathfrak{t}}\}_{t\in \mathfrak{L}},\{T^{\mathfrak{t}}\}_{t\in \mathfrak{L}}, L)$ and a model $Z = (\Pi, \Gamma)$, we define for $\frakt\in \mathfrak{L}$, $\gamma\in \mathbb{R}$ the space $\mathcal{D}_Z^\gamma(T^\frakt)$ to consist of all continuous sections $f$ of $T^{\frakt,\gamma}= \bigoplus_{\alpha<\gamma, \delta>\gamma } T^\frakt_{\alpha,\delta}$ satisfying for every compact set $K\subset M$

\begin{equation}\label{norm on modelled distr.}
\interleave f \interleave_{\gamma ; K}:=\| f\|_{\gamma ; K} +\sup_{\substack{p,q\in K,\\ d_\fraks(p,q)<1}} \sup_{\beta <\gamma} \frac{|f(q)-Q_{<\gamma}\Gamma_{q,p} f (p)|_{\beta}}{d_\fraks(p,q)^{\gamma-\beta}} <+\infty\ ,
\end{equation}

$$\| f\|_{\gamma ; K} := \sup_{p\in K} \sup_{\alpha<\gamma} |f(p)|_\alpha \ .$$

For a second model $\tilde Z = (\tilde \Pi, \tilde{\Gamma})$ and $\tilde{f}\in \mathcal{D}_{\tilde{Z}}^\gamma(T^\frakt)$, we define

$$ \interleave f ; \tilde{f} \interleave_{\gamma ; K} := \| f-\bar{f}\|_{\gamma ; K}+\sup_{\substack{p,q\in K,\\ d_\fraks(p,q)<1}} \sup_{\beta <\gamma} \frac{|f(q)-Q_{<\gamma}\Gamma_{q,p} f(p)-(\tilde f (q)-Q_{<\gamma}\tilde{\Gamma}_{q,p}\tilde{f}(p))|_{\beta}}{d_\fraks(p,q)^{\gamma-\beta}} \ .$$

Note that in the setting of Definition <ref>, one has the obvious notion of the push-forward of a modelled distributions

$$\mathcal{D}_Z^\gamma(T^\frakt)\to \mathcal{D}_{(\Phi_*Z)}^\gamma(\Phi_* T^\frakt),\qquad f(\cdot) \mapsto \Phi_*f(\cdot)\ . $$

It is clear that this map is continuous.

Given a regularity structure ensemble $\mathcal{T}=(\{E^{\mathfrak{t}}\}_{t\in \mathfrak{L}},\{T^{\mathfrak{t}}\}_{t\in \mathfrak{L}}, L)$ and a model $Z = (\Pi, \Gamma)$, there exists for every $\gamma\in (0, +\infty)$ a unique operator
$\mathcal{R}_Z\in L( \mathcal{D}_Z^\gamma, \mathcal{D}'(E))$, called reconstruction operator associated to $Z$, such that

\begin{equation}\label{reconstruction cond}
\sup_{p\in K} \sup_{\lambda<1} \sup_{\phi_p^\lambda\in \mathcal{B}^{r,\lambda}_p} \frac{|\langle \mathcal{R}_Z f-\Pi_p f(p),\phi_p^\lambda
\rangle|}{\lambda^\gamma}  \lesssim_{c,K} \interleave f\interleave_{\gamma ; K} \|Z\|_{\bar K}
\end{equation}

uniformly over all $\frakt\in \mathfrak{L}$, $f\in \mathcal{D}^\gamma_Z(T^\frakt)$ and $\phi^\lambda_p \in \mathcal{B}_p^{r,\lambda}(E^\frakt)$ as well as all models $Z=(\Pi,\Gamma)$ satisfying $\|Z\|_{\bar{K}}\leq c$.
Furthermore, the map $Z\mapsto \mathcal{R}_Z$ is continuous and if we denote by $\bar{Z}=(\bar{\Pi}, \bar{\Gamma})$ a second model for $\mathcal{T}$, $\mathcal{R}_{\bar{Z}}$ the associated reconstruction operator, and $\bar{\mathcal{D}}^{{\gamma}}_{\tilde{Z}}$ the corresponding space of modelled distributions, then the inequality

