#
Singular SPDEs on Homogeneous Lie Groups

Avi Mayorcas  Institute of Mathematics, Technische Universität Berlin, Str.
des 17. Juni 136, 10587 Berlin, Germany.
Email<EMAIL_ADDRESS>ORCID iD: 0000-0003-4133-9740  Harprit Singh
Department of Mathematics, Imperial College London, South Kensington Campus,
SW7 2AZ, UK.
Email<EMAIL_ADDRESS>ORCID iD: 0000-0002-9991-8393

###### Abstract

The aim of this article is to extend the scope of the theory of regularity
structures in order to deal with a large class of singular SPDEs of the form

$\partial_{t}u=\mathfrak{L}u+F(u,\xi)\ ,$

where the differential operator $\mathfrak{L}$ fails to be elliptic. This is
achieved by interpreting the base space $\mathbb{R}^{d}$ as a non-trivial
homogeneous Lie group $\mathbb{G}$ such that the differential operator
$\partial_{t}-\mathfrak{L}$ becomes a translation invariant hypoelliptic
operator on $\mathbb{G}$. Prime examples are the kinetic Fokker-Planck
operator $\partial_{t}-\Delta_{v}-v\cdot\nabla_{x}$ and heat-type operators
associated to sub-Laplacians. As an application of the developed framework, we
solve a class of parabolic Anderson type equations

$\partial_{t}u=\sum_{i}X^{2}_{i}u+u(\xi-c)$

on the compact quotient of an arbitrary Carnot group.

Keywords: Regularity structures, homogeneous Lie groups, hypoelliptic
operators, stochastic partial differential equations.

2020 MSC: 60L30 (Primary); 60H17, 35H10, 35K70 (Secondary)

###### Contents

  1. 1 Introduction
    1. 1.1 Related Literature
    2. 1.2 A Motivating Class of Examples
    3. 1.3 Structure and Contributions of the Paper
    4. 1.4 Open Problems and Wider Context
  2. 2 Analysis on Homogeneous Lie Groups
    1. 2.1 Derivatives and Polynomials
    2. 2.2 Distributions and Convolution
    3. 2.3 Discrete Subgroups
    4. 2.4 Concrete Examples
  3. 3 Regularity Structures and Models
    1. 3.1 The Polynomial Regularity Structure
    2. 3.2 Modelled Distributions
    3. 3.3 Singular Modelled Distributions
    4. 3.4 Convolution with Singular Kernels
    5. 3.5 Local Operations
    6. 3.6 Symmetries
    7. 3.7 Bounds on Models
  4. 4 Applications to Semilinear Evolution Equations
    1. 4.1 Short Time Behaviour of Kernel Convolutions
    2. 4.2 Initial Conditions
    3. 4.3 An Example Fixed Point Theorem
    4. 4.4 Concrete Examples of Differential Operators and Kernels
  5. 5 A Regularity Structure for Anderson Equations
    1. 5.1 Smooth Models
  6. 6 The Anderson Equation on Stratified Lie Groups
    1. 6.1 Stochastic Estimates for the Renormalised Model

##  1 Introduction

The theory of regularity structures, [39], provides a framework for the study
of subcritical stochastic partial differential equations (SPDEs) of the form

$\partial_{t}u-{\mathfrak{L}}u=F(u,\xi)\ ,$ (1.1)

when the operator ${\mathfrak{L}}$ is uniformly elliptic. This article extends
the theory of regularity structures in order to solve equations where the
differential operator stems from a large class of hypoelliptic operators. This
is achieved by building on the fundamental idea of Folland [28] to reinterpret
the differential operator in question as a differential operator on a
homogeneous Lie group and extending the theory of regularity structures to
these non-commutative spaces.

The treatment of singular SPDE via the theory of regularity structures can be
divided into two parts, an analytic step and an algebraic and stochastic step.

  1. 1.

The first, analytic step, is to introduce the notions of a regularity
structure, models and modelled distributions; generalised Taylor expansions of
both functions and distributions. Given this set-up, sufficient analytic tools
are then developed to allow for the study of abstract, fixed-point equations
in the spaces of modelled distributions. Crucially, a reconstruction operator
maps modelled distributions to genuine distributions (or functions) on the
underlying domain. For the case of constant coefficient, hypoelliptic
equations on compact quotients of Euclidean domains, this aspect of the theory
was already worked out in full generality in [39].

  2. 2.

The second step, which was carried out in a case by case basis in [39], was
then fully automated in the subsequent works [17, 14, 13]. In [14] the authors
construct concrete (equation-dependent) regularity structures and models
together with a large enough renormalisation group. Then, in [13], the
question of how this renormalisation group acts on the SPDE in question was
answered. Convergence of renormalised models for an extremely large class of
noises, known as the BPHZ theorem, was proved in [17].

In much of the above work translation invariance on $\mathbb{R}^{d}$, of both
the equation and the driving noise, plays a major role. In this paper we will
instead work with equations that are translation invariant with respect to a
homogeneous Lie group $\mathbb{G}$ (Euclidean spaces being special case).

The bulk of this paper is dedicated to implementing the analytic aspects of
the theory in the case when the underlying space is a general (non-Abelian)
homogeneous Lie group. The second step is then carried out for a specific
class of equations. While the theory developed herein remains, somewhat
surprisingly, close to that of [39], there are a number of key conceptual and
technical obstacles to overcome. The main cause of these deviations are the
non-commutative structure of the base space and the fact that the notion of
Taylor expansions, a crucial element of the theory, heavily depends on the
underlying group structure. We discuss these issues in more detail in Section
1.3 along with some wider context in Section 1.4. For the readers convenience
the article aims at being self-contained and thus arguments that adapt
directly from [39] are clearly marked as such.

The central motivation for this extension to homogeneous Lie groups is to
allow for the study of singular SPDE with linear part given by a hypoelliptic
operator, which may fail to be uniformly parabolic. Two motivating examples
are the heat operator associated to the sub-Laplacian on the Heisenberg group
and the kinetic Fokker-Planck operator on $\mathbb{R}\times\mathbb{R}^{2n}$,

$u(t,x,y,z)\mapsto\partial_{t}u-(\Delta_{x}+\Delta_{y}+(x^{2}+y^{2})\Delta_{z})u\text{
and }u(t,x,v)\mapsto\partial_{t}u-\Delta_{v}u-v\cdot\nabla_{x}u\ .$ (1.2)

We will carry these two operators and their associated homogeneous Lie groups,
throughout the paper as working examples of the theory. In the final sections
we develop a full solution theory for Anderson type equations associated to
sub-Laplacians on general stratified Lie groups (Carnot groups), of which the
Heisenberg group is a well-studied example.

More generally, however, the analytic results of this paper apply to any
differential operator, $\bar{\mathfrak{L}}$ on $\mathbb{R}^{d-1}$, such that
the following combination of results by Folland applies to
$\mathfrak{L}:=\partial_{t}-\bar{\mathfrak{L}}$, with respect to some
homogeneous Lie group structure on $\mathbb{R}^{d}$.

######  Theorem 1.1 ([28, Thm. 2.1 & Cor. 2.8]).

Let $\mathbb{G}$ be a homogeneous Lie group of homogeneous dimension
$|\mathfrak{s}|$ and $\mathfrak{L}$ be a left-translation invariant (with
respect to $\mathbb{G}$), homogeneous differential operator of degree
$\beta\in(0,|\mathfrak{s}|)$ on $\mathbb{G}$, such that $\mathfrak{L}$ and its
adjoint $\mathfrak{L}^{\ast}$ are both hypoelliptic. Then there exists a
unique homogeneous distribution $K$, of order $\beta-|\mathfrak{s}|$ such that
for any distribution, $\varphi\in\mathcal{D}^{\prime}(\mathbb{G})$

$\mathfrak{L}(\varphi\ast K)=(\mathfrak{L}\varphi)\ast K=\varphi.$

where the convolution is with respect to the given Lie structure.

Referring to Section 2 for a more thorough discussion of homogeneous Lie
groups and their properties, we recall here that a differential operator
$\mathfrak{D}$ on $\mathbb{R}^{d}$ is called hypoelliptic, if for every open
subset $\Omega\subset\mathbb{R}^{d}$ one has that,

$\mathfrak{D}u\in C^{\infty}(\Omega)\Rightarrow u\in C^{\infty}(\Omega)\ .$

The following celebrated result of Hörmander (almost) entirely classifies the
family of second order hypoelliptic operators.

######  Theorem 1.2 ([47, Thm. 1.1]).

Let $r\leq d$ and $\\{X_{i}\\}_{i=0}^{r}$ be a collection of first order
differential operators (i.e. vector fields) on $\mathbb{R}^{d}$ and
recursively define $\mathcal{V}_{1}=\text{span}\\{X_{i}\,:\,i=0,\ldots,r\\}$,
$\mathcal{V}_{n+1}=\mathcal{V}_{n}\cup\left\\{[V,W]\,:V\in\mathcal{V}_{1},\,W\in\mathcal{V}_{n}\right\\}$.
If a differential operator can be written in the form,

$\mathfrak{D}=\sum_{i=1}^{r}X_{i}^{2}+X_{0}+c,$ (1.3)

for some $c\in\mathbb{R}$, and there exists an $N\geq 1$ such that
$\dim(\mathcal{V}_{N})=d$ at every point in $\Omega\subseteq\mathbb{R}^{d}$
then $\mathfrak{D}$ is hypoelliptic on $\Omega$.

######  Remark 1.3.

By Frobenius’s theorem, [33], if the condition of Theorem 1.2 fails in some
open set then $\mathfrak{D}$ is not hypoelliptic on that set. However, the
statement is not truly sharp; for example the Grushin operator
$\partial^{2}_{vv}-v\partial_{x}$ is hypoelliptic, while the conditions of
Theorem 1.2 are violated on sets intersecting $\\{v=0\\}$. On the other hand
this type of exception cannot occur if the sections $\mathcal{V}_{i}$ are
continuous, since in this case it can be shown that the map
$x\mapsto\mathbb{1}_{\\{\dim\mathcal{V}_{i}(x)=d\\}}$ is upper semi-
continuous.

While Hörmander’s theorem gives an almost sharp characterisation of second
order, hypoelliptic, operators it does not say much about fine properties of
the fundamental solution. For example, it gives no direct route to a refined
regularity theory for hypoelliptic equations. This observation highlights the
contribution of Folland’s theorem (Theorem 1.1); since translation invariance
allows access to many additional tools in the Euclidean setting, such as
harmonic analysis and singular integral methods. Viewing the class of
translation invariant operators satisfying Folland’s theorem as analogous to
constant coefficient, elliptic operators on $\mathbb{R}^{d}$ a programme was
successfully carried out in the works [29, 30, 28, 61], establishing a full
$L^{p}$ regularity theory for general, smooth coefficient, hypoelliptic
operators. We refer to [12, Ch. 3] for a concise introduction to this
programme.

###  1.1 Related Literature

Non-translation invariant and non-uniformly elliptic SPDEs have been well-
studied in the classical, i.e. non-singular, regime and we do not attempt to
present this large literature here. We refer to standard texts such as [21,
54, 23, 59] for a general overview and references to more specific works
contained therein. However, in the more specialised setting of semi-linear
SPDEs on homogeneous Lie groups, we mention the works [63, 64, 58] which treat
both hypoelliptic and parabolic SPDEs on some classes of Lie groups and sub-
Riemannian manifolds. A solution theory for conservative SPDEs based on the
kinetic Fokker–Planck equation (see Section 4.4) in the Itô case was developed
in [26]. Using the theory of paracontrolled calculus this was extended to the
singular regime in [46]. The kinetic Fokker–Plank operator and its associated
homogeneous Lie group fall into the analytic framework developed in this
paper, however, we postpone a concrete application of this theory towards a
kinetic Anderson type equation to a future work. Recently, in [10], an
Anderson type equation on the Heisenberg group with white in time and coloured
in space noise was studied using Itô calculus techniques, up the to the full
sub-critical regime of the noise regularity. We treat a closely related
problem in the final sections of this paper. A fuller discussion of the
similarities and differences between the results of [10] and those of our
approach is postponed to Remark 6.7.

Singular SPDEs with non-translation invariant, but uniformly parabolic or
elliptic linear part, have also been considered, especially using the recently
developed pathwise techniques of [39, 37, 56, 62]. Some of these works are
discussed in more detail in Section 1.4 below. Quasilinear SPDEs have been
studied using regularity structures, rough path based methods and
paracontrolled distribution theory, see [36, 57, 9, 8]. Recently, an approach
inspired by the theory of regularity structures, but technically distinct, has
been developed in [56, 53, 52]. SPDEs on domains with boundaries have been
treated using both regularity structures and paracontrolled distribution based
methods, [50, 35, 20, 34]. Finally, a number of works have considered
parabolic, singular SPDEs on Riemannian manifolds; a paracontrolled approach
using the spectral decomposition associated to the Laplace–Beltrami operator
has been developed in [4, 5, 3]. An approach via regularity structures has
been applied to the $2$d parabolic Anderson equation on a Riemannian manifold
in [22], while [6, 62] develop some aspects required to treat non-translation
invariant, uniformly parabolic equations. Finally, the work [44] gives a
comprehensive extension of regularity structures to singular SPDEs on
Riemannian manifolds, with only the renormalisation of suitable stochastic
objects left to a case by case treatment.

###  1.2 A Motivating Class of Examples

In this paper we restrict ourselves to linear operators satisfying the
criteria of Theorem 1.1 and use the Anderson equation as a main motivating
example, although we stress that our main analytic results apply in the full
generality treated by [39]. The parabolic Anderson model

$\partial_{t}u=\Delta u-u\xi,\quad u|_{t=0}=u_{0},$ (1.4)

on $\mathbb{R}_{+}\times\mathbb{R}^{d}$ describes the conditional, expected
density of particles, where each particle moves according to an independent
Brownian motion and branches at a rate proportional to the random environment
$\xi$, see [49, 16]. Rigorously, this description is derived by discrete
approximations and it is well known that in the case $\xi$ is a spatial white
noise, when $d=2$ and $d=3$ one needs to recentre the potential in order to
obtain a non-trivial limit as the discretisation is removed, [39, 41, 42, 37].
We point out that if the environment is also allowed to depend on time and is
for example, white in time, then martingale methods can be used instead to
develop a probabilistic solution theory, see [65, 24, 25, 19].

If one replaces the Brownian motions with a general diffusion,

$\mathop{}\\!\mathrm{d}x_{t}=\sqrt{2}\sum_{i=1}^{r}X_{i}(x_{t})\mathop{}\\!\mathrm{d}W^{i}_{t},$

where the vector fields satisfy Hörmander’s rank condition (Theorem 1.2),
then, formally the conditional expected density of particles is described by
the hypoelliptic Anderson equation,

$\partial_{t}u-\bar{\mathfrak{L}}u:=\partial_{t}u-\sum_{i=1}^{r}X^{2}_{i}u=u\xi\
.$ (1.5)

When realisations of the environment are sufficiently singular then a pathwise
solution theory for (1.5) is expected to require renormalisation. Since the
vector fields $\\{X_{i}\\}_{i=1}^{r}$ satisfy Hörmander’s condition, the
operator $\bar{\mathfrak{L}}$ is smoothing and therefore in principle, an
extension of the theory of regularity structures should be applicable to find
a renormalised, pathwise, solution theory for (1.5). In Section 6 we apply the
analytic tools developed in the paper to demonstrate such an extension to the
case where $\bar{\mathfrak{L}}$ is the sub-Laplacian on a compact quotient of
stratified Lie group (Carnot group).

We show a result analogous to those of [39, 42], finding a notion of
renormalised solution to (1.5), when $\xi$ is a coloured periodic noise on a
stratified Lie group. More precisely, we show that when $\xi$ is replaced with
a mollified, recentred noise $\xi_{\varepsilon}-c_{\varepsilon}$, for
(specific) diverging constants $\\{c_{\varepsilon}\\}_{\varepsilon\in(0,1)}$,
solutions $u_{\varepsilon}$ to (1.5) converge in probability to a unique limit
independent of the specific choice of mollification-scheme. We stress that
this result does not cover the full subcritical regime of (1.5). The treatment
of this full regime would require an analogue of the BPHZ theorem on
homogeneous Lie groups, see [17, 45].

A notable example of a stratified Lie group is the Heisenberg group,
$\mathbb{H}^{n}\cong\mathbb{R}^{2n}\times\mathbb{R}$, see Section 2.4.1 for a
description. In this case the collection of left-translation invariant vector
fields, which generate the associated Lie algebra are,

$A_{i}(x,y,z)=\partial_{x_{i}}+y_{i}\partial_{z},\quad
B_{i}(x,y,z)=\partial_{y_{i}}-x_{i}\partial_{z},\quad\text{for
}\,i=1,\ldots,n.$

It is readily checked that the collections $\\{A_{i},B_{i}\\}_{i=1}^{n}$ are
left-translation invariant with respect to the group action described in
Section 2.4.1 and that one has $C(x,y,z):=[A_{i},B_{i}]=-2\partial_{z}$ for
all $i=1,\ldots,n$. The associated sub-Laplacian is the linear differential
operator,

$\bar{\mathfrak{L}}u=\sum_{i=1}^{n}(A_{i}^{2}+B_{i}^{2})u,$

naturally extended to a heat type operator as in (1.2) and (1.5), c.f. Section
4.4.1 for further discussion. A phenomenon of interest for Anderson equations
is that of localisation, the concentration of the solution at large times,
taking arbitrarily large values on islands of arbitrarily small size, [16,
49]. Since one expects the geometry of the underlying domain to have effect on
the emergence of this phenomenon, as also noted in [10] and since pathwise
approaches to the parabolic Anderson model have proved fruitful in studying
finer properties of solutions, see e.g. [2, 41, 42, 38, 50, 7], it is our hope
that the tools developed herein may prove useful in analysing similar
equations on more complex domains. The solution theory exposited in Section 6
applies to precisely this example on a compact quotient of the Heisenberg
group.

###  1.3 Structure and Contributions of the Paper

While it is possible to develop a bespoke solution theory for specific
hypoelliptic SPDEs, this rapidly becomes technically challenging and
computationally heavy, c.f. [46]. Furthermore, for more complicated equations
this bespoke approach quickly becomes impossible. Thus, in our view, the
overarching contribution of this article is to show that by taking a more
abstract and general point of view one can treat a large class of such
equations at once in a framework (surprisingly) close to that of [39]. In
addition, this work highlights the dependence on the algebraic structure of
the underlying space and thus contributes to the understanding of solution
theories in more general settings, see Section 1.4.

We begin, in Section 2, with a recap of analysis on homogeneous Lie groups
that is sufficient for our purpose. The most important result in this section
is Theorem 2.13 which provides a precise intrinsic version of Taylor’s theorem
on homogeneous Lie groups and is a sharper version of existing results, see
Remark 2.14.