\begin{align}
\sup_{p\in K} \sup_{\lambda<1} \sup_{\eta\in \mathcal{B}_p^{r,\lambda}} &\frac{|\langle \mathcal{R}_Z f-\Pi_p \bar{f}(p)-\mathcal{R}_{\bar{Z}}\bar{f}-\bar{\Pi}_p f(p),\eta^\lambda\rangle|}{\lambda^\gamma}  \nonumber\\
\lesssim_{c,K} & \interleave f,\bar{f} \interleave_{\gamma ; K}  \|Z\|_{\bar{K}}
+\interleave \bar{f}\interleave_{\gamma ; K}  \|Z;\bar{Z}\|_{\bar{K}}
\end{align}

holds uniformly over models $Z,\bar{Z}\in\mathcal{M}_\mathcal{T}$, satisfying $\|\tilde Z\|_{\bar{K}},\|\tilde Z\|_{\bar{K}} \leq c$ and ${f}\in \mathcal{D}^{{\gamma}}_Z,\  \bar{f}\in \bar{\mathcal{D}}^{\gamma}_{\bar{Z}}$.

We divide the proof into several steps.
First, we argue that it suffices to show the bound for $\lambda$ small enough, instead of $\lambda\leq 1$, i.e. the following claim.

Assume that there exists an operator $\mathcal{R}_Z: \mathcal{D}_Z^\gamma\to \mathcal{D}'(E)$, such that for every $K$ there exists $c_K$ for which

\begin{equation}\label{easier to prove reconstruction cond}
\sup_{p\in K} \sup_{\lambda<c_K} \sup_{\eta^{\lambda}_{p}\in \mathcal{B}^{r,\lambda}_p} \frac{|\langle \mathcal{R}_Z f-\Pi_p f(p),\eta_p^\lambda
\rangle|}{\lambda^\gamma}  \lesssim_{c,K} \interleave f\interleave_{\gamma ; \bar K} \|Z\|_{ \bar K}
\end{equation}

Then $\mathcal{R}_Z f$ satisfies (<ref>).

We argue similarly to Lemma <ref> and use Remark <ref> to write

\begin{align*}
\langle\mathcal{R}_Z f-\Pi_p f(p),\eta_p^\lambda \rangle &= \sum_{i=1}^N \Big(\mathcal{R}_Z f-\Pi_p f(p)\Big) (\eta_{p_i}^{c_{\bar K}\lambda})\\
&= \sum_i \big(\mathcal{R}_Z f-\Pi_{p_i} f(p_i)\big) (\eta_{p_i}^{c_{\bar K}\lambda})\\
&+\sum_i \big(\Pi_{p_i} f(p_i) - \Pi_{p_i}\Gamma_{p_i,p} f(p)\big)(\eta_{p_i}^{c_{\bar K}\lambda})\\
&+\sum_i \big(\Pi_{p_i}\Gamma_{p_i,p} f(p)-\Pi_p f(p)\big) (\eta_{p_i}^{c_{\bar K}\lambda})
\end{align*}

* For summands of the first type, we can use the assumption to obtain

$$\Big|\big(\mathcal{R}_Z f-\Pi_{p_i} f(p_i)\big) (\eta_{p_i}^{c_{\bar K}\lambda})\Big|\lesssim_{c,K}\lambda^\gamma \interleave f \interleave_{\gamma ; \bar K}\|Z\|_{\bar K} \ .$$

* For the terms in the second sum one has

$$\Big|\big(\Pi_{p_i} f(p_i) - \Pi_{p_i}\Gamma_{p_i,p} f(p)\big)(\eta_{p_i}^{c_{\bar K}\lambda})\Big|\leq \lambda^\gamma \|\Pi\|_{ \bar{K}} \interleave f\interleave_{\gamma ; \bar K} \ .$$

* The summands in the last line can be bounded by

$$\Big | \big(\Pi_{p_i}\Gamma_{p_i,p} -\Pi_p \big)f(p) (\eta_{p_i}^{c_{\bar K}\lambda}) \Big |\leq \lambda^{\gamma} \llbracket Z  \rrbracket_{\bar K} \interleave f\interleave_{\gamma ; \bar K} .$$

* The rest of the proof essentially adapts from [19]. The argument goes as follows.

*  First recall the theorem on the trivial bundle $\mathbb{R}^d\times \mathbb{R}^m$, c.f. [28], [1], <cit.>, [3] or [4], all proofs adapt easily to our slightly more general notion of models.

Next prove reconstruction for models compactly supported on the domain $U\subset M$ of a trivialisation $\Phi: \pi^{-1}(U)\to \mathbb{R}^d\times \RR^m$. This can be done by push-forward:
Assume $Z=(\Pi,\Gamma)$ is such a model, and denote by $\bar Z= (\bar \Pi, \bar \Gamma ):= \Phi_*Z$ the model pushed forward to $\mathbb{R}^d$. Let $f\in \mathcal{D}^\gamma_{Z}$ and denote by $\bar f \in \mathcal{D}^\gamma_{\bar Z} $ the pushed forward modelled distribution. Set

$$\mathcal{R}_Z f := (\Phi^{-1})_* \mathcal{R}_{\bar Z} \bar f \ .$$

We check that this satisfies (<ref>). Let $x= \phi (p)$ , then

$$\big(\mathcal{R}_Z-\Pi_p f(p)\big)(\phi_p^\lambda)= \big(\mathcal{R}_{\bar Z} \bar f - \bar{\Pi}_x \bar f(x) \big) ((\Phi^{-1})^*\phi^\lambda_p) \ .$$

Thus the claim follows by combining the previous step (<ref>) with Remark <ref> and the bounds on the pushforward of models and modelled distributions in Lemma <ref> and Remark <ref>.

* Fix a locally finite partition $\mathbb{1}_i$ of unity subordinate to trivialisations $\Phi_i$. For a general model $Z=(\Pi,\Gamma)$, define the compactly supported models
$Z_i:=(\mathbb{1}_i(\cdot)\Pi, \Gamma )$
and set

$$\mathcal{R}_Z:=\sum_i \mathcal{R}_{Z_i} \ .$$

Observe, that $\mathcal{D}^\gamma_Z\subset \mathcal{D}^\gamma_{Z_i} $.
Then the estimate (<ref>) follows from the identity

$$\mathcal{R}_Zf -\Pi_pf(p)= \sum_i \big(\mathcal{R}_{Z_i}f - \mathbb{1}_i(\cdot) \Pi_pf(p)\big) $$

combined with the previous step.

By a straightforward adaptation of the proof in [28] and by going to $\fraks$-charts on $M$ one finds the following.

Under the assumptions of Theorem <ref> one has the bound

$$ |(\mathcal{R}f -\Pi_p f(p) )(\phi^\lambda_p)|\lesssim \lambda^\gamma \sup_{p,q\in B_{\lambda}(p)} \sup_{\alpha<\gamma} \frac{|f(p)-\Gamma_{p,q}f(q)|_\alpha}{d_\fraks(p,q)^{\gamma-\alpha}} \ ,$$

where the implicit constant is of order $1+\|Z\|_{B_p(2\lambda)}$ . Furthermore, the analogue bound holds for the difference of two models.

Suppose, in the setting of Theorem <ref>, the model $Z=(\Pi,\Gamma)$ has the property that that there exists an $\epsilon>0$ such that for every $\tau_p\in T^{\frakt}$
$\Pi_p \tau_p \in \mathcal{C}^\epsilon (E^\frakt)$, then for every $f\in \mathcal{D}^\gamma_Z$ one has the identity

$$\mathcal{R}_Z f (p)= \left(\Pi_p f(p)\right) (p)\ .$$

§.§ More on jet models

The following observation is trivial for polynomials in the flat setting, but for admissible realisations on manifolds worth pointing out.

Let $R$ be an admissible realisation of $JE$. Then, for every compact set $K$ and $k\in \mathbb{N}$, there exist constants $c,C>0$, such that uniformly over $p\in K$, $\tau_p\in (JE)_{<k}$ and $\lambda\in (0,1)$

$$c \sum_{l=1}^k \lambda^l |Q_l \tau_p | \leq \sup_{\phi^\lambda_p \in \mathcal{B}^\lambda_p } |\langle R\tau_p, \phi^\lambda_p\rangle | \leq C\sum_{l=1}^k \lambda^l |Q_l \tau_p |  $$

First we proof the lemma in the scalar valued case.
The upper bound follows directly from the fact that the jet model is a model. The lower bound can be obtained as follows:

* It follows from Remark <ref>, that if $(\phi^\lambda_p)_{\lambda\in (0,1), p\in M}$ is a family of Dirac sequences, such that for some $c>0$,  $c\phi_p^\lambda\in \mathcal{B}^{r+k,\lambda}_p$ and we denote by $\partial_i$ the coordinate vector fields of a $\fraks$-exponential chart at $p$, then

\begin{equation}\label{convergence}
|Q_n \tau_p|\leq \max_{|l|_\fraks=n} \lim_{\lambda\to 0} |\langle R\tau_p, (\partial^l)^* \phi^\lambda_p\rangle |\ .
\end{equation}

Fix $(\phi^\lambda_p)_{\lambda\in \mathbb{N}, p\in M}$ such that for $\lambda\to 0$,
$\phi^\lambda_p\to \delta_p$ converges uniformly in $p$ over compact sets (in some distribution space), then the same also holds for $(\partial^l)^* \phi^\lambda_p\to (\partial^l)^*\delta_p$ . Therefore the limit in (<ref>) can  be taken to be uniform over $p$ in compacts.