Section 3 contains the bulk of the paper and establishes an extension of the
analytic aspects of regularity structures to the setting of homogeneous Lie
groups. In Section 3.1 we introduce the appropriate notion of abstract
polynomials on homogeneous Lie groups which are the backbone of much of our
analysis. In Section 3.2 we discuss modelled distributions. In order to prove
reconstruction we extend ideas from [32], constructing a suitable replacement
for the test functions, $\rho$ and $\varphi$ employed therein. This
construction avoids the need for a general theory of wavelets or Fourier
analysis on homogeneous Lie groups. In a similar vein, the set-up of singular
modelled distributions in Section 3.3, while conceptually similar to that of
[39, Sec. 6] requires a number technical novelties. In particular, the
singular hyperplane of [39] is replaced with a homogeneous Lie-subgroup and
thus statements and proofs are adapted to the underlying (non-commutative)
structure. In Section 3.4 we provide a surprisingly compact proof of the
Schauder estimates, building on Theorem 2.13 in combination with the setup of
Sections 3.1 & 3.2 in order to reduce to estimates already obtained in [39,
Sec. 5]. The final three sections, Section 3.5, 3.6 & 3.7, carry over almost
mutatis mutandis from [39] and are included for the reader’s convenience.

In Section 4 we demonstrate the application of this general theory to semi-
linear evolution equations. We again take an abstract point of view and start
with a general homogeneous Lie group and only postulate a “time" direction as
a one dimensional subspace of its Lie algebra. Then, we provide an example
fixed point theorem for multiplicative stochastic heat equations in this
setting. We note that there is no difficulty in extending the general fixed
point theorem of [39, Thm. 7.8] to our setting, we only specialise for the
sake of brevity. In Section 4.4 we present examples of singular kernels on
homogeneous Lie groups for which our theory applies, such as the Green’s
function of heat operators on the Heisenberg group and Kolmogorov type
operators associated to inertial systems.

In Section 5 we explicitly describe a regularity structure for treating
linear, Anderson-type equations on homogeneous Lie groups. For this specific
class of equations it is notable that one may reuse the algebraic structures
developed in [14]. However, an algebraic question left open by this work is
the extension of the results in [14] to homogeneous Lie groups in full
generality.

Finally, in Section 6, we employ the theory developed in the rest of the paper
to demonstrate a solution theory for Anderson-type equations on a compact
quotient $S$ of a stratified Lie group $\mathbb{G}$. Since we do not limit
ourselves to quotients by normal subgroups, the space $S$ may fail be a group
and therefore it is not possible to directly regularise the noise in the usual
manner. This last obstacle is circumvented by making use of the fact that $S$
is a $\mathbb{G}$-homogeneous space allowing us to define an intrinsic
regularisation of the noise.

###  1.4 Open Problems and Wider Context

As discussed above, translation invariant (and for example) parabolic
operators on Euclidean domains serve as a starting point from which non-
translation invariant parabolic problems on Euclidean domains as well as
parabolic problems on Riemannian manifolds can be studied. A somewhat parallel
progression holds starting from translation invariant operators on homogeneous
Lie groups, moving to general Hörmander operators as well as heat type
equations on sub-Riemannian manifolds, see [12]. This parallel progression in
PDE analysis leads one to ask, how far the theory of regularity structures can
be extended in these directions. The table below gives a schematic
presentation of the progress so far, presents some open problems and places
this work within the context of the study of subcritical parabolic-type SPDEs.
The two rows describe the two parallel progressions outlined above in the
context of regularity structures.

Translation invariant operators on $\mathbb{R}^{d}$ |  Non-translation invariant operators on $\mathbb{R}^{d}$ |  On Riemannian manifolds
---|---|---
The works [39, 14, 13, 17] present a general and complete picture. |  Analytic step in [39]; aspects of renormalisation addressed in [6, 62]. |  In [22] 2d-PAM is treated. A general account in [44].
Translation invariant operators on homogeneous Lie groups |  General Hörmander operators |  On sub-Riemannian manifolds
Analytic theory covered in this work. Renormalisation and convergence by hand. |  Open problem A |  Open problem B

Each problem in the second row is closely related to its equivalent problem in
the first row, we discuss the posed open problems in a little more detail.

  * •

Open Problem A is expected to be more involved than its counterpart in the
Euclidean setting; going from translation invariant operators to non-
translation invariant operators by local approximations. In the case of
general Hörmander operators, the underlying Lie group structure would in
general vary from point to point. An interesting application would be the
study of general hypoelliptic Anderson models (1.5) where the underling
particles perform a general diffusion with generator of the form (1.3), c.f.
Theorem 1.2.

  * •

Open Problem B is motivated by the fact that analogous to the tangent space
being an appropriate local approximations of a Riemannian manifold, the non-
holonomic tangent space is an appropriate local approximations of a sub-
Riemannian manifold and each fibre of the non-holonomic tangent space is a
stratified Lie group, c.f. [1]. Note that in view of Remark 1.3 the
difficulties in Problem B are expected to be somewhat complementary to those
of Problem A.

##### Notation:

Given $\alpha\in\mathbb{R}$ we let
$[\alpha]:=\max\\{r\in\mathbb{Z}\,:\,r\leq\alpha\\}$ denote the integer part
of $\alpha$. We often write $\lesssim$ to mean that an inequality holds up to
multiplication by a constant which may change from line to line but is uniform
over any stated quantities. In general we reserve the notation $D$ for the
usual derivative on Euclidean space, and $X,\,Y$ for elements of a Lie algebra
thought of as vector fields on the associated Lie group, see Section 2. We
will use $|\,\cdot\,|$ to denote the size of various quantities; elements of a
Lie group, the Haar measure of subsets of the group, the homogeneity of
members of the structure space in a regularity structure, traces of linear
maps and the absolute value function on $\mathbb{R}$ etc. The meaning will
usually be clear from the context but in cases where it is not we will make
sure to specify in the text. For integrals we will use the standard notation,
$\mathop{}\\!\mathrm{d}x$, for integration against the Haar measure on a given
homogeneous, Lie group, see Proposition 2.1. Throughout most of the article we
take an intrinsic point of view and do not equip the homogeneous Lie group
with an explicit chart.

##### Acknowledgements:

The authors wish to thank Martin Hairer, Rhys Steele, Ilya Chevyrev and Ajay
Chandra for helpful and insightful conversations during the preparation of
this manuscript.

AM wishes to thank the INI and DPMMS for their support and hospitality which
was supported by Simons Foundation (award ID 316017) and by Engineering and
Physical Sciences Research Council (EPSRC) grant number EP/R014604/1 as well
as DFG Research Unit FOR2402 for ongoing support.

HS acknowledges funding by the Imperial College London President’s PhD
Scholarship. He wishes to thank Josef Teichmann and Máté Gerencsér for their
hospitality during his visit to ETH Zürich and TU Wien.

##  2 Analysis on Homogeneous Lie Groups

We collect some basic facts on homogeneous Lie groups and smooth function on
them. Let $\mathfrak{g}$ be a Lie algebra and $\mathbb{G}$ be the unique, up
to Lie algebra isomorphism, corresponding simply connected Lie group. We write
$[\cdot,\cdot]$ for its Lie bracket and inductively set,
$\mathfrak{g}_{(1)}=\mathfrak{g}$ and
$\mathfrak{g}_{(n)}=[\mathfrak{g}_{(n-1)},\mathfrak{g}]$. Recalling the
exponential map $\exp:\mathfrak{g}\rightarrow\mathbb{G}$, we have the
Baker–Campbell–Hausdorff formula

$\exp(X)\exp(Y)=\exp(\operatorname{H}(X,Y))\ ,$ (2.1)

where $\operatorname{H}$ is given by
$\operatorname{H}(X,Y)=X+Y+\frac{1}{2}[X,Y]+...$ with the remaining terms
consisting of higher order iterated commutators of $X$ and $Y$; crucially
$\operatorname{H}$ is universal, i.e. it does not depend on the underlying Lie
algebra.

######  Proposition 2.1 ([31, Prop. 1.2]).

Assume that $\mathfrak{g}$ is nilpotent, i.e. $\mathfrak{g}_{(n)}={0}$ for
some $n\in\mathbb{N}$. Then, the exponential map is a diffeomorphism and

  * •

through this identification of $\mathfrak{g}$ with $\mathbb{G}$, the map

$\mathbb{G}\times\mathbb{G}\ni(x,y)\mapsto xy\in\mathbb{G}$

becomes a polynomial map (between vector spaces).

  * •

the pull-back to $\mathbb{G}$ of the Lebesgue measure on $\mathfrak{g}$ is a
bi-invariant Haar measure on $\mathbb{G}$.

######  Definition 2.1.

A dilation on $\mathfrak{g}$ is a group of algebra automorphisms
$\\{\operatorname{D}_{r}\\}_{r>0}$ of the form
$\operatorname{D}_{r}X=\exp(\log r\cdot\mathfrak{s})X$ with
$\mathfrak{s}:\mathfrak{g}\to\mathfrak{g}$ being a diagonalisable linear
operator with $1$ as its smallest eigenvalue.

######  Remark 2.2.

The requirement that the smallest eigenvalue of $\mathfrak{s}$ be $1$ is
purely cosmetic. Otherwise, denoting the smallest eigenvalue by
$\mathfrak{s}_{1}$, one can work with the new operator
$\tilde{\mathfrak{s}}=\frac{1}{\mathfrak{s}_{1}}\mathfrak{s}$.

######  Remark 2.3.

For $a\in\mathbb{R}$, denote by $W_{a}\subset\mathfrak{g}$ the eigenspace of
$\mathfrak{s}$ with eigenvalue $a$. Thus for $X\in W_{a}$, $Y\in W_{b}$ one
has the condition

$\operatorname{D}_{r}[X,Y]=[\operatorname{D}_{r}X,\operatorname{D}_{r}Y]=r^{a+b}[X,Y]$

which in particular implies $[W_{a},W_{b}]\subset W_{a+b}$ and since
$W_{a}=\\{0\\}$ for $a<1$

$\mathfrak{g}_{(j)}\subset\bigoplus_{a\geq j}W_{a}\ .$

In particular, if $\mathfrak{g}$ admits a family of dilations, it is
nilpotent. The converse is not necessarily true.

######  Definition 2.2.

A homogeneous Lie group $\mathbb{G}$ is a simply connected, connected Lie
group where its Lie algebra $\mathfrak{g}$ is endowed with a family of
dilations $\\{\operatorname{D}_{r}\\}_{r>0}$. For $r>0$, we define the group
automorphism

$x\mapsto r\cdot x:=\exp\circ\operatorname{D}_{r}\circ\exp^{-1}x\ .$

A homogeneous norm on $\mathbb{G}$ is a continuous function
$|\cdot|:\mathbb{G}\to[0,\infty)$ satisfying the following properties for all
$x\in\mathbb{G}$, $r\geq 0$

  1. 1.

$|x|=0$ if and only if $x=e$, the neutral element,

  2. 2.

$|x|=|x^{-1}|\ ,$

  3. 3.

$|r\cdot x|=r|x|\ .$

The homogeneous norm naturally induces a topology generated by the open sets
and in turn a Borel $\sigma$-algebra. From now on, we will always assume that
$\mathbb{G}$ is equipped with this topology and $\sigma$-algebra. Furthermore,
given a homogeneous group $\mathbb{G}$ we denote by
$\\{X_{j}\\}_{j=1}^{d}\in\mathfrak{g}$ a basis of eigenvectors of
$\mathfrak{s}$ with eigenvalues
$1=\mathfrak{s}_{1}\leq\mathfrak{s}_{2}\leq...\leq\mathfrak{s}_{d}$ and such
that

$\mathfrak{s}X_{j}=\mathfrak{s}_{j}X_{j}\ .$ (2.2)

Given a measurable subset $E\subset\mathbb{G}$ we write $|E|$ for its Haar
measure which we assume to be normalized such that the set
$B_{1}=:\\{x\in\mathbb{G}\ :\ |x|\leq 1\\}$ has measure $1$, c.f. Proposition
2.1. In integrals we use the standard notation $\mathop{}\\!\mathrm{d}x$. We
define $|\mathfrak{s}|:=\text{trace}(\mathfrak{s})$ as the homogeneous
dimension of $\mathbb{G}$, since for any measurable subset
$E\subset\mathbb{G}$ and $r>0$, one has

$|r\cdot E|=r^{|\mathfrak{s}|}|E|\ .$

We also define balls of radius $r>0$,

$B_{r}(x):=\\{y\in\mathbb{G}\ :\ |x^{-1}y|<r\\}\ .$

The topology induced by these balls agrees with the topology of $\mathbb{G}$
as a Lie group, see [27, Sec. 3.1.6]. Note that due to the non-commutativity
of $\mathbb{G}$, in general $|x^{-1}y|\neq|yx^{-1}|$. We consistently work
with the following choice of a semi-metric on the group,

$\mathop{}\\!\mathrm{d}_{\mathbb{G}}(x,y):=|x^{-1}y|=|y^{-1}x|\ .$

For $\mathfrak{K}\subset\mathbb{G}$ we write
$\bar{\mathfrak{K}}:=\\{z\in\mathbb{G}\,:\,\mathop{}\\!\mathrm{d}_{\mathbb{G}}(z,\mathfrak{K}):=\inf_{y\in\mathfrak{K}}|y^{-1}z|\leq
1\\}$ for the $1$-fattening of $\mathfrak{K}$.

######  Remark 2.4.

A function $f:\mathbb{G}\rightarrow\mathbb{R}$ is called homogeneous of degree
$\lambda\in\mathbb{R}$, if $f(r\cdot x)=r^{\lambda}f(x)$ for all
$x\in\mathbb{G}$. One can show, c.f. [31, Prop. 1.5 & Prop. 1.6], that for any
homogeneous norm on $\mathbb{G}$ there exists $\mu>0$ such that

  * •

$|xy|\leq\mu(|x|+|y|)$ for any $x,y\in\mathbb{G}$

  * •

$\left||xy|-|x|\right|\leq\mu|y|$ for any $x,y\in\mathbb{G}$ such that
$|y|\leq\frac{1}{2}|x|$ .

Furthermore all homogeneous norms are mutually equivalent and we may always
choose a homogeneous norm that is smooth away from $e\in\mathbb{G}$, [27, Sec.
3.1.6].

###  2.1 Derivatives and Polynomials

We identify $\mathfrak{g}$ with the left invariant vector fields
$\mathfrak{g}_{L}$ on $\mathbb{G}$ and write $\mathfrak{g}_{R}$ for the right
invariant vector fields. We write $X_{i}$ for the the basis elements as in
(2.2) seen as elements of $\mathfrak{g}_{L}$ and $Y_{i}$ for the basis of
$\mathfrak{g}_{R}$ satisfying $Y_{i}|_{e}=X_{i}|_{e}$ . Thus we can write

$X_{j}f(y)=\partial_{t}f(y\exp(tX_{j}))|_{t=0}\quad\text{ and }\quad
Y_{j}f(y)=\partial_{t}f(\exp(tY_{j})y)|_{t=0}\ $

for any smooth function $f\in C^{\infty}(\mathbb{G})$.

A map $P:\mathbb{G}\to\mathbb{R}$ is called a polynomial if
$P\circ\exp:\mathfrak{g}\rightarrow\mathbb{R}$ is a polynomial on
$\mathfrak{g}$.111Recall that the space of polynomial functions on
$\mathfrak{g}$ is canonically isomorphic to
$\bigoplus_{n}(\mathfrak{g}^{\ast})^{\otimes_{s}n}$ where $\otimes_{s}$
denotes the symmetric tensor product. Let $\zeta_{i}$ be the basis dual to the
basis $X_{i}$ of $\mathfrak{g}$. We set $\eta_{j}=\zeta_{j}\circ\exp^{-1}$,
which maps $\mathbb{G}$ to $\mathbb{R}$. Note that
$\eta=(\eta_{1},...,\eta_{d})$ forms a global coordinate system
222Occasionally we use the corresponding notation
$\frac{\partial}{\partial\eta_{i}}$. and furthermore any polynomial map on
$\mathbb{G}$ can be written in terms of coefficients $a_{I}\in\mathbb{R}$ as

$P=\sum_{I}a_{I}\eta^{I}$

with the sum running over a finite subset of $\mathbb{N}^{d}$ and where for a
multi-index $I=(i_{1},...,i_{d})\in\mathbb{N}^{d}$ we write
$\eta^{I}=\eta_{1}^{i_{1}}\cdot...\cdot\eta_{d}^{i_{d}}$. Define
$d(I)=\sum_{j}\mathfrak{s}_{j}i_{j}$ and $|I|=\sum_{j}i_{j}$, we call
$\max\\{d(I)\ :\ a_{I}\neq 0\\}$ the homogeneous degree and $\max\\{|I|\ :\
a_{I}\neq 0\\}$ the isotropic degree of $P$. For $a>0$, we denote by
$\mathcal{P}_{a}$ the space of polynomials of homogeneous degree strictly less
than $a$ and define $\triangle=\\{d(I)\in\mathbb{R}\ :\
I\in\mathbb{N}^{d}\\}$. 333We point out a possibly counter-intuitive quirk of
our definition; for $k\in\triangle$ the set $\mathcal{P}_{k}$ does not contain
polynomials of degree $k$ but only those of degree less than $k$.  We can
rewrite the group law on $\mathbb{G}$ explicitly in terms of
$\eta=(\eta_{1},...,\eta_{d})$.

######  Proposition 2.5.

For $j\in\\{1,\ldots,d\\}$ and multi-indices $I,\,J$ s.t.
$d(I)+d(J)=\mathfrak{s}_{j}$, there exist constants $C^{I,J}_{j}\in\mathbb{R}$
such that the following formula holds,

$\eta_{j}(xy)=\eta_{j}(x)+\eta_{j}(y)+\sum_{I,J\neq 0,\
d(I)+d(J)=\mathfrak{s}_{j}}C_{j}^{I,J}\eta^{I}(x)\eta^{J}(y)\ .$

###### Proof.

By the Baker–Campbell–Hausdorff formula (2.1) one has

$\eta_{j}(xy)=\eta_{j}(x)+\eta_{j}(y)+\sum_{|I|+|J|\geq
2}C^{I,J}_{j}\eta^{I}(x)\eta^{J}(y)\ .$

By setting either $x=e$ or $y=e$, we find that $C^{I,J}_{j}=0$ if $I=0$ or
$J=0$. Since furthermore $\eta_{j}((rx)(ry))=r^{\mathfrak{s}_{j}}\eta_{j}(xy)$
the claim follows. ∎

######  Remark 2.6.

Proposition 2.5 implies that for $\mathfrak{s}_{j}<2$ one has
$\eta_{j}(xy)=\eta_{j}(x)+\eta_{j}(y)$ while for $\mathfrak{s}_{j}=2$ one has
$\eta_{j}(xy)=\eta_{j}(x)+\eta_{j}(y)+\sum_{\mathfrak{s}_{k}=\mathfrak{s}_{l}=1}C_{j}^{k,l}\eta_{k}(x)\eta_{l}(y)$
.