* Thus by the previous step, for every $K$ compact, there exists $\lambda_K>0$, such that for all $\lambda \leq \lambda_{K,k}$, $p\in K$

\begin{align*}
\sum_{n\leq k} \lambda^n  |Q_n \tau_p| &
< 2\sum_{n\leq k} \lambda^n \max_{|l|_\fraks=n}  |\langle R\tau_p, \partial^l \phi^\lambda_p\rangle |\\
&\lesssim_k \sup_{n}\big( \lambda^n \max_{|l|_\fraks=n} |\langle R\tau_p, \partial^l \phi^\lambda_p\rangle |\big)\\
&=\max_{\psi^\lambda\in \tilde{\mathcal{B}}^\lambda_{p}} |\langle R\tau_p, \psi^\lambda_p\rangle | \ ,
\end{align*}

where we set $\tilde{\mathcal{B}}^\lambda_{p}= \big\{ \lambda^{|l|_\fraks} \partial^l \phi^\lambda_p \ | \ |l_\fraks|\leq k\big\}\subset\mathcal{ B}^{r,\lambda}_p.$
To see the first inequality one first observe it for the case $\tau_p\in T_n$.

* The claim for the case $\lambda=1$ is straight forward, thus we conclude the general case by interpolation.

This completes the proof in the scalar valued case. The bundle valued case follows by combining the lemma in this case with Remark <ref> and Remark <ref>.

The following is a direct application of Lemma <ref> .

Let $\delta\in (0,+\infty)$ and $\bar{\mathcal{T}}:=(E, \bar T, L)$ be a jet regularity structure such that $\triangle \subset (\delta,+\infty]$ and $Z=(\Pi,\Gamma)$ a jet model. Let $\gamma\notin \NN$ s.t. $\gamma<\delta$ and suppose $f$ is a continuous section of $\bar T_{<\gamma,;}$ .

* If $f$ satisfies for every $K$ compact a bound of the form

$$|\big(\Pi_p f(p) - \Pi_p \Gamma_{p, q} f(q) \big)(\phi^\lambda_p) |\lesssim_K \lambda^{\gamma}$$
uniformly over $p,q\in K$ and $\lambda\in [2^{-1}d_\fraks(p,q), d_\fraks(p,q)]$, then the following bound holds

$$|f(p)-\Gamma_{p, q} f(q)|_m\lesssim_K d_\fraks(p,q)^{\gamma-m} \ $$

for $m<\gamma$
and thus $f\in \mathcal{D}^\gamma_Z$ .

* If $f \in \mathcal{D}^\gamma_Z(\bar{T})$ satisfies uniformly over $K$ compact and $p\in K$ a bound of the form

$$|\Pi_p f(p) (\phi_p^\lambda)|\lesssim \lambda^\gamma$$

then $f=0$.

§.§ Hölder functions

A function $f: \RR^d\mapsto \mathbb{R}$ is said to belong to the Hölder space[We point out that for integer $\alpha\in \mathbb{N}$ this space is different from the usual notion of $C^\alpha$ spaces.] $\mathcal{C}^\alpha_{\mathfrak{s}}(\RR^d)$ for some $\alpha>0$, if for every compact set $K\subset \RR$ the following semi-norm is finite

$$\| f\|_{{c}^\alpha_K} := \sup_{x\in K} \inf_{P\in \mathcal{P}^{\lfloor \alpha \rfloor} } \sup_{y\in K : |x-y|_\fraks \leq 1} \frac{|f(y)-\Pi_x P(y)|}{|x-y|_\fraks^{\alpha}} \ ,$$

where $\mathcal{P}^{\lfloor \alpha \rfloor}$ is as in Section <ref> and $\Pi_x: \mathcal{P}^{\lfloor \alpha \rfloor}\to \mathcal{C}^\infty(\RR)$ is the linear map specified by $\Pi_x X^k(y) =  (y-x)^k$.  It is a classical fact that for $f\in \mathcal{C}^\alpha_{\mathfrak{s}}$, there exists for each $x\in \RR$ a unique $P_x(f)\in \mathcal{P}^{\lfloor \alpha \rfloor}$ such that

$$\sup_{y : |x-y|_\fraks \leq 1} \frac{|f(y)-\Pi_xP_x(f) (y)|}{|x-y|_\fraks^{\alpha}} <+\infty \ .$$

Fix a norm $| \cdot |$ on the finite dimensional vector space $\mathcal{P}^{\lfloor \alpha \rfloor}$ and define

$$\| f\|_{\mathcal{C}^\alpha(K)}:= \sup_{x\in K} | P_x(f) |  + \| f\|_{{c}^\alpha_K} \ .$$

(Note that $\| f\|_{\mathcal{C}^\alpha_K}$ is a norm on functions in ${\mathcal{C}_\mathfrak{s}^\alpha}$ which are supported in $K$. )
We extend the definition of $\mathcal{C}^\alpha_{\mathfrak{s}}$ to manifolds, resp. vector bundles in the usual manner.

Let $E\to M$ be an $n$-dimensional vector bundle. Fix a locally finite partition of unity $\{\mathbb{1}_i\}_{i\in \NN}$ of $M$ subordinate to an atlas of $\fraks$-charts $\{(U_i,\phi_i)\}_{i\in \mathbb{N}}$ and let $\{\Phi_i\}_{i\in \mathbb{N}}$ be subordinate trivialisations, i.e. $\Phi_i= (G_i,\phi_i)$.
We say that $f\in \mathcal{C}^\alpha (E)$, if for any compact set $K \subset M$

$$\|f\|_{\mathcal{C}^\alpha_{\mathfrak{s}}(K)}:=\sum_{j=1}^n \sum_{i\in \mathbb{N}:  \supp \mathbb{1}_i\cap K \neq \emptyset} \left\|\big((\Phi_i^{-1})^* f_i\big)_j\right\|_{\mathcal{C}^\alpha_{\mathfrak{s}}(\phi^{-1}(K\cap \supp \mathbb{1}_i) )} <\infty \ , $$

where $f_i:= \mathbb{1}_i \cdot f$.

It is straightforward to see that the semi-norms obtained by using a different partition of unity and local trivialisations are equivalent.

Fix $\alpha \in \mathbb{R}_+\setminus \mathbb{N}$. For a continuous section $f\in \mathcal{C}(E)$ the following properties are equivalent:

*  $f\in \mathcal{C}^\alpha_{\mathfrak{s}}(E)$

*  There exists a unique continuous section $\{F_p\}_{p\in M}$ of $(JE)_{<\alpha}$, such that for any admissible realisation $R$

\begin{equation}\label{equation jet characterisation}
\sup_{p\in K} \inf_{F_p\in (J_p E)_{<\alpha} } \sup_{q\in M : d_\fraks(p,q) \leq 1} \frac{|f(q)-RF_p(q)|}{d_\fraks(p,q)^{\alpha}}<\infty
\end{equation}

for every compact set $K\subset M$.

*  There exists a unique section $F$ of $JE_{<\alpha}$ such that for any jet regularity structure $\bar{\mathcal{T}}=(E,\bar{T},L)$ such that $\triangle\subset(\alpha, +\infty]$ and $JE_{<\alpha}\subset \bar{T}$ one has $F\in \mathcal{D}_Z^\alpha(\bar{T})$ for any jet model $Z$ and furthermore

$$\mathcal{R}_ZF=f\ .$$

One easily sees that  (<ref>) and (<ref>) are equivalent.
To show (<ref>) implies (<ref>) we check that $F$ as defined in (<ref>) is a modelled distribution and $\mathcal{R}F=f$ .
Let $p,q\in M$ and $\lambda\in [2^{-1}d_\fraks(p,q), 2 d_\fraks(p,q)]$, since

\begin{align*}
\Pi_p(F(p)- \Gamma_{p,q}F(q))(\phi^\lambda_p)=& (\Pi_pF(p)-f)(\phi^\lambda_p)+  (f-\Pi_q F(q))((\phi^\lambda_p))\\
(\Pi_q F(q)- \Gamma_{p,q}F(q))(\phi^\lambda_p)\ ,
\end{align*}

one obtains

$$|\Pi_p(F(p)- \Gamma_{p,q}F(q))(\phi^\lambda_p)|\lesssim \lambda^{\alpha}$$

by using (<ref>) for the first term, (<ref>) combined with Remark <ref> for the second term and the bound in the definition of a model for the third term.
Thus the claim follows from Lemma <ref>.