######  Remark 2.7.

It is also noteworthy to realise that Proposition 2.5 implies that
$\mathcal{P}_{a}$ is invariant under right and left-translations. (This is not
true if one replaces homogeneous degree by isotropic degree, except if
$\mathbb{G}$ is abelian or $a=0$).

######  Remark 2.8.

If follows from Proposition 2.5 that one can write

$\eta^{K}(xy)=\eta^{K}(x)+\eta^{K}(y)+\sum_{I,J\neq 0,\
d(I)+d(J)=d(K)}C_{K}^{I,J}\eta^{I}(x)\eta^{J}(y)\ ,$

where the constants $C_{K}^{I,J}$ can be written in terms of the constants
$C_{j}^{I,J}$.

######  Lemma 2.9.

For $i,\,j\in\\{1,\ldots,d\\}$,

$X_{i}\eta_{j}=\delta_{i,j}+\sum_{I\neq 0,\
d(I)=\mathfrak{s}_{j}-\mathfrak{s}_{i}}C_{j}^{I,e_{i}}\eta^{I}\ ,$ (2.3)

where $e_{i}$ denotes the multi-index $(0,\ldots,0,1,0,\ldots,0)$ with the $1$
being in the $i$-th slot.

###### Proof.

This follows directly from Proposition 2.5, applying $X_{i}$ to the function

$y\mapsto\eta_{j}(xy)\ ,$

evaluating at $y=0$ and using the fact that
$X_{i}|_{0}=\frac{\partial}{\partial\eta_{i}}|_{0}$. ∎

######  Proposition 2.10 ([31, Prop. 1.26]).

One has $X_{j}=\sum P_{j,k}\left(\frac{\partial}{\partial\eta_{k}}\right)$
where

$P_{j,k}=\left\\{\begin{array}[]{ll}1&\mbox{if }k=j\\\ 0&\mbox{ if
}\mathfrak{s}_{k}\leq\mathfrak{s}_{j},k\neq j\end{array}\right.$

and $P_{j,k}$ is a homogeneous polynomial of degree
$\mathfrak{s}_{k}-\mathfrak{s}_{j}$ if $\mathfrak{s}_{k}>\mathfrak{s}_{j}$ .
The analogous statement holds for the vector fields $Y_{j}$,

For a multi-index $I=(i_{1},...,i_{d})\in\mathbb{N}^{d}$ we introduce the
notation $X^{I}=X_{1}^{i_{1}}...X_{d}^{i_{d}}$. Note that the order of the
composition matters since $\mathfrak{g}$ is not in general Abelian. It is a
well known fact that any left invariant differential operator on $\mathbb{G}$
can (uniquely) be written as a linear combination of
$\\{X^{I}\\}_{I\in\mathbb{N}^{d}}$. The next proposition follows as a direct
consequence.

######  Proposition 2.11 ([31, Prop. 1.30]).

The following maps from $\mathcal{P}_{a}\to\mathbb{R}^{\dim\mathcal{P}_{a}}$
are linear isomorphisms.

  1. 1.

$P\mapsto\left\\{\left(\frac{\partial}{\partial_{\eta}}\right)^{I}P(e)\right\\}_{d(I)<a}$
,

  2. 2.

$P\mapsto\left\\{X^{I}P(e)\right\\}_{d(I)<a}$ ,

  3. 3.

$P\mapsto\left\\{Y^{I}P(e)\right\\}_{d(I)<a}$ ,

The same holds replacing $e\in\mathbb{G}$ with any other point
$x\in\mathbb{G}$.

######  Definition 2.3.

For a smooth function $f:\mathbb{G}\to\mathbb{R}$, a point $x\in\mathbb{G}$
and $a\in(0,\infty)$, we define the left Taylor polynomial of homogeneous
degree (less than) $a$ of $f$ at $x$ to be the unique polynomial
$\operatorname{P}^{a}_{x}[f]\in\mathcal{P}_{a}$ such that
$X^{I}\operatorname{P}^{a}_{x}[f](e)=X^{I}f(x)$ for all $I$ such that
$d(I)<a$. The right Taylor Polynomial can be defined similarly, replacing
$X^{I}$ by $Y^{I}$.

######  Remark 2.12.

One observes that for $I\in\mathbb{N}^{d}$ and $a>d(I)$ one has that

$X^{I}\operatorname{P}^{a}_{x}[f]=\operatorname{P}^{a-d(I)}_{x}[X^{I}f]$

for all $f\in C^{\infty}(\mathbb{G})$. Indeed, this follows from the fact that
for any $d(J)<a-d(I)$ one has
$X^{J}(X^{I}\operatorname{P}^{a}_{x}[f])(e)=X^{J}(X^{I}f)(x)$, since
$X^{J}X^{I}$ can be written as a linear combination of $\\{X^{K}\\}_{d(K)<a}$.

######  Theorem 2.13 (Taylor’s Theorem).

For each $a>0$ and every $f\in C^{\infty}(\mathbb{G})$ it holds that

$f(xy)-\operatorname{P}^{a}_{x}[f](y)=\sum_{|I|\leq[a]+1,d(I)\geq
a}\int_{\mathbb{G}}X^{I}f(xz)Q^{I}(y,\mathop{}\\!\mathrm{d}z)\ ,$

where for each multi-index $I$ and $y\in\mathbb{G}$ the measure
$Q^{I}(y,\,\cdot\,)$ is supported on $B_{\beta^{[a]+1}|y|}(e)$ for some
$\beta>0$ depending only on $\mathbb{G}$ and satisfies
$\int_{\mathbb{G}}|Q^{I}(y,\mathop{}\\!\mathrm{d}z)|\lesssim|y|^{d(I)}$.

######  Remark 2.14.

Note that there is a more common version of Taylor’s Theorem on homogeneous
Lie groups, c.f. [31, Thm. 1.37], which states that the remainder satisfies
the estimate

$|f(xy)-\operatorname{P}^{a}_{x}[f](y)|\lesssim_{a}\sum_{|I|\leq[a]+1,d(I)\geq
a}|y|^{d(I)}\sup_{|z|\leq\beta^{[a]+1}|y|}|X^{I}f(xz)|\ $ (2.4)

and which follows as a trivial consequence of Theorem 2.13. For us the
explicit formula of the reminder is needed in order establish Schauder
estimates in Section 3.4.

######  Remark 2.15.

The analogous claim for the right Taylor polynomial,
$(\operatorname{P}_{R})^{a}_{x}[f]$, holds. In this case the analogue of (2.4)
reads

$|f(yx)-(\operatorname{P}_{R})^{a}_{x}[f](y)|\lesssim_{a}\sum_{|I|\leq[a]+1,d(I)\geq
a}|y|^{d(I)}\sup_{|z|\leq\beta^{[a]+1}|y|}|Y^{I}f(zx)|\ .$ (2.5)

######  Remark 2.16.

If we define
$\tilde{\operatorname{P}}^{a}_{x}[f](y):=\operatorname{P}^{a}_{x}[f](x^{-1}y)$,
then it follows that

$f(y)-\tilde{\operatorname{P}}^{a}_{x}[f](y)=\sum_{|I|\leq[a]+1,d(I)\geq
a}\int_{\mathbb{G}}X^{I}f(xz)Q^{I}(x^{-1}y,dz)$

and in particular that

$|f(y)-\tilde{\operatorname{P}}^{a}_{x}[f](y)|\lesssim_{a}\sum_{\begin{subarray}{c}|I|\leq[a]+1\\\
d(I)\geq a\end{subarray}}|x^{-1}y|^{d(I)}\sup_{|z|\leq
C|x^{-1}y|}|X^{I}f(xz)|\ .$

In the particular case when $f$ is compactly supported, one can rewrite this
for the rescaled function
$f^{\lambda}(x)=\frac{1}{\lambda^{|\mathfrak{s}|}}f(\lambda\cdot x)$ as

$\displaystyle|f^{\lambda}(y)-\tilde{\operatorname{P}}^{a}_{x}[f^{\lambda}](y)|$
$\displaystyle\lesssim_{a}\sum_{\begin{subarray}{c}|I|\leq[a]+1\\\ d(I)\geq
a\end{subarray}}\lambda^{-d(I)-|\mathfrak{s}|}|x^{-1}y|^{d(I)}\sup_{z\in\mathbb{G}}|X^{I}f(z)|$
(2.6) $\displaystyle\lesssim_{a,f}\sum_{\delta\geq
0}\lambda^{-(a+\delta)-|\mathfrak{s}|}|x^{-1}y|^{a+\delta}\ ,$

where the sum over $\delta$ runs over some finite subset of $[0,\infty)$.

######  Remark 2.17.

Given $f\in C^{\infty}(\mathbb{G})$, if we define $F\in
C^{\infty}(\mathbb{G}\times\mathbb{G})$ by setting $F(y,z)=f(z^{-1}y)$, we
find that

$\tilde{\operatorname{P}}^{a}_{x}[F(\,\cdot\,,z)](y)=\tilde{\operatorname{P}}^{a}_{z^{-1}x}[f](z^{-1}y)\
.$

######  Remark 2.18.

For any $p\in\mathcal{P}_{\gamma}$ one has

$\tilde{\operatorname{P}}_{x}^{\gamma}[p](y)=p(y)\ .$

######  Remark 2.19.

In the proof of Theorem 2.13, given below, we use the following elementary
observations

  * •

The map

$\displaystyle\Phi:\mathbb{R}^{d}$ $\displaystyle\to\mathbb{G},$
$\displaystyle(t_{1},...,t_{d})$
$\displaystyle\mapsto\exp(t_{1}X_{1})\cdot...\cdot\exp(t_{d}X_{d})$

is a diffeomorphism. To see this note that it is clearly a local
diffeomorphism since one has

$\displaystyle D\Phi|_{0}:T\mathbb{R}^{d}|_{0}\sim\mathbb{R}^{d}$
$\displaystyle\to T\mathbb{G}|_{0}\sim\mathfrak{g}$
$\displaystyle(t_{1},...,t_{d})$ $\displaystyle\mapsto\sum_{i}t_{i}X_{i}.$

Then since,

$\Phi(r^{\mathfrak{s}_{1}}t_{1},...,r^{\mathfrak{s}_{d}}t_{d})=r\cdot\Phi(t_{1},...,t_{d})$

it is seen to be a global diffeomorphism.

  * •

Setting for $t\in\mathbb{R}^{d}$,

$|t|_{\mathfrak{s}}:=\sum_{i=1}^{d}|t_{i}|^{1/\mathfrak{s}_{i}}\ ,$

one finds that there exists some $\beta=\beta(\mathbb{G})>0$ such that

$\frac{1}{\beta}|t|_{\mathfrak{s}}\lesssim|\Phi(t)|\lesssim\beta|t|_{\mathfrak{s}}$
(2.7)

uniformly over $t\in\mathbb{R}^{d}$.

  * •

By repeated use of the commutator, for any $I,J\in\mathbb{N}^{d}$ and
$i\in\mathbb{N}$ there exist coefficients $\lambda_{i,J,I}$ such that

$X_{i}X^{J}=\sum_{I}\lambda_{i,J,I}X^{I}$ (2.8)

and one has $\lambda_{i,J,I}=0$ whenever $d(J)+\mathfrak{s}_{i}\neq d(I)$ .

  * •

If $a\in\triangle$, then $m_{a}:=\max\\{|I|\ :\ d(I)\leq a\\}=[a]$ where $[a]$
denotes the integer part of $a$. This follows from the fact that $d(I)\leq a$
implies $m_{a}\leq[a]$ and since $([a],0,...,0)$ is in the set over which the
maximum is taken.

###### Proof of Theorem 2.13.

Define for $i\in\\{1,...,d\\}$ the following measure on $\mathbb{R}^{d}$ with
support on $B^{\mathfrak{s}}_{|t|_{\mathfrak{s}}}(0)$

$\tilde{Q}^{e_{i}}(t,ds)=\prod_{j<i}\delta_{t_{j}}(ds_{j})\cdot\mathbf{1}_{[0,t_{i}]}(ds_{i})\prod_{j>i}\delta_{0}(ds_{j})$

and define
${Q}^{e_{i}}(y,\,\cdot\,)=\Phi_{*}\tilde{Q}^{e_{i}}(\Phi^{-1}(y),\,\cdot\,)$
to be the push-forward measure. First we show that,

$f(xy)-f(x)=\sum_{i=1}^{n}\int_{\mathbb{G}}(X_{i}f)(xz){Q}^{e_{i}}(y,dz)\ .$
(2.9)

Indeed one can write for $y=y_{1}y_{2}\ldots y_{d}$, where
$y_{i}=\exp(t_{i}X_{i})$ and $t=(t_{1},...,t_{d})=\Phi^{-1}(y)$

$\displaystyle f(xy)-f(x)$ $\displaystyle=\sum_{i=1}^{d}f(xy_{1}\ldots
y_{i-1}y_{i})-f(xy_{1}...y_{i-1})$
$\displaystyle=\sum_{i=1}^{d}\int_{0}^{t_{i}}\partial_{s}f(xy_{1}\ldots
y_{i-1}\exp(sX_{i}))|_{s=s^{\prime}}ds^{\prime}$
$\displaystyle=\sum_{i=1}^{d}\int_{0}^{t_{i}}(X_{i}f)(xy_{1}\ldots
y_{i-1}\exp(sX_{i}))ds$
$\displaystyle=\sum_{i=1}^{d}\int_{\mathbb{R}^{d}}(X_{i}f)(x\Phi(s))\tilde{Q}^{e_{i}}(t,ds)$
$\displaystyle=\sum_{i=1}^{d}\int_{\mathbb{G}}(X_{i}f)(xz){Q}^{e_{i}}(y,dz)\
.$

Using the fact that $\Phi$ is a diffeomorphism one easily checks that
${Q}^{e_{i}}(y,dz)$ is supported on $B_{\beta|y|}(0)$ where $\beta$ is as in
(2.7), and satisfies $\int_{\mathbb{G}}|Q^{e_{i}}|(y,dz)\lesssim|y|^{d_{i}}$,
thus we have proved the theorem in the special case $a\in(0,1]$, also known as
mean-value theorem. We turn to the proof in the general case.

######  Claim 2.20.

Set $g(y)=f(xy)-\operatorname{P}^{a}_{x}[f](y)$, we note that $X^{J}g(e)=0$
for $d(J)<a$ while $X^{J}g(y)=X^{J}f(xy)$ for $d(J)\geq a$. We shall prove
that for any multi-index $J$ it holds that one can write

$X^{J}g(y)=\sum_{|I|\leq[a]+1,d(I)\geq
a}\int_{\mathbb{G}}X^{I}f(xz)Q^{I,J}(y,dz)\ ,$

where the measures $Q^{I,J}(x,\cdot)$ satisfy the following properties

  * •

$Q^{I,J}(x,\cdot)$ is supported on $B_{\beta^{m(J)}|y|}(e)\subset\mathbb{G}$
where $m(J)=\min\\{m\in\mathbb{N}:\ a-m<d(J)\\}\ ,$

  * •

$\int_{\mathbb{G}}|Q^{I,J}|(x,dy)\lesssim|x|^{d(I)-d(J)}$ .

We shall prove by induction on $m\in\\{1,...,\lceil a\rceil\\}$, that for
multi-indices $J$ satisfying $a-m\leq d(J)<a$ the claim holds.

  * •

The case $m=1$ follows from (2.8) and (2.9),

$\displaystyle X^{J}g(y)=X^{J}g(y)-X^{J}g(e)$
$\displaystyle=\sum_{i=1}^{d}\int_{\mathbb{G}}(X_{i}X^{J}g)(z){Q}^{e_{i}}(y,dz)$
$\displaystyle=\sum_{i=1}^{d}\int_{\mathbb{G}}\sum_{I\ :\
d(I)=d(J)+\mathfrak{s}_{i}}\lambda_{i,J,I}(X^{I}g)(z){Q}^{e_{i}}(y,dz)$
$\displaystyle=\sum_{i=1}^{d}\int_{\mathbb{G}}\sum_{I\ :\
d(I)=d(J)+\mathfrak{s}_{i}}\lambda_{i,J,I}(X^{I}f)(xz){Q}^{e_{i}}(y,dz)$
$\displaystyle=\sum_{|I|\leq[a]+1,d(I)\geq
a}\int_{\mathbb{G}}X^{I}f(xz)Q^{I,J}(y,dz)$

The properties of $Q^{I,J}(y,dz)$ follow from the corresponding properties of
${Q}^{e_{i}}(y,dz)$, completing the case $m=1$.

  * •

Suppose the claim holds for $m-1$. Then by the same argument as above

$\displaystyle X^{J}g(y)=$
$\displaystyle\sum_{i=1}^{d}\sum_{d(K)=d(J)+\mathfrak{s}_{i}}\lambda_{i,J,K}\int_{\mathbb{G}}(X^{K}g)(xz){Q}^{e_{i}}(y,dz)$
$\displaystyle=$
$\displaystyle\sum_{i,K:d(K)=d(J)+\mathfrak{s}_{i}<a}\lambda_{i,J,K}\int_{\mathbb{G}}(X^{K}g)(xz){Q}^{e_{i}}(y,dz)$
$\displaystyle\quad+\sum_{i,K:d(K)=d(J)+\mathfrak{s}_{i}\geq
a}\lambda_{i,J,K}\int_{\mathbb{G}}(X^{K}g)(xz){Q}^{e_{i}}(y,dz).$

We can apply the induction hypothesis to the terms in the first sum, since
$d(K)=d(J)+\mathfrak{s}_{i}\geq d(J)+1\geq a-(m-1)$ and find

$(X^{K}g)(xz)=\sum_{|I|\leq[a]+1,d(I)\geq
a}\int_{\mathbb{G}}X^{I}f(x\tilde{z})Q^{I,K}(z,d\tilde{z})$

and therefore

$\displaystyle X^{J}g(y)=$
$\displaystyle\sum_{i,K:d(K)=d(J)+\mathfrak{s}_{i}<a}\int_{\mathbb{G}}(X^{K}g)(xz){Q}^{e_{i}}(y,dz)$
$\displaystyle+\sum_{i,K:d(K)=d(J)+\mathfrak{s}_{i}\geq
a}\lambda_{i,J,K}\int_{\mathbb{G}}(X^{K}g)(xz){Q}^{e_{i}}(y,dz)$
$\displaystyle=$
$\displaystyle\sum_{i,K:d(K)=d(J)+\mathfrak{s}_{i}<a}\lambda_{i,J,K}\sum_{|I|\leq
k+1,d(I)\geq
a}\int_{\mathbb{G}}\int_{\mathbb{G}}X^{I}f(x\tilde{z})Q^{I,K}(z,d\tilde{z}){Q}^{e_{i}}(y,dz)$
$\displaystyle+\sum_{i,K:d(K)=d(J)+\mathfrak{s}_{i}\geq
a}\lambda_{i,J,K}\int_{\mathbb{G}}(X^{K}g)(xz){Q}^{e_{i}}(y,dz).$
$\displaystyle=$
$\displaystyle\sum_{i,K:d(K)=d(J)+\mathfrak{s}_{i}<a}\lambda_{i,J,K}\sum_{|I|\leq
k+1,d(I)\geq
a}\int_{G}X^{I}f(x\tilde{z})\big{(}Q^{I,K}*{Q}^{e_{i}}\big{)}(y,d\tilde{z})$
$\displaystyle+\sum_{i,K:d(K)=d(J)+\mathfrak{s}_{i}\geq
a}\lambda_{i,J,K}\int_{\mathbb{G}}(X^{K}g)(xz){Q}^{e_{i}}(y,dz)$
$\displaystyle=$ $\displaystyle\sum_{|I|\leq[a]+1,d(I)\geq
a}\int_{\mathbb{G}}X^{I}f(x\tilde{z})Q^{I,J}(y,d\tilde{z})\,$

where $Q^{I,J}$ is defined such that the last line holds true. Concerning the
measures $Q^{I,J}(y,\cdot)$, we note that

    1. 1.

$\operatorname*{\text{supp}}Q^{I,J}(y,\cdot)\subset B_{\beta^{m(J)}|y|}$ since

$\operatorname*{\text{supp}}\big{(}Q^{I,K}*{Q}^{e_{i}}(y,\cdot)\big{)}\subset
B_{\beta^{m(K)+1}|y|}$

as $\operatorname*{\text{supp}}Q^{I,K}(y,\cdot)\subset B_{\beta^{m(K)}|y|}(e)$
and $\operatorname*{\text{supp}}{Q}^{e_{i}}(y,dz)\subset B_{\beta|y|}(e)$.

    2. 2.

$\int_{\mathbb{G}}|Q^{I,J}(y,\cdot)|\lesssim|x|^{d(I)-d(J)}$ since

$\int|Q^{I,K}*{Q}^{e_{i}}|(y,dz)\leq\int|Q^{I,K}|(y,dz)\,\cdot\,\int|{Q}^{e_{i}}|(y,dz)\lesssim|y|^{d(I)-d(K)}\cdot|y|^{\mathfrak{s}_{i}}\
.$

This completes the proof. ∎

###  2.2 Distributions and Convolution

We define $\mathcal{D}(\mathbb{G}):=C^{\infty}_{c}(\mathbb{G})$ to be the
space of compactly supported smooth functions on $\mathbb{G}$ equipped with
the natural Fréchet topology. The space of distributions on $\mathbb{G}$ is
given by the dual $\mathcal{D}^{\prime}(\mathbb{G})$ of
$\mathcal{D}(\mathbb{G})$ and for $\xi\in\mathcal{D}^{\prime}(\mathbb{G})$ and
$\phi\in\mathcal{D}(\mathbb{G})$ we either write $\langle\phi,\xi\rangle$ or
$\xi(\phi)$ for the canonical pairing. Given $r\in\mathbb{R}_{\geq 0}$ we
introduce the following useful space of test functions

$\mathfrak{B}^{\lambda}_{r}(x):=\left\\{\phi\in
C^{\infty}_{c}(B_{\lambda}(x))\ :\
|X^{I}\phi|\leq\frac{1}{\lambda^{|\mathfrak{s}|+d(I)}}\ \forall I:d(I)\leq
r\right\\}\ .$

We will often use the shorthand $\mathfrak{B}_{r}:=\mathfrak{B}^{1}_{r}(e)$ .
Given $\phi\in\mathcal{D}(\mathbb{G})$ we extensively use the following
notations,

$\phi^{\lambda}_{x}(z):=\frac{1}{\lambda^{|\mathfrak{s}|}}\phi\left(\frac{1}{\lambda}\cdot(x^{-1}z)\right).$

Recalling that we use the notation $\mathop{}\\!\mathrm{d}x$ for the Haar
measure on $\mathbb{G}$ we define the norms,

$\|f\|_{L^{p}}:=\begin{cases}\left(\int_{\mathbb{G}}|f(x)|^{p}\mathop{}\\!\mathrm{d}x\right)^{1/p},&\text{
for }1\leq p<\infty,\\\
\operatorname*{ess\,sup}_{x\in\mathbb{G}}|f(x)|,&p=\infty.\end{cases}$

By left and right translation invariance of the Haar measure, for $f\in
L^{1}(\mathbb{G})$ it holds that

$\int_{\mathbb{G}}f(yx)\mathop{}\\!\mathrm{d}y=\int_{\mathbb{G}}f(xy)\mathop{}\\!\mathrm{d}y=\int_{\mathbb{G}}f(y)\mathop{}\\!\mathrm{d}y=\int_{\mathbb{G}}f(y^{-1})\mathop{}\\!\mathrm{d}y.$

See the discussion after [27, Thm. 1.1.1].

######  Remark 2.21.

From the scaling properties of $\operatorname{D}$ it follows that

$\mathfrak{B}^{\lambda}_{r}(x)=\left\\{\phi^{\lambda}_{x}\
:\phi\in\mathfrak{B}^{1}_{r}(e)\right\\}.$

One defines the convolution of two functions
$\phi,\psi:\mathbb{G}\to\mathbb{R}$ as

$\psi*\phi(x)=\int\psi(y)\phi(y^{-1}x)dy=\int\psi(xy^{-1})\phi(y)dy\ .$ (2.10)

Note that in general $\psi*\phi(x)\neq\phi*\psi(x)$ but still many of the
usual inequalities hold, in particular the Young inequality, c.f. [31, Prop.
1.18]. One can write

$\psi*\phi(x)=\int\psi(y)\phi_{y}(x)dy\ .$

We extend convolution to generalised function, i.e. elements of
$\mathcal{D}^{\prime}(\mathbb{G})$, by duality in the usual manner. From now
on we will use the notation $\tilde{\phi}(z):=\phi(z^{-1})$, one notes that
the following identities hold

$\displaystyle\langle f,g*\phi\rangle=\langle\tilde{g}*$ $\displaystyle
f,\phi\rangle=\langle f*\tilde{\phi},g\rangle$ (2.11)
$\displaystyle\widetilde{f*g}$ $\displaystyle=\tilde{g}*\tilde{f}$ (2.12)
$\displaystyle\psi^{\lambda}*\phi^{\lambda}=(\psi*\phi)^{\lambda}\quad$
$\displaystyle\text{and}\quad(\psi\ast\phi)_{x}=\psi_{x}\ast\phi.$

Recalling the notation $X^{I}=X_{1}^{i_{1}}...X_{d}^{i_{d}}$ for any multi-
index $I=(i_{1},...,i_{d})\in\mathbb{N}^{d}$, we introduce the analogous
notation $Y^{I}=Y_{d}^{i_{d}}....Y_{1}^{i_{1}}$ for the right invariant basis
vectors. Note the reversal in the composition. Since for any right invariant
vector field $Y$ and left invariant vector field $X$ such that $X|_{e}=Y|_{e}$
it holds that $\langle Xf,g\rangle=\langle f,Yg\rangle$ for $f,\,g\in
C^{\infty}_{c}(\mathbb{G})$, it follows from the definition of the convolution
that

  * •

$X^{I}(\psi*\phi)=\psi*(X^{I}\phi)$,

  * •

$Y^{I}(\psi*\phi)=(Y^{I}\psi)*\phi$,

  * •

$(X^{I}\psi)*\phi=\psi*Y^{I}\phi\ $.

######  Definition 2.4.

For $\alpha\in\mathbb{R}$ we define the space
$\mathcal{C}^{\alpha}(\mathbb{G})$ of $\alpha$-Holder continuous distributions
to be those $f\in\mathcal{D}^{\prime}(\mathbb{G})$ such that for every compact
set $\mathfrak{K}\subset\mathbb{G}$,

  * •

If $\alpha\leq 0$

$\|f\|_{\mathcal{C}^{\alpha}(\mathfrak{K})}=\sup_{x\in\mathfrak{K}}\sup_{\phi\in\mathfrak{B}_{\lceil{-\alpha\rceil}}}\sup_{\lambda\in(0,1)}\frac{|\langle
f,\phi^{\lambda}_{x}\rangle|}{\lambda^{\alpha}}<+\infty\ .$

  * •

If $\alpha>0$, there exists a continuous map
$\mathbb{G}\to\mathcal{P}_{\alpha},\ x\mapsto\tilde{P}_{x}$ such that

$\|f\|_{\mathcal{C}^{\alpha}(\mathfrak{K})}=\sup_{x\in\mathfrak{K}}\sup_{\phi\in\mathfrak{B}_{0}}\sup_{\lambda\in(0,1)}\frac{|\langle
f-\tilde{P}_{x},\phi_{x}^{\lambda}\rangle|}{\lambda^{\alpha}}+\sup_{x\in\mathfrak{K}}|\tilde{P}_{x}|_{\mathcal{P}_{\alpha}}<+\infty\
,$ (2.13)

where $|\,\cdot\,|_{\mathcal{P}_{\alpha}}$ denotes any norm on the finite
dimensional vector space $\mathcal{P}_{\alpha}$.

Furthermore, $C(\mathbb{G})$ denotes the space of continuous functions (note
that only the strict inclusion
$C(\mathbb{G})\subsetneq\mathcal{C}^{0}(\mathbb{G})$ holds).

######  Remark 2.22.

We note that as in [39], for $\alpha\in\triangle$ the
$\mathcal{C}^{\alpha}(\mathbb{G})$ spaces do not agree with the usual notion
of continuously $\alpha$-differentiable functions. This definition, however,
is more canonical in the context of regularity structures, c.f. Theorem 3.12.

The following proposition gives some useful properties that also mirror those
of Euclidean Hölder spaces.

######  Proposition 2.23.

Given any real number $\alpha\in\mathbb{R}$ the following holds.

  1. 1.

$\mathcal{C}^{\alpha}(\mathbb{G})\subset\mathcal{C}^{\beta}(\mathbb{G})$ for
every $\beta<\alpha$.

  2. 2.

For any multiindex $I$ the map $X^{I}:\
\mathcal{C}^{\alpha}(\mathbb{G})\to\mathcal{C}^{\alpha-d(I)}(\mathbb{G}),\quad
f\mapsto X^{I}f$ is well defined.

  3. 3.

If $\alpha>0$, then any distribution $f\in\mathcal{C}^{\alpha}(\mathbb{G})$
agrees444As usual, we say a distribution agrees with a continuous functions,
if it lies in the image of the canonical embedding
$C(\mathbb{G})\to\mathcal{D}^{\prime}(\mathbb{G})$. with a continuous
function.

Note that this proposition in particular implies that
$C^{\infty}(\mathbb{G})=\cap_{\alpha>0}\mathcal{C}^{\alpha}(\mathbb{G})$.

###### Proof.

Points 1 and 2 follow directly. For Point 3, denote by $\tilde{P}_{x}$ the
polynomial in Definition 2.4, we set $F=f-\tilde{P}_{\cdot}(\cdot)$, then for
any $\psi\in\mathfrak{B}_{0}$ such that $\int_{\mathbb{G}}\psi=c^{-1}>0$ and
smooth compactly supported function $g$ we find

$\displaystyle\langle F,g\rangle=c\lim_{\lambda\to 0}\langle
F,g*\psi^{\lambda}\rangle=c\lim_{\lambda\to 0}\left\langle
F,\int_{\mathbb{G}}g(y)\psi_{y}^{\lambda}dy\right\rangle=c\lim_{\lambda\to
0}\int_{\mathbb{G}}\langle F,\psi_{y}^{\lambda}\rangle g(y)dy=0\ ,$

where in the last step we used that since $y\to\tilde{P}_{y}$ is continuous we
have

$\langle F,\psi_{y}^{\lambda}\rangle=\langle
f-\tilde{P}_{y}(\,\cdot\,),\psi_{y}^{\lambda}\rangle+\int\big{(}\tilde{P}_{y}(z)-\tilde{P}_{z}(z)\big{)}\psi_{y}^{\lambda}(z)\,dz\to
0\ ,$

as $\lambda\to 0$. Thus we find that the distribution $f$ agrees with the
continuous function $x\mapsto\tilde{P}_{x}(x)$, concluding the proof. ∎

######  Remark 2.24.

Note that by Proposition 2.23 for $\alpha>0$ and any
$f\in\mathcal{C}^{\alpha}(\mathbb{G})$ we know that $X^{I}f$ agrees with a
continuous function whenever $d(I)<\alpha$. Thus, in particular Definition 2.3
of Taylor polynomials $\operatorname{P}^{\alpha}_{x}[f]$ extends to
$\mathcal{C}^{\alpha}(\mathbb{G})\supset C^{\infty}(\mathbb{G})$.

###  2.3 Discrete Subgroups

Recall that, given a discrete subgroup $\mathfrak{G}\subset\mathbb{G}$ acting
on $\mathbb{G}$ (say) on the left, then the quotient space
$S:=\mathbb{G}/\mathfrak{G}=\\{\mathfrak{G}x\ :\ x\in\mathbb{G}\\}$ is a
smooth manifold. Furthermore, the quotient map $\pi:\mathbb{G}\to S$ is a
smooth normal covering map , c.f. [51, Thm. 21.29] and the canonical
$\mathbb{G}$ right action

$S\times\mathbb{G}\to
S,\qquad(\mathfrak{G}x,x^{\prime})\mapsto\mathfrak{G}xx^{\prime}\ $

makes it a homogeneous space. We call $\mathfrak{G}$ a lattice if
$S=\mathbb{G}/\mathfrak{G}$ is a compact space. This is equivalent to $S$
carrying a finite $\mathbb{G}$ invariant measure, which we shall denote by
$d(\mathfrak{G}x)$, c.f. [60, Thm. 2.1]. Throughout this article we assume
that every homogeneous Lie group we work with carries a lattice, the following
theorem, [60, Thm. 2.12], gives sufficient conditions for this to be the case,
which all our examples satisfy.

######  Theorem 2.25.

Let $\mathbb{G}$ be a simply connected nilpotent Lie group and $\mathfrak{g}$
its Lie algebra. Then, $\mathbb{G}$ admits a lattice if and only if
$\mathfrak{g}$ admits a basis with respect to which the structure constants of
$\mathfrak{g}$ are rational.

Let us point out that there do exist nilpotent Lie groups that do not admit a
basis with respect to which the structure constants are rational, see [60,
Rem. 2.14] for an example.

Denote by $\pi^{\ast}:C(S)\to C(\mathbb{G})$ the pull-back under $\pi$. We
define a convolution map

$*_{S}:C^{\infty}(S)\times C_{c}^{\infty}(\mathbb{G})\to
C^{\infty}(S),\quad(f,\phi)\mapsto f*_{S}\phi$

as the unique map, such that the following diagram commutes

${C^{\infty}(S)\times
C_{c}^{\infty}(\mathbb{G})}$${C^{\infty}(S)}$${C^{\infty}(\mathbb{G})\times
C_{c}^{\infty}(\mathbb{G})}$${C^{\infty}(\mathbb{G})\
,}$$\scriptstyle{*_{S}}$$\scriptstyle{\pi^{\ast}\times\mathrm{id}}$$\scriptstyle{\pi^{\ast}}$$\scriptstyle{*}$

where convolution on the bottom row is convolution on the group as defined in
2.10. In order to check that this map is well defined, we use the following
terminology. A function $f\in C^{\infty}(\mathbb{G})$ is called left
$\mathfrak{G}$ periodic, if for every $x\in\mathbb{G}$ and $n\in\mathfrak{G}$
it holds that

$f(x)=f(nx)\ .$

Thus, we only need to check that for any $\phi\in C_{c}^{\infty}(\mathbb{G})$
and any left $\mathfrak{G}$ periodic function $f\in C^{\infty}(\mathbb{G})$,
the function $f*\phi$ is left $\mathfrak{G}$ periodic. Indeed for any
$n\in\mathfrak{G}$ and $x\in\mathbb{G}$, it holds that

$\displaystyle f*\phi(nx)$
$\displaystyle=\int_{\mathbb{G}}f(y)\phi(y^{-1}nx)dy$
$\displaystyle=\int_{\mathbb{G}}f(y)\phi((n^{-1}y)^{-1}x)dy$
$\displaystyle=\int_{\mathbb{G}}f(ny)\phi(y^{-1}x)dy$
$\displaystyle=\int_{\mathbb{G}}f(y)\phi(y^{-1}x)dy\ .$

By duality, one naturally extends the notion of convolution on $S$ to pairs,
$(\xi,\zeta)\in\mathcal{D}^{\prime}(S)\times\mathcal{D}^{\prime}_{c}(\mathbb{G})$,
where $\mathcal{D}^{\prime}_{c}(\mathbb{G})$ denotes distributions on
$\mathbb{G}$ with compact support.

###  2.4 Concrete Examples

In order to cement ideas we present two concrete examples of non-abelian,
homogeneous Lie groups, both of which are identified with fixed global charts.
These will be revisited in Section 4.4 when we discuss the heat operator on
the Heisenberg group and Kolmogorov type operators.

####  2.4.1 The Heisenberg group

Given $n\geq 1$ we define the Heisenberg group $\mathbb{H}^{n}$ as the set
$\mathbb{R}^{2n}\times\mathbb{R}$ with the group law,

$(x,y,z)(x^{\prime},y^{\prime},z^{\prime})=\left(x+x^{\prime},y+y^{\prime},z+z^{\prime}+\sum_{i=1}^{n}\left(x^{\prime}_{i}y_{i}-x_{i}y^{\prime}_{i}\right)\right).$

######  Remark 2.26.

Note that one may equally define the complex Heisenberg group on
$\mathbb{C}^{n}\times\mathbb{R}$ equipped with the group law,

$(u,z)(u^{\prime},z^{\prime})=\left(u+u^{\prime},z+z^{\prime}+\Im\sum_{i=1}^{n}u_{i}\bar{u}^{\prime}_{i}\right).$

One sees that these definitions are equivalent after identifying
$\mathbb{C}^{n}$ with $\mathbb{R}^{2n}$.