Clearly (<ref>) implies (<ref>) by the reconstruction theorem, Theorem <ref>.

Let $E\to M$ be a vector bundle over a compact manifold $M$ and $\alpha\notin \NN$ as well as $\bar{\mathcal{T}}=(E,\bar{T},L)$ be a jet regularity structure such that $\triangle\subset(\alpha, +\infty]$ and $JE_{<\alpha}\subset \bar{T}$. It follows from the previous proposition that the reconstruction operator $\mathcal{R}_Z: \mathcal{D}^\alpha \to \mathcal{C}^\alpha$ is a bijective bounded map.
Therefore by the open mapping theorem of functional analysis it has a bounded inverse. It follows from the proof of Lemma <ref> that this inverse is explicitly given by the map

$$\mathcal{C}^\alpha\to \mathcal{D}^\alpha, \quad f\mapsto (i_{[\alpha],\infty}\circ j^{[\alpha]}) f \ .$$

§ LOCAL OPERATIONS ON MODELLED DISTRIBUTIONS

In this section we introduce linear and non-linear operations on modelled distributions.

§.§ Products and traces

Given a regularity structure ensemble $\mathcal{T}=(\{E^\mathfrak{t}\}_{\mathfrak{t}\in \mathfrak{L}},\{T^\mathfrak{t}\}_{\mathfrak{t}\in \mathfrak{L}}, L )$, $\mathfrak{t}_3\in \mathfrak{L}$ and two sectors $V^{\mathfrak{t}_1}\subset T^{{\mathfrak{t}_1}}, {V}^{\mathfrak{t}_2}\subset T^{\mathfrak{t}_2}$ of regularity $\underline{\alpha}_{1}$ resp. $\underline{\alpha}_{2}$,
a bilinear bundle morphism ${\star: V^{\mathfrak{t}_1}\times V^{\mathfrak{t}_2}\to T^{\mathfrak{t}_3}}$ is called an abstract  (tensor) product if the following properties are satisfied.

*  $E^{\frakt_1} \otimes E^{\frakt_2}=E^{\frakt_3}$ .

* For any $\tau_p^1 \in V^{\mathfrak{t}_1}_{\alpha}$, $\tau_p^2\in V^{\mathfrak{t}_2}_{\alpha'}$ ones has $\tau_p^1\star  \tau_p^2\in T^{\mathfrak{t}_3}_{\alpha+\alpha'}$ .

The product $\star$ is called $\gamma$-precise, if furthermore the sectors $V^{\mathfrak{t}_1}_{\alpha}$, $V^{\mathfrak{t}_2}_{\alpha'}$ are $\gamma$-precise and for every $\tau_p^1 \in V^{\mathfrak{t}_1}_{\alpha}$, $\tau_p^2\in V^{\mathfrak{t}_2}_{\alpha'}$ such that $\alpha+\alpha'<\gamma$ one has

* $\tau_p^1\star  \tau_p^2\in T^{\mathfrak{t}_3}_{\alpha+\alpha', >\gamma}\ $,

for all $q\in M$ and $\Gamma_{q,p}\in L_{q,p}$
$$Q_{<\gamma,:}(\Gamma_{q,p}\tau_p^1\star \Gamma_{q,p}\tau_p^2)= Q_{<\gamma,:}\Gamma_{q,p}(\tau_p^1\star \tau_p^2)\ .$$

Natural extensions of this definition are given as follows.

* Products between more than two sectors $V^{\mathfrak{t}_1}\times\ldots\times V^{\mathfrak{t}_n}$ are defined in the analogous manner.

* When $E^\frakt_1=E^\frakt_2$ above, one defines symmetric tensor products by replacing the tensor product in (<ref>) by a symmetric tensor product and further enforcing the product $\star$ be commutative.

We shall often use the notation $\tau_p^1\star_\gamma  \tau_p^2:=Q_{<\gamma} (\tau_p^1\star  \tau_p^2)$.

This definition is analogous to the definition of a product in [28] except for the following differences. On the one hand we need to keep track of the vector bundles, which is straightforward and is done using sector ensembles, secondly, we need to introduce the notion of the product being $\gamma$-precise.[In [28] the notion of being $\gamma$-regular is introduced. We prefer to to use different nomenclature, since this notion though similar, is distinct from being $\gamma$-precise.]
At first glance it would seem natural to impose that $\star$ maps $V^{\mathfrak{t}_1}_{\alpha,\delta}\times V^{\mathfrak{t}_2}_{\alpha',\delta'}$ into $T^{\mathfrak{t}_3}_{\alpha+\alpha', \bar{\delta}}$, where ${\bar{\delta}=\big(\delta+(\underline{\alpha}_{2}\wedge 0)\big)\wedge\big( \delta'+(\underline{\alpha}_{1}\wedge 0)\big)}$. But the additional freedom gained by not enforcing this choice is crucial when discussing aspects of renormalisation, c.f. Section <ref>.

Given a regularity structure ensemble $\mathcal{T}=(\{E^\mathfrak{t}\}_{\mathfrak{t}\in \mathfrak{L}},\{T^\mathfrak{t}\}_{\mathfrak{t}\in \mathfrak{L}}, L )$ and two sectors $V^{\mathfrak{t}_1}\subset T^{{\mathfrak{t}_1}}, {V}^{\mathfrak{t}_2}\subset T^{{\mathfrak{t}_2}}$ of regularity $\underline{\alpha}_1$ resp. $\underline{\alpha}_2$, one can always define a tensor product between $V^{\mathfrak{t}_1}$ and ${V}^{\mathfrak{t}_2}$ as follows.

* Extend the set $\mathfrak{L}$ by a new element $\mathfrak{t}_3$, i.e. let $\bar{\mathfrak{L}}= \mathfrak{L}\cup \{\mathfrak{t}_3\}$,

* set $E^{\frakt_3}:=E^{\frakt_1} \otimes E^{\frakt_2}$,

* set for example

\begin{equation}\label{eq:example_prod_construction}
T^{\frakt_3}_{\alpha,\delta}:=\bigoplus_{\substack{\alpha'+\alpha''=\alpha,\\ \big(\delta'+(\underline{\alpha}_{2}\wedge 0)\big)\wedge\big( \delta''+(\underline{\alpha}_{1}\wedge 0)\big)=\delta }} V^{\frakt_1}_{\alpha',\delta'} \otimes V^{\frakt_2}_{\alpha'',\delta''}\ ,
\end{equation}

* extend $ L_{p,q}$ to act on $T^{\frakt_3}$ multiplicatively,

* set $\bar{\mathcal{T}}:=(\{E^\mathfrak{t}\}_{\mathfrak{t}\in \bar{\mathfrak{L}}},\{T^\mathfrak{t}\}_{\mathfrak{t}\in \bar{\mathfrak{L}}}, L )$.

Observe that in general there is no canonical extension of a non smooth model $Z$ on $\mathcal{T}$ to a model on $\bar{\mathcal{T}}$. An important exception occurs when $V^{\mft_1}$ or $V^{\mft_2}$ is a jet sector and $Z$ is a model that acts on that sector as a jet model.

Note that while the construction in the previous remark is rather canonical, in certain situations there is more structure present which makes other constructions more natural. A very important example is the case of jet regularity structures.
Assume in the setting of the previous remark that
$V^{\mathfrak{t}_1}\subset J{E^{\mathfrak{t}_1}}$ and ${V}^{\mathfrak{t}_2}\subset J{E^{\mathfrak{t}_2}}$, then it is natural define the product to be the restriction of the canonical product

$$ m: J{E^{\mathfrak{t}_1}}\otimes  J{E^{\mathfrak{t}_2}} \to  J ( {E^{\mathfrak{t}_1}}\otimes{E^{\mathfrak{t}_2}})$$

introduced in Section <ref>.
In order to satisfy Property (<ref>) of being a $\gamma$-regular product for some $\gamma>0$, needs to restrict the space $L$ accordingly.
This in turn makes the space of jet models smaller, but by Point <ref> in Section <ref> it is non-empty.
We shall call the product $m$ on jet sectors the canonical product on jets and write $\star$ for it.

The following can be proved exactly as <cit.>.