The origin $e=(0,0,0)$ is clearly the identity and for
$(x,y,z)\in\mathbb{R}^{2n}\times\mathbb{R}$ one has $(x,y,z)^{-1}=(-x,-y,-z)$.
The Lie algebra, $\mathfrak{h}$ of $\mathbb{H}^{n}$ is identified with
$\mathbb{R}^{2n+1}$ and spanned by the basis of left-invariant vector fields,

$A_{i}(x,y,z)=\partial_{x_{i}}+y_{i}\partial_{z},\quad
B_{i}(x,y,z)=\partial_{y_{i}}-x_{i}\partial_{z},\quad C(x,y,z)=\partial_{z}$

and equipped with the Lie bracket $[A,B]=AB-BA$. We observe that for any
$(x,y,z)\in\mathbb{H}^{n}$ and $i=1,\ldots,n$,

$[A_{i},B_{i}](x,y,z)=-2C.$ (2.14)

We say that a Lie algebra is graded if there exist vector spaces
$\\{W_{k}\\}_{k\geq 1}$ where only finitely many $W_{k}$ are non-zero,
$\mathfrak{g}=\bigoplus_{k=1}^{\infty}W_{k}$ and $[W_{i},W_{j}]\subset
W_{i+j}$.

######  Definition 2.5.

A homogeneous Lie group $\mathbb{G}$ is called stratified (or a Carnot group)
if its Lie algebra is graded and generated by $W_{1}$.

Due to (2.14), we see that if we equip $\mathbb{H}^{n}$ with the dilation map,

$\lambda\cdot(x,y,z):=(\lambda x,\lambda y,\lambda^{2}z),$

then $\mathbb{H}^{n}$ becomes a stratified group. Refer to [12, Sec. 3.3.6]
for a discussion of generalisations of this structure.

A simple example of a lattice on the Heisenberg group is the set of integer
vectors $(a,b,c)\in\mathfrak{H}^{n}:=\mathbb{Z}^{2n}\times\mathbb{Z}$ equipped
with the group law as defined above. Note that it is not a normal subgroup and
thus the quotient space $\mathbb{H}^{n}/\mathfrak{H}^{n}$ is not a group.
However, since it is a lattice the homogeneous space
$\mathbb{H}^{n}/\mathfrak{H}^{n}$ is compact, c.f. Subsection 2.3.

####  2.4.2 Matrix Exponential Groups

For $n\geq 1$, let $B$ be a rational $n\times n$ block matrix of the form,

$B=\begin{pmatrix}0&B_{1}&0&\cdots&0\\\ 0&0&B_{2}&\cdots&0\\\
\vdots&\vdots&\ddots&\ddots&\vdots\\\ 0&0&\cdots&0&B_{k}\\\
0&0&\cdots&0&0\end{pmatrix}$

with each $B_{i}$ a $p_{i-1}\times p_{i}$ block matrix of rank $p_{i}$, where
$n\geq p_{0}\geq p_{1}\geq\cdots\geq p_{k}$ and $\sum_{i=0}^{k}p_{i}=n$. Note
that this implies the zero blocks on the diagonal are all $p_{i}\times p_{i}$
square matrices. We can equip $\mathbb{R}\times\mathbb{R}^{n}$ with an
associated Lie structure by defining the group law

$(t,z)(s,z^{\prime}):=\left(t+s,z^{\prime}+\exp\left(sB^{\top}\right)z\right).$

To define the dilation we begin by decomposing according to the structure of
the block matrix $B$, writing

$\mathbb{R}^{n}=\mathbb{R}^{p_{0}}\times\cdots\times\mathbb{R}^{p_{n}}.$

Thus the action of $B^{\top}$ on
$z=(z_{0},...,z_{k})\in\mathbb{R}^{p_{0}}\times\cdots\times\mathbb{R}^{p_{k}}$
is written
$B^{\top}z=(B^{\top}_{1}z_{0},\,B^{\top}_{2}z_{1},...B^{\top}_{k}z_{k-1})$
etc. Then we set

$\lambda\cdot(t,z_{0},\ldots,z_{k}):=(\lambda^{2}t,\lambda
z_{0},\ldots,\lambda^{2k+1}z_{k}).$

The origin $e=(0,0)$ is again the identity element and
$(t,z)^{-1}=(-t,-\exp(-tB^{\top})z)$. The Lie algebra is spanned by the
translation invariant vector fields,

$X_{i}(t,z)=\partial_{z_{i}}\quad\text{for
}i=1,\ldots,p_{0}\quad\text{and}\quad
Y(t,z)=\partial_{t}-(Bz)\cdot\nabla=\partial_{t}-\sum_{i,j=p_{0}+1}^{n}b_{ij}z_{i}\partial_{z_{i}}.$

This defines the matrix exponential group associated to $B$ and one sees that
it is not stratified.

The simplest non-trivial example is to set $n=2$ and

$B=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix},$

so that using the suggestive notation
$(t,v,x)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}$, the group law becomes,

$(t,v,x)(s,w,y)=(t+s,v+w,x+y+sv).$

The equivalent scaling as above is to set
$\lambda\cdot(t,x,v)=(\lambda^{2}t,\lambda v,\lambda^{3}x)$. We refer to [55]
for more details and Section 4.4 below for a discussion of natural, second
order, hypoelliptic, linear operators associated to these groups.

##  3 Regularity Structures and Models

######  Definition 3.1.

A regularity structure is a pair
$\mathcal{T}=(\operatorname{T},\operatorname{G})$ consisting of the following
elements:

  1. 1.

A graded vector space $\operatorname{T}=\bigoplus_{\alpha\in
A}\operatorname{T}_{\alpha}$ where

     * •

the index set $A\subset\mathbb{R}$ is discrete, bounded from below and
contains zero,

     * •

each $\operatorname{T}_{\alpha}$ is finite dimensional with a fixed norm
$|\cdot|_{\alpha}$. We write
$\mathcal{Q}_{\alpha}:\operatorname{T}\to\operatorname{T}_{\alpha}$ for the
canonical projection,

     * •

$\operatorname{T}_{0}$ is isomorphic to $\mathbb{R}$ with a distinguished
element $\mathbf{1}\in\operatorname{T}_{0}$, such that $|\mathbf{1}|_{0}=1$.

The space $\operatorname{T}$ is called the structure space.

  2. 2.

A group $\operatorname{G}$ of linear operators acting on $\operatorname{T}$,
such that for every $\Gamma\in\operatorname{G}$ it holds that
$\Gamma|_{\operatorname{T}_{0}}$ is the identity map and for all
$\tau\in\operatorname{T}_{\alpha}$:

$\Gamma\tau-\tau\in\bigoplus_{\beta<\alpha}\operatorname{T}_{\beta}.$

The group $\operatorname{G}$ is called the structure group of $\,\mathcal{T}$.

A sector is a $\operatorname{G}$ invariant subspace $V\subset\operatorname{T}$
and if $V\neq\\{0\\}$ the regularity of the sector $V$ is defined as
$\min\\{\alpha\in A\ :\ V\cap\operatorname{T}_{\alpha}\neq\\{0\\}\\}$.

We make use of the natural shorthands,
$\operatorname{T}_{>\alpha},\,\operatorname{T}_{\geq\alpha},\,\operatorname{T}_{<\alpha},\,\operatorname{T}_{\leq\alpha}$
and the projections $\mathcal{Q}_{>\alpha}$ etc.

######  Definition 3.2.

Given a regularity structure $\mathcal{T}=(\operatorname{T},\operatorname{G})$
and $r\in\mathbb{N}$ such that $r>|\min A|$, a model for $\mathcal{T}$ is a
pair $M=(\Pi,\Gamma)$, consisting of

  * •

a realisation map $\Pi:\mathbb{R}^{d}\to
L(T,\mathcal{D}^{\prime}(\mathbb{G})),\ x\mapsto\Pi_{x}$, such that for any
compact set $\mathfrak{K}\subset\mathbb{G}$ one has
$\|\Pi\|_{\gamma;\mathfrak{K}}:=\sup_{x\in\mathfrak{K}}\|\Pi\|_{\gamma,x}<+\infty$
for all $\gamma>0$, where

$\|\Pi\|_{\gamma,x}:=\sup_{\zeta\in A\cap(-\infty,\gamma)}\sup_{\tau\in
T_{\zeta}}\sup_{\lambda<1}\sup_{\phi\in\mathfrak{B}_{r}}\frac{|\langle\Pi_{x}\tau,\phi^{\lambda}_{x}\rangle|}{|\tau|_{\zeta}\lambda^{\zeta}},$

  * •

a re-expansion map
$\Gamma:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\operatorname{G},\
(x,y)\mapsto\Gamma_{x,y}$, which satisfies the algebraic conditions

$\Pi_{x}\Gamma_{x,y}=\Pi_{y},\qquad\Gamma_{x,y}\Gamma_{y,z}=\Gamma_{x,z}$

and the analytic condition
$\|\Gamma\|_{\gamma;\mathfrak{K}}:=\sup_{x,y\in\mathfrak{K}:\
{|y^{-1}x|}<1}\|\Gamma\|_{x,y,\gamma}<+\infty$ for all $\gamma>0$, where

$\|\Gamma\|_{x,y,\gamma}:=\sup_{\zeta\in
A\cap(-\infty,\gamma)}\sup_{A\ni\beta<\zeta}\sup_{\tau\in
T_{\zeta}}\frac{|\Gamma_{x,y}\tau|_{\beta}}{{|y^{-1}x|}^{\zeta-\beta}|\tau|_{\zeta}}.$

Lastly, we denote by $\mathcal{M}_{\mathcal{T}}$ the space of models for
$\mathcal{T}$ equipped with the semi-norms
$\|M\|_{\gamma;\mathfrak{K}}:=\|\Pi\|_{\gamma;\mathfrak{K}}+\|\Gamma\|_{\gamma;\mathfrak{K}}$.

######  Remark 3.1.

In the rest of the article, often without any further comment, we will write

$r:=\min\left\\{n\in\mathbb{N}\ :\ n>|\min A|\right\\}\ .$

###  3.1 The Polynomial Regularity Structure

As an important example we describe the polynomial regularity structure and
canonical model which are crucial in the analysis of singular SPDEs on
homogeneous Lie groups. We define the structure space $\bar{\operatorname{T}}$
to be the symmetric tensor (Hopf) algebra generated by
$\\{\boldsymbol{\eta}_{i}\\}_{i=1}^{d}$, which we think of as abstract lifts
of the monomials $\\{\eta_{i}\\}_{i=1}^{d}$. For a multi-index $I$, we write
$\boldsymbol{\eta}^{I}:=\boldsymbol{\eta}_{1}^{i_{1}}\cdot...\cdot\boldsymbol{\eta}_{d}^{i_{d}}$
as well as $\mathbf{1}:=\boldsymbol{\eta}^{0}$. The group
$\bar{\operatorname{G}}$ is given by a copy of $\mathbb{G}$ acting by

$g\mapsto\Big{(}\Gamma_{g}:\boldsymbol{\eta}_{j}\mapsto\boldsymbol{\eta}_{j}+\sum_{I,J\neq
0,\ d(I)+d(J)=\mathfrak{s}_{j}}C_{j}^{I,J}\eta^{I}(g)\boldsymbol{\eta}^{J}\
+\eta_{j}(g)\mathbf{1}\Big{)}$

and
$\Gamma_{g}\boldsymbol{\eta}^{I}=\big{(}\Gamma_{g}\boldsymbol{\eta}\big{)}^{I}$
. The canonical polynomial model is defined by setting

$\Pi_{x}\boldsymbol{\eta}_{j}(z)=\eta_{j}(x^{-1}z)$

and

$\Gamma_{x,y}\boldsymbol{\eta}_{j}=\boldsymbol{\eta}_{j}+\sum_{I,J\neq 0,\
d(I)+d(J)=\mathfrak{s}_{j}}C_{j}^{I,J}\eta^{I}(y^{-1}x)\boldsymbol{\eta}^{J}\
+\eta_{j}(y^{-1}x)\mathbf{1}\ .$

These maps are extended multiplicatively to all of $\bar{\operatorname{T}}$.
Using Prop. 2.5 in the third equality,

$\displaystyle\Pi_{x}\Gamma_{x,y}\boldsymbol{\eta}_{j}(z)$
$\displaystyle=\Pi_{x}\Big{(}\boldsymbol{\eta}_{j}+\sum_{I,J\neq 0,\
d(I)+d(J)=\mathfrak{s}_{j}}C_{j}^{I,J}\eta^{I}(y^{-1}x)\boldsymbol{\eta}^{J}\
+\eta_{j}(y^{-1}x)\Big{)}(z)$ $\displaystyle=\eta_{j}(x^{-1}z)+\sum_{I,J\neq
0,\ d(I)+d(J)=\mathfrak{s}_{j}}C_{j}^{I,J}\eta^{I}(y^{-1}x)\eta^{J}(x^{-1}z)\
+\eta_{j}(y^{-1}x)$ $\displaystyle=\eta_{j}(y^{-1}xx^{-1}z)$
$\displaystyle=\eta_{j}(y^{-1}z)$
$\displaystyle=\Pi_{y}\boldsymbol{\eta}_{j}(z).$

####  3.1.1 Derivatives and abstract Polynomials

Next we lift the vector fields $X_{i}$ to abstract differential operators
$\mathcal{X}^{i}$ on $\bar{\operatorname{T}}$ of degree $\mathfrak{s}_{j}$,
c.f. Section 3.5. We set

$\mathcal{X}^{i}\boldsymbol{\eta}^{j}=\delta_{i,j}\mathbf{1}+\sum_{I\neq 0,\
d(I)=\mathfrak{s}_{j}-\mathfrak{s}_{i}}C_{j}^{I,e_{i}}\boldsymbol{\eta}^{I}$

and extend this definition to the whole regularity structure by the Leibniz
rule. The conditions to be an abstract differential operator are checked
directly:

  1. 1.

Indeed $\mathcal{X}^{i}:\operatorname{T}_{\alpha}\mapsto
T_{\alpha-\mathfrak{s}_{i}}$.

  2. 2.

By the Leibinz rule the second property is checked by showing that

$\mathcal{X}^{i}\Gamma_{x,y}\boldsymbol{\eta}^{j}=\Gamma_{x,y}\mathcal{X}^{i}\boldsymbol{\eta}^{j}\
.$ (3.1)

Note that
$\Pi_{x}\mathcal{X}^{i}\boldsymbol{\eta}^{j}=X_{i}\Pi_{x}\boldsymbol{\eta}^{j}\
,$ since

$\Pi_{x}\mathcal{X}^{i}\boldsymbol{\eta}_{j}(z)=\Pi_{x}\Big{(}\delta_{i,j}\mathbf{1}\,+\sum_{I\neq
0,\
d(I)=\mathfrak{s}_{j}-\mathfrak{s}_{i}}C_{j}^{I,e_{i}}\boldsymbol{\eta}^{I}\Big{)}=\delta_{i,j}\,+\sum_{I\neq
0,\ d(I)=\mathfrak{s}_{j}-\mathfrak{s}_{i}}C_{j}^{I,e_{i}}\eta^{I}(x^{-1}z)$

and using left invariance of the vector field $X_{i}$ as well as (2.3)

$X_{i}\Pi_{x}\boldsymbol{\eta}_{j}(z)=(X_{i}\eta_{j}(x^{-1}\cdot))(z)=(X_{i}\eta_{j})(x^{-1}z)=\delta_{i,j}+\sum_{I\neq
0,\ d(I)=\mathfrak{s}_{j}-\mathfrak{s}_{i}}C_{j}^{I,e_{i}}\eta^{I}(x^{-1}z)\
.$

Thus (3.1) follows from the injectivity of the maps $\Pi_{x}$ and the relation
$\Pi_{x}\Gamma_{x,y}=\Pi_{y}$.

######  Remark 3.2.

In addition to showing that $\mathcal{X}_{i}$ is an abstract differential
operator, we have shown that it also realizes $X_{i}$ for the polynomial
model.

####  3.1.2 Abstract Taylor Expansions

The following operator, which sends a smooth function to its abstract Taylor
expansion, will be used throughout the article.

######  Definition 3.3.

For $x\in\mathbb{G}$, we define the family of maps

$\boldsymbol{\operatorname{{P}}}^{a}_{x}:\mathcal{C}^{a}(\mathbb{G})\to\bar{\operatorname{T}}$

where for $a>0$ $\boldsymbol{\operatorname{{P}}}^{a}_{x}[f]$ is characterised
by the fact that
$\Pi_{e}\boldsymbol{\operatorname{{P}}}^{a}_{x}[f]=\operatorname{P}^{a}_{x}[f]\in\mathcal{P}_{a}$
and for $a\leq 0$ it satisfies $\boldsymbol{\operatorname{{P}}}^{a}_{x}[f]=0$.

######  Remark 3.3.

Note that the map $\boldsymbol{\operatorname{{P}}}^{a}_{x}$ is well defined by
Remark 2.24 . We shall almost exclusively use it, when its argument is a
smooth function $f\in
C^{\infty}(\mathbb{G})\subset\mathcal{C}^{a}(\mathbb{G})$.

######  Lemma 3.4.

Let $f\in C^{\infty}(\mathbb{G})$ and $a\geq 0$. Then for each multi-index $I$
the coefficient of $\boldsymbol{\eta}^{I}$ in
$\boldsymbol{\operatorname{{P}}}^{a}_{x}[f]$ is given by a linear combination
(depending only on $\mathbb{G}$) of $\\{X^{K}f(x)\\}_{d(K)\leq d(I)}$.
Furthermore, for $b\geq a$ one has

$\boldsymbol{\operatorname{{P}}}_{x}^{a}[f]=\mathcal{Q}_{<a}\boldsymbol{\operatorname{{P}}}^{b}_{x}[f]\
.$ (3.2)

###### Proof.

We first observe that (3.2) holds since the polynomial
$\Pi_{e}\mathcal{Q}_{\leq a}\boldsymbol{\operatorname{{P}}}^{b}_{x}[f]$
satisfies the property required in Definition 2.3. The first part of the lemma
follows by combining Proposition 2.11 with (3.2). ∎

######  Lemma 3.5.

In the setting of the above lemma one has

$|X^{I}\Pi_{e}\boldsymbol{\operatorname{{P}}}^{a}_{x}[f](z)|=|X^{I}\operatorname{P}^{a}_{x}[f](z)|\lesssim\sum_{d(I)\leq
d(J)<a}\sup_{d(J^{\prime})\leq d(J)}|X^{J^{\prime}}f(x)||z|^{d(J)-d(I)}\ .$

###### Proof.

Since the coefficient of ${\eta}^{J}$ in ${\operatorname{P}}^{a}_{x}[f]$ is
bounded by a universal multiple of $\sup_{d(J^{\prime})\leq
d(J)}|X^{J^{\prime}}f(x)|$ by Lemma 3.4 the claim follows using the fact that
$|X^{I}\Pi_{e}\boldsymbol{\eta}^{J}(z)|\lesssim|z|^{{d(J)-d(I)}}$ . ∎

The next lemma will be useful when proving Schauder estimates in Section 3.4,
since it allows us to circumvent the issue of having non-explicit Taylor
polynomials.

######  Lemma 3.6.

Let $\delta>0$ and $P\in\bar{\operatorname{T}}_{<\delta}$. If, for some
$\varepsilon\in[0,1]$,

$\left|(X^{I}\Pi_{e}P)(e)\right|\leq\varepsilon^{\delta-d(I)}$

for all $I\in\mathbb{N}^{d}$ such that $\delta>d(I)$, then it holds that
$|P|_{a}\lesssim_{\delta,\mathbb{G}}\varepsilon^{\delta-a}$ for all
$a\leq\delta$.

###### Proof.

We observe that $\operatorname{P}_{x}^{\delta}[\Pi_{e}P]=\Pi_{e}P$. Thus, the
claim follows directly from Lemma 3.4. ∎

###  3.2 Modelled Distributions

######  Definition 3.4.

Given a regularity structure
$\mathcal{T}=(\operatorname{T},\operatorname{G})$, a model $M=(\Pi,\Gamma)$
and $\gamma\in\mathbb{R}$ we define $\mathscr{D}_{M}^{\gamma}$ as the space of
all continuous maps $f:\mathbb{G}\to\operatorname{T}_{<\gamma}$, such that for
all $\zeta\in A\cap(-\infty,\gamma)$ the following bounds hold for every
compact set $\mathfrak{K}\subset\mathbb{G}$

$\sup_{x\in\mathfrak{K}}|f(x)|_{\zeta}<+\infty,\qquad\sup_{\begin{subarray}{c}x,y\in\mathfrak{K},\\\
0<{|y^{-1}x|}\leq
1\end{subarray}}\frac{|f(y)-\Gamma_{x,y}f(x)|_{\zeta}}{{|y^{-1}x|}^{\gamma-\zeta}}<+\infty\
.$

We define the corresponding semi-norm $\|\,\cdot\,\|_{\gamma;\mathfrak{K}}$ on
$\mathscr{D}_{M}^{\gamma}$ by setting,

$\|f\|_{\gamma;\mathfrak{K}}:=\sup_{x\in\mathfrak{K}}\sup_{\zeta<\gamma}|f(x)|_{\zeta}+\sup_{\zeta<\gamma}\sup_{\begin{subarray}{c}x,y\in\mathfrak{K},\\\
0<{|y^{-1}x|}\leq
1\end{subarray}}\frac{|f(y)-\Gamma_{x,y}f(x)|_{\zeta}}{{|y^{-1}x|}^{\gamma-\zeta}}.$

Given two models $M=(\Pi,\Gamma),\,\bar{M}=(\bar{\Pi},\bar{\Gamma})$, two
modelled distributions
$f\in\mathscr{D}^{\gamma}_{M},\,\bar{f}\in\mathscr{D}^{\gamma}_{\bar{M}}$ and
a compact set $\mathfrak{K}\subset\mathbb{G}$ we define the quantity,

$\|f;\bar{f}\|_{\gamma;\mathfrak{K}}:=\sup_{x\in\mathfrak{K}}\sup_{\zeta<\gamma}|f(x)-\bar{f}(x)|_{\zeta}+\sup_{\begin{subarray}{c}x,y\in\mathfrak{K}\\\
0<{|y^{-1}x|}\leq
1\end{subarray}}\sup_{\zeta<\gamma}\frac{|f(y)-\bar{f}(y)-\Gamma_{x,y}f(y)+\bar{\Gamma}_{y,x}\bar{f}(x)|_{\zeta}}{{|y^{-1}x|}^{\gamma-\zeta}}.$
(3.3)

For the set of modelled distributions taking values in a sector
$V\subset\operatorname{T}$ we write $\mathscr{D}^{\gamma}_{M}(V)$ and if the
regularity of the sector is $\alpha\in A$ we often use the shorthand
$\mathscr{D}^{\gamma}_{\alpha;M}$. We will freely drop the explicit dependence
on the model, image sector and its regularity when the context is clear.

####  3.2.1 Reconstruction

######  Theorem 3.7 (Reconstruction Theorem).

Let $\mathcal{T}=(\operatorname{T},\operatorname{G})$ be a regularity
structure with $\underline{\alpha}=\min A$. Then for every $\gamma>0$ and
$M=(\Pi,\Gamma)\in\mathcal{M}_{\mathcal{T}}$, there exists a unique,
continuous linear map
$\mathcal{R}_{M}:\mathscr{D}_{M}^{\gamma}\rightarrow\mathcal{C}^{\underline{\alpha}}$,
called the reconstruction operator associated to $M$, such that for any
compact $\mathfrak{K}$ and $\lambda\in(0,1]$

$\sup_{\psi\in\mathfrak{B}_{r}}|\langle\mathcal{R}_{M}f-\Pi_{x}f(x),\psi^{\lambda}_{x}\rangle|\lesssim_{\mathfrak{K}}\lambda^{\gamma}\|f\|_{\gamma;B_{2\mu\lambda}(x)}\|\Pi\|_{\gamma;B_{2\mu\lambda}(x)}\
,$ (3.4)

uniformly over $x\in\mathfrak{K}$, where $\mu\geq 1$ was defined in Remark
2.4. Furthermore the map $M\rightarrow\mathcal{R}_{M}$ is locally Lipschitz
continuous in the sense that for a second model
$\bar{M}=(\bar{\Pi},\bar{\Gamma})$ and
$\bar{f}\in\mathscr{D}_{\bar{M}}^{\gamma}$, for every $\lambda\in(0,1]$

$\displaystyle\sup_{\psi\in\mathfrak{B}_{r}}|\langle\mathcal{R}_{M}f-\mathcal{R}_{\bar{M}}\bar{f}-\Pi_{x}f(x)+\bar{\Pi}_{x}\bar{f}(x),\psi^{\lambda}_{x}\rangle|\lesssim_{\mathfrak{K}}\lambda^{\gamma}\bigg{(}$
$\displaystyle\|f;\bar{f}\|_{\gamma;B_{2\mu\lambda}(x)}\|\bar{\Pi}\|_{\gamma;B_{2\mu\lambda}(x)}$
(3.5)
$\displaystyle+\|f\|_{\gamma;B_{2\mu\lambda}(x)}\|\Pi-\bar{\Pi}\|_{\gamma;B_{2\mu\lambda}(x)}\bigg{)}\
,$

######  Remark 3.8.

Existence of a reconstruction operator for $\gamma\leq 0$ also holds, however,
uniqueness does not. In the case $\gamma=0$ the analogous bound to (3.4)
contains an additional logarithmic correction on the right hand side, c.f.
[15].

######  Remark 3.9.

It follows from the definition of the reconstruction operator, that if $f$ is
a modelled distributions with values in a sector $V\subset\operatorname{T}$ of
regularity $\alpha\geq\underline{\alpha}$, then $\mathcal{R}_{M}f\in
C^{\alpha}$.

In order to prove Theorem 3.7 we require two preliminary results. As in [32,
Sec. 13.4] we make the observation that for every $N\geq 0$ there exists a
$\rho:\mathbb{G}\rightarrow\mathbb{R}$, smooth and compactly supported in
$B_{1}(e)$ and such that

$\int\eta^{I}(x)\rho(x)\mathop{}\\!\mathrm{d}x=\delta_{I,0},\quad 0<d(I)\leq
N,$ (3.6)

where the $\delta$ here denotes the Kronecker delta applied componentwise to
the multi-index. For $\mathfrak{r}>1$ we define
$\rho^{(n)}(x):=\mathfrak{r}^{n|\mathfrak{s}|}\rho(\mathfrak{r}^{n}\cdot x)$
as well as,

$\rho^{(n,m)}=\rho^{(n)}\ast\rho^{(n+1)}\ast\cdots\ast\rho^{(m)},$

with the convention $\rho^{(n,n)}=\rho^{(n)}$. We then have the following
result.

######  Lemma 3.10.

If $\,\mathfrak{r}>\|\rho\|_{L^{1}}>1$, then there exists a smooth, compactly
supported function $\varphi^{(n)}=\lim_{m\rightarrow\infty}\rho^{(n,m)}$,
where the convergence takes place in $\mathcal{D}(\mathbb{G})$ and
$\operatorname*{\text{supp}}(\varphi^{(n)})\subset B_{C\mathfrak{r}^{-n}}$ for
$C=\frac{\mathfrak{r}}{\mathfrak{r}-1}$.

###### Proof.

First, note that since $\|\rho^{(m)}\|_{L^{1}}=\|\rho\|_{L^{1}}$ and
$\rho^{(m)}$ is supported on a ball of radius $\mathfrak{r}^{-m}$, it follows
from the mean value theorem on $\mathbb{G}$ (c.f. (2.4) with $a=0$) that

$|f\ast\rho^{(m)}(x)-f(x)|=\left|\int_{\mathbb{G}}(f(y)-f(x))\rho^{(m)}(y^{-1}x)\mathop{}\\!\mathrm{d}y\right|\lesssim\max_{i=1,...,d}\|Y_{i}f\|_{{L^{\infty}}}\|\rho\|_{L^{1}}\mathfrak{r}^{-m}\
.$

Secondly, since

$Y^{I}\rho^{(n,m)}=Y^{I}(\rho^{(n)}\ast\rho^{(n+1,m)})=(Y^{I}\rho^{(n)})\ast\rho^{(n+1,m)}$
(3.7)

one finds by applying Young’s convolution inequality $m$-times, c.f. Section
2.2, that

$\displaystyle\|Y_{i}\rho^{(n,m)}\|_{L^{\infty}}=\|(Y_{i}\rho^{(n)})\ast\rho^{(n+1,m)}\|_{L^{\infty}}\leq\|Y_{i}\rho^{(n)}\|_{\infty}\|\rho\|_{L^{1}}^{m-n-1}$
$\displaystyle\leq\mathfrak{r}^{n(\mathfrak{s}_{i}+|\mathfrak{s}|)}\|Y_{i}\rho\|_{L^{\infty}}\|\rho\|^{m-n-1}_{L^{1}}$
$\displaystyle\leq\mathfrak{r}^{2n|\mathfrak{s}|}\|Y_{i}\rho\|_{L^{\infty}}\|\rho\|^{m-n-1}_{L^{1}}$

and therefore

$\displaystyle\|\rho^{(n,m)}-\rho^{(n,m-1)}\|_{L^{\infty}}=\|\rho^{(n,m-1)}\ast\rho^{(m)}-\rho^{(n,m-1)}\|_{L^{\infty}}$
$\displaystyle\lesssim\max_{i=1,...,d}\|Y_{i}\rho^{(n,m-1)}\|_{{L^{\infty}}}\|\rho\|_{L^{1}}\mathfrak{r}^{-m}$
$\displaystyle\leq\mathfrak{r}^{2n|\mathfrak{s}|}\max_{i=1,...,d}\|Y_{i}\rho\|_{L^{\infty}}\|\rho\|^{m-n-2}_{L^{1}}\mathfrak{r}^{-m}$
$\displaystyle\leq\mathfrak{r}^{2n|\mathfrak{s}|}\frac{\max_{i=1,...,d}\|Y_{i}\rho\|_{L^{\infty}}}{\|\rho\|^{n+2}_{L^{1}}}\left(\frac{\|\rho\|_{L^{1}}}{\mathfrak{r}}\right)^{m}$

which is summable in $m$ since we assumed that
$\mathfrak{r}>\|\rho\|_{L^{1}}$. Thus we may write,

$\rho^{(n,m)}=\rho^{(n)}+\sum_{k=0}^{m-n-1}\rho^{(n,m-k)}-\rho^{(n,m-1-k)},$

and it follows that $\rho^{(n,m)}$ converges uniformly as $m\to+\infty$. Using
(3.7) we obtain convergence in $\mathcal{D}(\mathbb{G})$. It remains to check
the support of $\varphi^{(n)}$. For two functions $f_{1},f_{2}$ such that
$\operatorname*{\text{supp}}(f_{i})\in B_{r_{i}}$ one has
$\operatorname*{\text{supp}}(f_{1}\ast f_{2})\in B_{r_{1}+r_{2}}$, hence it
follows that $\varphi^{(n)}$ is supported in a ball of radius
$\sum_{m=n}^{\infty}\mathfrak{r}^{-m}=\frac{\mathfrak{r}^{-n}}{1-\mathfrak{r}^{-1}}$.
∎

It follows from the definitions that
$\varphi^{(n)}=\rho^{(n)}\ast\varphi^{(n+1)}$. We set

$\tilde{\rho}^{(m,n)}:=\tilde{\rho}^{(m)}\ast\tilde{\rho}^{(m-1)}\ast\cdots\ast\tilde{\rho}^{(n)}$

and using (2.12) we note that
$\tilde{\varphi}^{(m+1)}\ast\tilde{\rho}^{(m)}=\tilde{\varphi}^{(m)}$ and
$\tilde{\rho}^{(m,n)}\to\tilde{\varphi}^{(n)}$ in $\mathcal{D}(\mathbb{G})$ as
$m\to\infty$.

######  Lemma 3.11.

Let $\mathfrak{r}>1$ and $\rho$ be as in Lemma 3.10 Let $\alpha>0$ and
$\xi_{n}:\mathbb{G}\rightarrow\mathbb{R}$ be a sequence of functions such that
for every compact $\mathfrak{K}\subset\mathbb{G}$ there exists a
$C_{\mathfrak{K}}$ such that $\sup_{x\in\mathfrak{K}}|\xi_{n}(x)|\leq
C_{\mathfrak{K}}\mathfrak{r}^{\alpha n}$ and such that
$\xi_{n}=\xi_{n+1}\ast\tilde{\rho}^{(n)}$. Then the sequence $\xi_{n}$ is
Cauchy in $\mathcal{C}^{-\beta}(\mathbb{G})$ for every $\beta>\alpha$ and the
limit $\xi$ satisfies $\xi_{n}=\xi\ast\tilde{\varphi}^{(n)}$. If furthermore,
for some $x\in\mathbb{G}$ and $\gamma>-\alpha$ one has the bound

$|\xi_{n}(y)|\leq\mathfrak{r}^{\alpha
n}\left(|x^{-1}y|^{\gamma+\alpha}+\mathfrak{r}^{-(\gamma+\alpha)n}\right)$

uniformly over $n\geq 0$ and $y\in\mathbb{G}$ such that $|x^{-1}y|\leq 1$,
then $|\langle\xi,\psi^{\lambda}_{x}\rangle|\lesssim\lambda^{\gamma}$ for all
$\lambda\leq 1$ and $\phi\in\mathfrak{B}_{r}$, where $r=-[-\alpha]$.

###### Proof.

The proof follows along the same steps as that of [32, Lem. 13.24], but one
has to be careful since convolution is non-commutative in our setting. Let
$\lambda\in(0,1]$ and $\psi\in\mathfrak{B}_{r}$, we first establish the bound

$|\langle\xi_{n}-\xi_{n+1},\psi^{\lambda}_{x}\rangle|\lesssim\lambda^{-\beta}\mathfrak{r}^{(\alpha-\beta)n}$
(3.8)

uniformly over $\psi\in\mathfrak{B}_{r}$, $\lambda\in(0,1]$ and locally
uniformly over $x\in\mathbb{G}$. First observe the trivial bound,

$|\langle\xi_{n}-\xi_{n+1},\psi^{\lambda}_{x}\rangle|\leq\sup_{x\in\bar{\mathfrak{K}}}\left(|\xi_{n}(x)|+|\xi_{n+1}(x)|\right)\|\psi^{\lambda}_{x}\|_{L^{1}}\leq(1+\mathfrak{r}^{\alpha})C_{\bar{\mathfrak{K}}}\bar{C}\mathfrak{r}^{\alpha
n},$

where $\bar{C}:=\sup\\{\int|\psi^{\lambda}(x)|\mathop{}\\!\mathrm{d}x\ :\
\psi\in\mathfrak{B}_{r}\\}<+\infty$ . Hence, when
$\lambda\leq\mathfrak{r}^{-n}$ the bound (3.8) holds directly.

In the case $\mathfrak{r}^{-n}\leq\lambda$, using (2.11) we rewrite

$|\langle\xi_{n}-\xi_{n+1},\psi^{\lambda}_{x}\rangle|=\langle\xi_{n+1}\ast\tilde{\rho}^{(n)}-\xi_{n+1},\psi^{\lambda}_{x}\rangle|=|\langle\xi_{n+1},\psi_{x}^{\lambda}*\rho^{(n)}-\psi_{x}^{\lambda}\rangle|.$

By Taylor’s theorem and in particular Remark 2.16

$|\psi_{x}^{\lambda}(z)-\tilde{\operatorname{P}}^{r}_{y}[\psi_{x}^{\lambda}](z)|\,\lesssim_{r}\sum_{\delta>0}\lambda^{-(r+\delta)-|\mathfrak{s}|}|y^{-1}z|^{r+\delta}\
,$ (3.9)

where the sum runs over a finite set. It follows from (2.10), Remark 2.8 and
(3.6) that we have

$\tilde{\operatorname{P}}^{r}_{y}[\psi^{\lambda}_{x}]*\rho^{(n)}(z)=\int\tilde{\operatorname{P}}^{r}_{y}[\psi^{\lambda}_{x}](zw^{-1})\rho^{(n)}(w)dw=\int\tilde{\operatorname{P}}^{0}_{y}[\psi^{\lambda}_{x}](zw^{-1})\rho^{(n)}(w)dw=\psi_{x}^{\lambda}(y)$

and thus

$\psi_{x}^{\lambda}*\rho^{(n)}(y)-\psi_{x}^{\lambda}(y)=(\psi_{x}^{\lambda}-\tilde{\operatorname{P}}^{r}_{y}[\psi^{\lambda}_{x}])\ast\rho^{(n)}(y),$

which by (3.9) is bounded uniformly by a multiple of
$\sum_{\delta>0}\lambda^{-(r+\delta)-|\mathfrak{s}|}\mathfrak{r}^{-n(r+\delta)}\
$ and supported on a ball of radius $\lambda+\mathfrak{r}^{-n}\leq 2\lambda$.
Using the bound $|\xi_{n+1}|\lesssim_{\alpha}\mathfrak{r}^{\alpha n}$ we
conclude that

$|\langle\xi_{n+1},\psi_{x}^{\lambda}*\rho^{(n)}-\psi_{x}^{\lambda}\rangle|\lesssim\sum_{\delta>0}\lambda^{-(r+\delta)}\mathfrak{r}^{-n(r+\delta)}\mathfrak{r}^{\alpha
n}\lesssim\lambda^{-\beta}\mathfrak{r}^{(\alpha-\beta)n}$

where we used $\mathfrak{r}^{-n}\leq\lambda$ and without loss of generality
assumed that $r\geq\beta$ in the last line. Hence $\\{\xi_{n}\\}_{n\geq 1}$ is
Cauchy in $\mathcal{C}^{-\beta}(\mathbb{G})$ and for any test function,

$\langle\xi_{n},\psi\rangle=\langle\xi_{n+1},\psi*\rho^{(n)}\rangle=\langle\xi_{m+1},\psi*\rho^{(n,m)}\rangle=\langle\xi,\psi*\varphi^{(n)}\rangle,$

showing that $\xi_{n}=\xi*\tilde{\varphi}^{(n)}$.