Let $V^{\mathfrak{t}_1}\subset T^{\mathfrak{t}_1}, {V}^{\frakt_2}\subset T^{\frakt_2}$ be two sectors of regularity $\alpha_1$ resp. $\alpha_2$ and let $\star: V^{\mathfrak{t}_1}\times  V^{\mathfrak{t}_2}\to T^{\mathfrak{t}_3}$ be an abstract tensor product. Let $Z$ be a model and $\gamma_1,\gamma_2>0$. Assuming the product $\star$ is $\gamma$-precise for $\gamma= (\gamma_1+\alpha_2) \wedge (\gamma_2+\alpha_1)$, the map

\star_\gamma:\ \mathcal{D}^{\gamma_1}(V^{\frakt_1})\times \mathcal{D}^{\gamma_2}(V^{\frakt_2}) \to \mathcal{D}^{(\gamma_2+\alpha_1) \wedge (\gamma_1+\alpha_2) }(T^{\frakt_3}), \qquad
(f_1,f_2)\mapsto f_1\star_\gamma f_2:= Q_{\gamma} f_1\star f_2

is well defined, continuous and the bound

$$\interleave f_1\star_\gamma f_2  \interleave_{\gamma ; K}\lesssim \interleave f_1\interleave_{\gamma_1 ; K}\interleave f_2 \interleave_{\gamma_2 ; K}(1+ \|\Gamma\|_{ K})$$

If furthermore $\tilde Z = (\tilde \Pi, \tilde{\Gamma})$ is a second model and $\tilde{f}_1\in \mathcal{D}_{\tilde{Z}}^{\gamma_1}(V^{\frakt_1})$, $\tilde{f}_2\in \mathcal{D}_{\tilde{Z}}^{\gamma_2}(V^{\frakt_2})$ one has the bound

$$\interleave f_1\star_\gamma f_2 ; \tilde{f}_1\star_\gamma \tilde{f}_2 \interleave_{\gamma ; K}\lesssim_C \interleave f_1 ; \tilde{f}_1 \interleave_{\gamma_1 ; K}+\interleave f_2 ;  \tilde{f}_2 \interleave_{\gamma_2 ; K}+ \|\Gamma-\tilde{\Gamma}\|_{ K}$$

uniformly over  $f_i, \tilde{f}_i$ satisfying $\interleave f_i \interleave_{\gamma_i ; K}, \interleave  \tilde{f}_i \interleave_{\gamma_i ; K}<C$ and models satisfying
$\|\Gamma\|_{ K}, \|\tilde{\Gamma}\|_{K}<C$.

§.§.§ Trace operator

Let $E, F$ be vector bundles, recall the trace operator $\tr: E^*\otimes E\otimes F\to F$  acting on elementary tensors as $e_p^*\otimes e_p \otimes f_p\mapsto (e_p^*,e_p) f$. Denote by $I$ the $E\otimes E^*$ tensor field, such that $(I_p,e_p^*\otimes e_p)= (e_p^*, e_p)$ and observe that the dual map of $\tr$ is given by
$$F\to E^*\otimes E\otimes F \quad f\mapsto I\otimes f\ . $$

Thus there is a natural extension of the trace operator

$$\tr:\mathcal{D}'(E^*\otimes E\otimes F)\to \mathcal{D}'(F) $$

characterised by the fact that for each $\xi\in \mathcal{D}'(E^*\otimes E\otimes F)$ and $\phi \in \mathcal{D}(F) $ one has

$$\langle \tr \xi, \phi \rangle =  \langle \xi, I \otimes \phi \rangle \ .$$

Let $E^{\frakt_1}= E^*\otimes E\otimes F$, $E^{\frakt_2}= F$. An abstract trace operator is a bi-graded bundle morphism $\mathfrak{\trr}: T^{\mathfrak{t}_1}\mapsto T^{\mathfrak{t}_2}$ satisfying

* for any $\tau_p\in T^{\frakt_1}_{\alpha,\delta}$ one has $\trr \tau_p \in T^{\frakt_2}_{\alpha,\delta}$

* writing $\iota_1: T^{\frakt_1}_0\to E^{\frakt_1}$, $\iota_2: T^{\frakt_2}_0\to E^{\frakt_2}$ for the canonical isomorphisms one has

$$\iota_2\circ \trr = \tr \circ \iota_1  ,$$

* for each $\Gamma_{p,q}\in L_{p,q}$ one has $\Gamma_{p,q} \trr =\trr\Gamma_{p,q} $

A model realizes an abstract trace operator if
$$\Pi_p(\trr \tau_p) = \tr (\Pi_p\tau_p) \ .$$

It is obvious from the definition that for $f\in \mathcal{D}^\gamma (V^{\mathfrak{t}_1})$ one has $\trr f\in \mathcal{D}^\gamma(T^{\mathfrak{t}_2})$ and that

$$\tr \mathcal{R}f= \mathcal{R}\trr f\ .$$