To prove the second claim, for any test function $\psi\in\mathfrak{B}_{r}$,
$\lambda>0$ and $x\in\mathbb{G}$ we write,

$\langle\xi,\psi^{\lambda}_{x}\rangle=\langle\xi_{n},\psi^{\lambda}_{x}\rangle+\sum_{k\geq
n}\langle\xi_{k+1}-\xi_{k},\psi^{\lambda}_{x}\rangle,$

where $n$ is chosen so that
$\lambda\in[\mathfrak{r}^{-(n+1)},\mathfrak{r}^{-n}]$ and as a consequence

$|\langle\xi_{n},\psi^{\lambda}_{x}\rangle|\leq\lambda^{-|\mathfrak{s}|}\mathfrak{r}^{\alpha
n}\int_{B_{\lambda}(x)}\left(|x^{-1}y|^{\gamma+\alpha}+\mathfrak{r}^{-(\gamma+\alpha)n}\right)\mathop{}\\!\mathrm{d}y,\lesssim\lambda^{\gamma+\alpha}\mathfrak{r}^{\alpha
n}+\mathfrak{r}^{-\gamma n}\lesssim\lambda^{\gamma}.$

To bound the summands $\langle\xi_{k+1}-\xi_{k},\varphi^{\lambda}_{x}\rangle$
we proceed as in the proof of the first claim to find that

$\displaystyle|\langle\xi_{k}-\xi_{k+1},\psi^{\lambda}_{x}\rangle|$
$\displaystyle=|\langle\xi_{k+1},\psi^{\lambda}_{x}*\rho^{(k)}-\psi^{\lambda}_{x}\rangle|$
$\displaystyle=|\langle\xi_{k+1},(\psi^{\lambda}_{x}-\tilde{\operatorname{P}}^{r}_{x}[\psi^{\lambda}_{x}]))*\rho^{(k)}\rangle|$
$\displaystyle\lesssim\sum_{\delta>0}\lambda^{-(r+\delta)-|\mathfrak{s}|}\mathfrak{r}^{-k(r+\delta)}\int_{B_{2\lambda}(x)}|\xi_{k+1}(y)|\mathop{}\\!\mathrm{d}y$
$\displaystyle\lesssim\sum_{\delta>0}\lambda^{-(r+\delta)-|\mathfrak{s}|}\mathfrak{r}^{k(\alpha-r-\delta)}\left(\int_{B_{2\lambda}(x)}|x^{-1}y|^{\gamma+\alpha}\mathop{}\\!\mathrm{d}y+\mathfrak{r}^{-(\gamma+\alpha)(k+1)}\lambda^{|\mathfrak{s}|}\right)$
$\displaystyle\lesssim\sum_{\delta>0}\left(\lambda^{\gamma+\alpha-r-\delta}\mathfrak{r}^{k(\alpha-r-\delta)}+\lambda^{-(r+\delta)}\mathfrak{r}^{-k(\gamma+r+\delta)}\right)$

where the sum in $\delta$ is again over a finite set. Since $r+\delta>\alpha$
the quantity on the left is summable over $k\geq n$ and is of order
$\lambda^{\gamma}$, concluding the proof. ∎

We are now ready to proof Theorem 3.7.

###### Proof of Theorem 3.7.

For $m>0$ we first define the operators
$\mathcal{R}^{(m,m)}:\mathscr{D}^{\gamma}\rightarrow C(\mathbb{G})$ by
setting,

$\left(\mathcal{R}^{(m,m)}f\right)(y):=\left(\Pi_{y}f(y)\ast\tilde{\varphi}^{(m)}\right)(y)=\langle\Pi_{y}f(y),{\varphi}^{(m)}_{y}\rangle.$

We then set, for $n<m$,

$\mathcal{R}^{(m,n)}f=\mathcal{R}^{(m,m)}f\ast\tilde{\rho}^{(m-1,n)}$

and recalling that
$\tilde{\varphi}^{(m+1)}\ast\tilde{\rho}^{(m)}=\tilde{\varphi}^{(m)}$ we find

$\displaystyle\mathcal{R}^{(m,n)}f-\mathcal{R}^{(m+1,n)}f$
$\displaystyle=\mathcal{R}^{(m,m)}f\ast\tilde{\rho}^{(m-1,n)}-\mathcal{R}^{(m+1,m+1)}f\ast\tilde{\rho}^{(m,n)}$
$\displaystyle=\big{(}\mathcal{R}^{(m,m)}f-\mathcal{R}^{(m+1,m+1)}f\ast\tilde{\rho}^{(m)}\big{)}\ast\tilde{\rho}^{(m-1,n)}\
.$

Using the identity

$(F\ast\tilde{\phi})(x)=\langle F,\phi_{x}\rangle=\int F(y)\phi_{x}(y)dy\ ,$

it follow by a straightforward computation that

$\displaystyle\big{(}\mathcal{R}^{(m,n)}f-\mathcal{R}^{(m+1,n)}f\big{)}(x)$
$\displaystyle=\int\big{(}\mathcal{R}^{(m,m)}f-\mathcal{R}^{(m+1,m+1)}f\ast\tilde{\rho}^{(m)}\big{)}(y){\rho}_{x}^{(m-1,n)}(y)dy$
$\displaystyle=\int\int\langle\Pi_{y}f(y)-\Pi_{z}f(z),{\varphi}_{z}^{(m+1)}\rangle{\rho}_{y}^{(m)}(z)dz{\rho}_{x}^{(m-1,n)}(y)dy\
.$

Therefore, using the fact that $\Pi_{z}=\Pi_{y}\Gamma_{yz}$, we have,

$\big{(}\mathcal{R}^{(m,n)}f-\mathcal{R}^{(m+1,n)}f\big{)}(x)=\int\int\langle\Pi_{y}(f(y)-\Gamma_{yz}f(z)),{\varphi}_{z}^{(m+1)}\rangle{\rho}_{y}^{(m)}(z)dz{\rho}_{x}^{(m-1,n)}(y)dy.$

Then successively applying the facts that,

  * •

$\sup_{y,\,z\in
B_{2\mu\mathfrak{r}^{-m}}(0)}|\langle\Pi_{y}\tau,\varphi^{(m+1)}_{z}|\lesssim\|\Pi\|_{\gamma;B_{2\mu\mathfrak{r}^{-m}}(0)}\mathfrak{r}^{-\alpha
m}|\tau|_{\alpha}$ for $\tau\in T_{\alpha}$,

  * •

$\|f(y)-\Gamma_{yz}f(z)\|_{\alpha}\lesssim\|f\|_{\gamma;B_{\mathfrak{r}^{-m}}(y)}\mathfrak{r}^{(\alpha-\gamma)m}$
uniformly over $|y^{-1}z|\lesssim\mathfrak{r}^{-m}$

  * •

$\|\rho_{x}^{(n,m-1)}\|_{L^{1}}\lesssim 1$ uniformly over $m>n\geq 0$,

we establish the bound,

$\|\mathcal{R}^{(m,n)}f-\mathcal{R}^{(m+1,n)}f\|_{L^{\infty}(\mathfrak{K})}\lesssim\|\Pi\|_{\gamma,\bar{\mathfrak{K}}}\|f\|_{\gamma;\bar{\mathfrak{K}}}\mathfrak{r}^{-\gamma
m}\ ,$

uniformly over $m\geq n\geq 0$, where $\bar{\mathfrak{K}}$ denotes the two
fattening of the set $\mathfrak{K}$. It follows directly from the definition
and properties of a model that we also have the bound,

$\|\mathcal{R}^{(n,n)}f\|_{L^{\infty}(\mathfrak{K})}\lesssim\|\Pi\|_{\gamma,\bar{\mathfrak{K}}}\|f\|_{\gamma;\bar{\mathfrak{K}}}\mathfrak{r}^{-\underline{\alpha}n},$
(3.10)

where $\underline{\alpha}=\min A$. It follows that $\mathcal{R}^{(m,n)}f$
converges uniformly on compacts as $m\rightarrow\infty$ to
$\mathcal{R}^{(n)}f$ which also satisfies the bound(3.10). Since it also holds
that for every $m\geq n+1$,

$\mathcal{R}^{(m,n)}f=\mathcal{R}^{(m,m)}f\ast\tilde{\rho}^{(m-1,n)}=\mathcal{R}^{(m,m)}f\ast\tilde{\rho}^{(m-1,n+1)}\ast\tilde{\rho}^{(n)}=\mathcal{R}^{(m,n+1)}f\ast\tilde{\rho}^{(n)}\
,$

we find that

$\mathcal{R}^{(n)}f=\mathcal{R}^{(n+1)}f\ast\tilde{\rho}^{(n)}\ .$

Therefore we may apply Lemma 3.11 to see that there exists a limit
$\mathcal{R}f:=\lim_{n\rightarrow\infty}\mathcal{R}^{(n)}f$.

With validity of the limit established we now turn to show the bounds (3.4)
and (3.5); this requires us to keep more careful track of the underlying sets
in the proof. We begin with (3.4), first noting that if we define
$f_{x}(y):=\Gamma_{y,x}f(x)$ then one has
$\mathcal{R}^{(n,n)}f_{x}=\Pi_{x}f(x)\ast\tilde{\varphi}^{(n)}$ so that (3.4)
can be written as the claim that for all $\lambda\in(0,1]$,

$\sup_{\psi\in\mathfrak{B}_{r}}|\langle\mathcal{R}(f-f_{x}),\psi^{\lambda}_{x}\rangle|\lesssim_{\mathfrak{K}}\lambda^{\gamma}\|f\|_{\gamma;B_{2\lambda}(x)}\|\Pi\|_{\gamma;B_{2\lambda}(x)}\
.$ (3.11)

Using that
$|(f-f_{x})(z)|_{\alpha}\lesssim\|f\|_{\gamma;B_{{|z^{-1}x|}}(x)}{|z^{-1}x|}^{\gamma-\alpha}$
for $x,\,z\in\mathfrak{K}$, it follows that for all $y\in B_{\lambda}(x)$ one
has

$\displaystyle|(\mathcal{R}^{(n,n)}(f-f_{x})(y)|=|\langle\Pi_{y}(f(y)-f_{x}(y)),\varphi^{(n)}_{y}\rangle|$
$\displaystyle=|\langle\Pi_{y}(f(y)-\Gamma_{y,x}f(x)),\varphi^{(n)}_{y}\rangle|$
$\displaystyle\lesssim\|\Pi\|_{\gamma;B_{\lambda}(y)}\|f\|_{\gamma;B_{C\mathfrak{r}^{-n}}(y)}\sum_{\underline{\alpha}\leq\alpha\leq\gamma}\mathfrak{r}^{-\alpha
n}|y^{-1}x|^{\gamma-\alpha}$
$\displaystyle\lesssim\|\Pi\|_{\gamma;B_{\lambda}(y)}\|f\|_{\gamma;B_{C\mathfrak{r}^{-n}}(y)}\mathfrak{r}^{-\underline{\alpha}n}({|y^{-1}x|}^{\gamma-\underline{\alpha}}+\mathfrak{r}^{(\underline{\alpha}-\gamma)n}),$
(3.12)

where $C(\mathfrak{r}):=\frac{\mathfrak{r}}{\mathfrak{r}-1}$ is as in Lemma
3.10. Given $n>n_{0}(\lambda)$ sufficiently larger, we have a uniform bound
$C\mathfrak{r}^{-n}\leq\lambda$. By the convergence of $\mathcal{R}^{(n,n)}$
in the first exponent, it follows that $\mathcal{R}^{(n)}$ satisfies the same
bound and so inspecting the proof of the second half of Lemma 3.11, in
particular noticing that we integrate the above estimate over $y\in
B_{\lambda}(x)$, we conclude that (3.11) holds for the limit $\mathcal{R}$.

Using the obvious notation we can also rewrite (3.5) as

$\displaystyle\sup_{\psi\in\mathfrak{B}_{r}}|\langle\mathcal{R}(f-f_{x})-\bar{\mathcal{R}}(\bar{f}-\bar{f}_{x}),\psi^{\lambda}_{x}\rangle|\lesssim\lambda^{\gamma}$
$\displaystyle\big{(}\|f;\bar{f}\|_{\gamma;B_{2\lambda}(x)}\|\bar{\Pi}\|_{\gamma;B_{2\lambda}(x)}$
$\displaystyle+\|f\|_{\gamma;B_{2\lambda}(x)}\|\Pi-\bar{\Pi}\|_{\gamma;B_{2\lambda}(x)}\big{)}\
,$

uniformly over $\lambda\in(0,1]$. This is seen very similarly to (3.11) but
using this time that for $n$ large enough,

$\displaystyle|\mathcal{R}^{(n,n)}(f-f_{x})-\bar{\mathcal{R}}^{(n,n)}(\bar{f}-\bar{f}_{x})(y)|$
$\displaystyle=|\langle\Pi_{y}(f(y)-\Gamma_{x,y}f(x))-\bar{\Pi}_{y}(\bar{f}(y)-\bar{\Gamma}_{y,x}\bar{f}(x)),\varphi^{(n)}_{y}\rangle|$
$\displaystyle=|\langle\Pi_{y}(f(y)-\Gamma_{x,y}f(x)-\bar{f}(y)+\bar{\Gamma}_{y,x}\bar{f}(x))+(\Pi_{y}-\bar{\Pi}_{y})(\bar{f}(y)-\bar{\Gamma}_{y,x}\bar{f}(x)),\varphi^{(n)}_{y}\rangle|$
$\displaystyle\lesssim\big{(}\|\Pi\|_{\gamma;B_{\lambda}(y)}\|f;\bar{f}\|_{\gamma;B_{\lambda}(y)}+\|\Pi-\bar{\Pi}\|_{\gamma;B_{\lambda}(y)}\|\bar{f}\|_{\gamma;B_{\lambda}(y)}\big{)}\sum_{\underline{\alpha}\leq\alpha\leq\gamma}\mathfrak{r}^{-\alpha
n}{|y^{-1}x|}^{\gamma-\alpha}\ .$

It remains to show that the reconstruction map is unique for $\gamma>0$ and
that $\mathcal{R}f$ is indeed an element of $C^{\underline{\alpha}}$. This is
done exactly as in [39, Sec. 3]. ∎

####  3.2.2 Functions as Modelled Distributions

Let $\bar{\mathcal{T}}=(\bar{\operatorname{T}},\bar{\operatorname{G}})$ be the
polynomial regularity structure equipped with the polynomial model. We show
that in this setting the reconstruction theorem and its inverse map Taylor
polynomials to Hölder functions and vice versa.

######  Theorem 3.12.

For $\gamma>0$ the reconstruction operator is an isomorphism between
$\mathscr{D}^{\gamma}(\mathbb{G})$ and $\mathcal{C}^{\gamma}(\mathbb{G})$. In
particular the inverse of the reconstruction map is given by

$\mathcal{C}^{\gamma}(\mathbb{G})\ni
f(\cdot)\mapsto\boldsymbol{\operatorname{{P}}}^{\gamma}_{(\,\cdot\,)}[f]\in\mathscr{D}^{\gamma}(\mathbb{G}),$

where $\boldsymbol{\operatorname{{P}}}^{a}$ is defined in Definition 3.3

###### Proof.

Given a modelled distribution in $\mathscr{D}^{\gamma}$ and since we are
working with the polynomial regularity structure equipped with its canonical
model, it follows directly from the bound satisfied by the image of the
reconstruction operator, i.e. Equation(3.4), and the definition of
$\mathcal{C}^{\gamma}$ in (2.13) that the reconstruction operator is a map
$\mathscr{D}^{\gamma}(\mathbb{G})\to\mathcal{C}^{\gamma}(\mathbb{G})$.
Continuity also follows from (3.4) and by linearity.

To see the other direction recall that given a Hölder continuous distribution
$f\in\mathcal{C}^{\gamma}(\mathbb{G})$ by Proposition 2.23 the
$\\{X^{I}f\\}_{d(I)<\gamma}$ are actually functions and, in particular, that
$\tilde{P}_{x}=\Pi_{x}\boldsymbol{\operatorname{{P}}}^{[\gamma]}_{x}[f]$.
Therefore, for $\lambda=|x^{-1}y|_{\mathbb{G}}$

$\left|\Pi_{x}\big{(}\boldsymbol{\operatorname{{P}}}^{[\gamma]}_{x}[f]-\Gamma_{x,y}\boldsymbol{\operatorname{{P}}}^{[\gamma]}_{y}[f]\big{)}(\psi^{\lambda}_{x})\right|=|\langle\tilde{P}_{x}-f,\psi^{\lambda}_{x}\rangle|+|\langle
f-\tilde{P}_{y},\psi^{\lambda}_{x}\rangle|\lesssim\lambda^{\gamma}\ $ (3.13)

uniformly in $\psi\in\mathfrak{B}_{r}$. On the other hand, writing
$\boldsymbol{\operatorname{{P}}}^{[\gamma]}_{x}[f]-\Gamma_{x,y}\boldsymbol{\operatorname{{P}}}^{[\gamma]}_{y}[f]=\sum_{d(I)<\gamma}c^{I}_{x,y}\boldsymbol{\eta}^{I}$,
we find that

$\Pi_{x}\big{(}\boldsymbol{\operatorname{{P}}}^{[\gamma]}_{x}[f]-\Gamma_{x,y}\boldsymbol{\operatorname{{P}}}^{[\gamma]}_{y}[f]\big{)}(\psi^{\lambda}_{x})=\sum_{d(I)<\gamma}c^{I}_{x,y}\Pi_{e}\boldsymbol{\eta}^{I}(\psi^{\lambda})=\sum_{d(I)<\gamma}c^{I}_{x,y}\lambda^{d(I)}\Pi_{e}\boldsymbol{\eta}^{I}(\psi)\
.$

Using that $\mathcal{P}_{\gamma}$ is a finite dimensional vector space and the
surjectivity of the linear map

$C_{c}^{\infty}(B_{1}(e))\to\mathbb{R}^{\dim\mathcal{P}_{\gamma}},\quad\psi\mapsto\\{\Pi_{e}\boldsymbol{\eta}^{I}(\psi)\\}_{d(I)<\gamma}$

we find that

$\sum_{d(I)<\gamma}|c^{I}_{x,y}|\lambda^{d(I)}\lesssim_{\gamma}\sup_{\psi\in\mathfrak{B}_{r}}\left|\sum_{d(I)<\gamma}c^{I}_{x,y}\lambda^{d(I)}\Pi_{e}\boldsymbol{\eta}^{I}(\psi)\right|\
.$ (3.14)

Together, (3.13) and (3.14) imply that
$|c^{I}_{x,y}|\lesssim\lambda^{\gamma-d(I)}$, we may then apply Lemma 3.6 to
see that
$\boldsymbol{\operatorname{{P}}}^{[\gamma]}_{(\,\cdot\,)}[f]\in\mathscr{D}^{\gamma}$
. ∎

####  3.2.3 Local Reconstruction

We will require the following further refinement of the reconstruction theorem
which is an analogue in our case of [39, Prop. 7.2].

######  Proposition 3.13.

In the setting of Theorem 3.7 one has the improved bound,

$\sup_{\psi\in\mathfrak{B}_{m}}|(\mathcal{R}f-\Pi_{x}f(x))(\psi^{\lambda}_{x})|\lesssim\lambda^{\gamma}\|\Pi\|_{\gamma;B_{2\mu\lambda}(x)}\sup_{y,\,z\in\operatorname*{\text{supp}}(\psi^{\lambda}_{x})}\sup_{\ell<\gamma}\frac{|f(z)-\Gamma_{zy}f(y)|_{\ell}}{{|y^{-1}z|}^{\gamma-\ell}}\
,$ (3.15)

as well as, given a second model $\bar{M}(\bar{\Pi},\bar{\Gamma})$ and a
modelled distribution $\bar{f}\in\mathscr{D}^{\gamma}_{\bar{M}}$, the
analogous bound, that for every $\lambda\in(0,1]$,

$\displaystyle\sup_{\psi\in\mathfrak{B}_{r}}|\langle\mathcal{R}_{M}f-\mathcal{R}_{\bar{M}}\bar{f}-\Pi_{x}f(x)+\bar{\Pi}_{x}\bar{f}(x),\psi^{\lambda}_{x}\rangle|\lesssim$
$\displaystyle\lambda^{\gamma}\bigg{(}\|f;\bar{f}\|_{\gamma;\operatorname*{\text{supp}}(\psi^{\lambda}_{x})}\|\bar{\Pi}\|_{\gamma;B_{2\mu\lambda}(x)}$
(3.16)
$\displaystyle+\|f\|_{\gamma;\operatorname*{\text{supp}}(\psi^{\lambda}_{x})}\|\Pi-\bar{\Pi}\|_{\gamma;B_{2\mu\lambda}(x)}\bigg{)}\
,$

for any $x\in\mathbb{G}$ and any $\lambda\in(0,1]$.

###### Proof.

Since the right hand side of (3.15) is linear in $f$, as in [39] we may assume
it to be equal to $1$. We use the functions $\rho^{(n)}$ and $\varphi^{(n)}$
from Section 3.2.1 and recall that they satisfy

$\varphi^{(n-1)}(x)=\int_{B_{\mathfrak{r}^{-n+1}}}\rho^{(n-1)}(y)\varphi_{y}^{(n)}(x)dy,\quad\varphi^{(n-1)}_{y}=\int_{B_{\mathfrak{r}^{-n+1}}}\rho^{(n-1)}(w)\varphi_{yw}^{(n)}(x)dw$

in particular since $\int\rho^{(n)}(x)\mathop{}\\!\mathrm{d}x=1$ we have
$\int\varphi_{y}^{(n)}(\,\cdot\,)\mathop{}\\!\mathrm{d}y=1$ for all $n\geq 0$.
Define, for fixed $\psi^{\lambda}_{x}$, the sets

$\Lambda_{n}=\left\\{y\in\mathbb{G}\ :\
\operatorname*{\text{supp}}(\varphi^{(n)}_{y})\cap\operatorname*{\text{supp}}(\psi^{\lambda}_{x})\neq\emptyset\right\\}\subset\mathbb{G}$

which by definition is contained in $B_{\lambda+C\mathfrak{r}^{-n}}(x)$ with
$C=\frac{\mathfrak{r}}{\mathfrak{r}{-1}}$ as in Lemma 3.10, as well as a
(measurable) function

$\pi_{n}:\Lambda_{n}\to\operatorname*{\text{supp}}(\varphi^{(n)}_{y})\cap\operatorname*{\text{supp}}(\psi^{\lambda}_{x}).$

Next, let

$\displaystyle R_{n}$
$\displaystyle=\int_{\Lambda_{n}}\big{(}\mathcal{R}f-\Pi_{\pi_{n}(y)}f(\pi_{n}(y))\big{)}(\psi_{x}^{\lambda}\varphi^{n}_{y})dy$
$\displaystyle=\langle\mathcal{R}f,\psi_{x}^{\lambda}\rangle-\int_{\Lambda_{n}}\Pi_{\pi_{n}(y)}f(\pi_{n}(y))(\psi_{x}^{\lambda}\varphi^{n}_{y})dy\
.$

It follows that for $n_{0}=\min\\{n\in\mathbb{N}\ :\
\mathfrak{r}^{-n}\leq\lambda\\}$, one has

$\left|(\mathcal{R}f-\Pi_{x}f(x))(\psi^{\lambda}_{x})-R_{n}\right|=\left|\int_{\Lambda_{n}}\big{(}\Pi_{x}f(x)-\Pi_{\pi_{n}(y)}f(\pi_{n}(y))\big{)}(\psi_{x}^{\lambda}\varphi^{n}_{y})dy\right|\lesssim\lambda^{\gamma}$
(3.17)

as well as for $n>n_{0}$

$\displaystyle R_{n-1}-R_{n}=$
$\displaystyle\int_{\Lambda_{n-1}}\Pi_{\pi_{n-1}(y)}f(\pi_{n-1}(y))(\psi_{x}^{\lambda}\varphi^{n-1}_{y})dy-\int_{\Lambda_{n}}\Pi_{\pi_{n}(z)}f(\pi_{n}(z))(\psi_{x}^{\lambda}\varphi^{n}_{z})dz$
$\displaystyle=$
$\displaystyle\int_{B_{\mathfrak{r}^{-n+1}}}\rho^{(n-1)}(w)\int_{\Lambda_{n}}\Big{(}\Pi_{\pi_{n-1}(zw^{-1})}f(\pi_{n-1}(zw^{-1}))-\Pi_{\pi_{n}(z)}f(\pi_{n}(z))\Big{)}(\psi_{x}^{\lambda}\varphi^{n}_{z})dzdw\
.$

Since $|\pi_{n}(z)^{-1}\pi_{n-1}(zw^{-1})|\leq\bar{C}\mathfrak{r}^{-n}$ and
for $\tau\in\operatorname{T}_{\alpha}$ such that $|\tau|\leq 1$

$|\Pi_{\pi_{n}(z)}\tau(\psi_{x}^{\lambda}\varphi^{n}_{z})|\lesssim\lambda^{|\mathfrak{s}|}\mathfrak{r}^{-\alpha
n}\ ,$

we find that

$\int_{\Lambda_{n}}\Big{|}\big{(}\Pi_{\pi_{n-1}(zw^{-1})}f(\pi_{n-1}(zw^{-1}))-\Pi_{\pi_{n}(z)}f(\pi_{n}(z))\big{)}(\psi_{x}^{\lambda}\varphi^{n}_{z})\Big{|}dz\lesssim\mathfrak{r}^{-\gamma
n}$

and thus conclude

$|R_{n}-R_{n-1}|\lesssim\mathfrak{r}^{-\gamma n}.$

A similar argument gives $|R_{n}|\to 0$ as $n\to\infty$ which combined with
(3.17) concludes the proof of (3.15). The proof of the analogous bound,
(3.16), follows in a similar manner. ∎

###  3.3 Singular Modelled Distributions

As in [39] we will eventually be concerned with solutions to SPDEs that take
values in spaces of modelled distributions with permissible singularities in
some regions of the domain. Our main example will be modelled distributions on
space-time domains that are allowed to be discontinuous at $\\{t=0\\}$, see
Section 4. However, as in [39] we build the notion of singular modelled
distributions allowing for singularities on more general sets, generalisations
of which have been used in [35, 34] to study singular equations with boundary
conditions.

We fix a homogeneous sub-Lie group $P\subset\mathbb{G}$ with associated Lie
algebra for which we write $\mathfrak{p}\subset\mathfrak{g}$. The assumption
that $P$ be a homogeneous sub-Lie group means that the scaling map
$\mathfrak{s}$ restricts to a map
$\bar{\mathfrak{s}}:=\mathfrak{s}|_{\mathfrak{p}}:\mathfrak{p}\rightarrow\mathfrak{p}$
which is diagonalisable. We fix a decomposition
$\mathfrak{g}=\mathfrak{p}\oplus\mathfrak{p}^{c}$ such that $\mathfrak{p}^{c}$
is also invariant under $\mathfrak{s}$ and define the homogeneous dimension of
$P$ and its complement, $P^{c}:=\exp(\mathfrak{p}^{c})$ as,

$|\bar{\mathfrak{s}}|:=\text{trace}(\mathfrak{s}|_{\mathfrak{p}})\quad\text{and}\quad|\mathfrak{m}|=\text{trace}(\mathfrak{s}|_{\mathfrak{p}^{c}})\
.$ (3.18)

Furthermore, we set

$|x|_{P}:=1\wedge d_{\mathbb{G}}(x,P)=1\wedge\inf\left\\{z\in
P\,:\,{|x^{-1}z|}\right\\},\quad|x,y|_{P}:=|x|_{P}\wedge|y|_{P}.$

Note that since $P$ is closed (being the image under $\exp$ of a linear
subspace of $\mathfrak{g}$), one sees easily that the infimum above is
actually a minimum. Given $\mathfrak{K}\subset\mathbb{G}$ we define the set

$\mathfrak{K}_{P}:=\left\\{(x,y)\in(\mathfrak{K}\setminus P)^{2}\,:\,x\neq
y\,\text{ and }\,{|x^{-1}y|}\leq|x,y|_{P}\right\\}.$

That is $\mathfrak{K}_{P}$ contains all the points in $\mathfrak{K}$ that are
closer to each other than they are to $P$.

######  Definition 3.5 (Singular Modelled Distributions).

Given a regularity structure $\mathcal{T}$ and a subgroup $P$ as above, for
any $\gamma>0$, $\eta\in\mathbb{R}$ and maps $f:\mathbb{G}\setminus
P\rightarrow\mathcal{T}$, we set

$\|f\|_{\gamma,\eta;\mathfrak{K}}:=\sup_{x\in\mathfrak{K}\setminus
P}\sup_{\zeta<\gamma}\frac{|f(x)|_{\zeta}}{|x|_{P}^{(\eta-\zeta)\wedge
0}},\quad\llbracket
f\rrbracket_{\gamma,\eta;\mathfrak{K}}:=\sup_{x\in\mathfrak{K}\setminus
P}\sup_{\zeta<\gamma}\frac{|f(x)|_{\zeta}}{|x|^{\eta-\zeta}_{P}}.$

Then given a model $M=(\Pi,\Gamma)$ as well as a sector $V$, the space
$\mathscr{D}^{\gamma,\eta}_{P,M}(V)$ consists of all functions
$f:\mathbb{G}\setminus P\rightarrow V_{<\gamma}$ such that for every compact
set $\mathfrak{K}\subset\mathbb{G}$,

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|f\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\gamma,\eta;\mathfrak{K}}:=\|f\|_{\gamma,\eta;\mathfrak{K}}+\sup_{(x,y)\in\mathfrak{K}_{P}}\sup_{\zeta<\gamma}\frac{|f(x)-\Gamma_{x,y}f(y)|_{\zeta}}{{|y^{-1}x|}^{\gamma-\zeta}|x,y|^{\eta-\gamma}_{P}}<+\infty.$

For two models $M=(\Pi,\Gamma),\,\bar{M}=(\bar{\Pi},\bar{\Gamma})$ and two
modelled distributions
$f\in\mathscr{D}^{\gamma,\eta}_{P,M},\,\bar{f}\in\mathscr{D}^{\gamma,\eta}_{P,\bar{M}}$
we also set,

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|f;\bar{f}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\gamma,\eta;\mathfrak{K}}:=\|f-\bar{f}\|_{\gamma,\eta;\mathfrak{K}}+\sup_{(x,y)\in\mathfrak{K}_{P}}\sup_{\zeta<\gamma}\frac{|f(x)-\bar{f}(x)-\Gamma_{x,y}f(y)+\bar{\Gamma}_{xy}\bar{f}(y)|_{\zeta}}{{|y^{-1}x|}^{\gamma-\zeta}|x,y|^{\eta-\gamma}_{P}}.$

If $V$ is a sector of regularity $\alpha\in A$, where appropriate we will use
the shorthand
$\mathscr{D}^{\gamma,\eta}_{\alpha;P,M}=\mathscr{D}^{\gamma,\eta}_{P,M}(V)$
and we will drop the dependence on the model when the context is clear.

######  Remark 3.14.

We refer to [39] for more intuition regarding the definition of these spaces
and their properties - all of which carry over to our setting. In particular
[39, Rem. 6.4] discusses the relationship between the spaces
$\mathscr{D}^{\gamma,\eta}_{P}$ and $\mathscr{D}^{\gamma}$.

######  Remark 3.15.

The family of norms $\llbracket f\rrbracket_{\gamma,\eta;\mathfrak{K}}$ and
the two following lemmas play a role when we consider fixed-point maps in
Section 4 below. Their utility is in allowing us to extract small scaling
parameters in terms of the distance to the subgroup. In the semi-linear
evolution equation setting this allows us to obtain fixed points on
sufficiently short time intervals.

######  Lemma 3.16.

Let $\mathfrak{K}\subset\mathbb{G}$ be a compact domain such that for every
$x\in\mathfrak{K}$ and $\bar{x}:=\operatorname*{arg\,min}_{y\in P}|x^{-1}y|$
one has that the points
$\bar{x}\big{(}\lambda\cdot(\bar{x}^{-1}x)\big{)}\in\mathfrak{K}$ for every
$\lambda\in[0,1]$. Also let $f\in\mathscr{D}^{\gamma,\eta}_{P}$ for some
$\gamma>0$ and assume that for every $\zeta<\eta$ the map
$x\mapsto\mathcal{Q}_{\zeta}f(x)$ extends continuously in such a way that for
$x\ \in P$ one has $\mathcal{Q}_{\zeta}f(x)=0$. Then, one has the bound,

$\llbracket
f\rrbracket_{\gamma,\eta;\mathfrak{K}}\lesssim{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|f\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\gamma,\eta;\mathfrak{K}},$

where the implied constant depends affinely on
$\|\Gamma\|_{\gamma;\mathfrak{K}}$ but is otherwise independent of
$\mathfrak{K}$. Similarly, if
$\bar{f}\in\mathscr{D}^{\gamma,\eta}_{P,\bar{M}}$ with respect to a different
model $\bar{M}=(\bar{\Pi},\bar{\Gamma})$ and is such that

$\lim_{x\to P}\mathcal{Q}_{\zeta}(f(x)-\bar{f}(x))=0$

for every $\zeta<\eta$, then one has the bound

$\llbracket
f-\bar{f}\rrbracket_{\gamma,\eta;\mathfrak{K}}\lesssim{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|f;\bar{f}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\gamma,\eta;\mathfrak{K}}+\|\Gamma-\bar{\Gamma}\|_{\gamma;\mathfrak{K}}\left({\left|\kern-1.07639pt\left|\kern-1.07639pt\left|f\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\gamma,\eta;\mathfrak{K}}+{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\bar{f}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\gamma,\eta;\mathfrak{K}}\right),$

with proportionality constant also depending affinely on
$\|\Gamma\|_{\gamma;\mathfrak{K}}$ and
$\|\bar{\Gamma}\|_{\gamma;\mathfrak{K}}$.

###### Proof.

The proof follows almost exactly as that of [39, Lem. 6.5] but here using the
sequence $x_{0}:=x,\,x_{\infty}:=\bar{x}=\operatorname*{arg\,min}_{z\in
P}{|x^{-1}z|}$ and
$x_{n}:={x_{\infty}\big{(}2^{-n}\cdot(x^{-1}_{\infty}x_{0})\big{)}}$.

∎

######  Lemma 3.17.

Let $\gamma>0$, $\kappa\in(0,1)$ and assume $f,\,\bar{f}$ satisfy the
assumptions of Lemma 3.16. Then, for every compact
$\mathfrak{K}\subset\mathbb{G}$, one has

${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|f;\bar{f}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{(1-\kappa)\gamma,\eta;\mathfrak{K}}\lesssim\llbracket
f-\bar{f}\rrbracket^{\kappa}_{\gamma,\eta;\mathfrak{K}}\left({\left|\kern-1.07639pt\left|\kern-1.07639pt\left|f\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\gamma,\eta;\mathfrak{K}}+{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|\bar{f}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\gamma,\eta;\mathfrak{K}}\right)^{1-\kappa}.$

###### Proof.

Follows by a direct adaptation of the proof of [39, 6.6]. One need only
replace $\mathbb{R}^{d}$ there by $\mathbb{G}$ here. ∎

####  3.3.1 Reconstruction Theorem for Singular Modelled Distributions

Since the reconstruction is purely local, it follows from our earlier proof
that for any singular modelled distribution,
$f\in\mathscr{D}^{\gamma,\eta}_{P}$, there exists a unique element
$\tilde{\mathcal{R}}f\in\mathcal{S}^{\prime}(\mathbb{G}\setminus P)$, i.e. in
the dual of smooth functions compactly supported away from $P$, such that,

$\langle\tilde{\mathcal{R}}f-\Pi_{x}f(x),\psi^{\lambda}_{x}\rangle\lesssim\lambda^{\gamma},$

for all $x\notin P$ and $\lambda\ll d_{\mathbb{G}}(x,P)$. However, we show
below that under appropriate assumptions there exists a natural extension of
$\tilde{\mathcal{R}}f$ to an actual distribution on $\mathbb{G}$ with
regularity $\mathcal{C}^{\underline{\alpha}}$.

######  Proposition 3.18 (Singular Reconstruction).

Let $f\in\mathscr{D}^{\gamma,\eta}_{P}(V)$, $\gamma>0$ and $\eta\leq\gamma$.
Then, provided $\alpha\wedge\eta>-|\mathfrak{m}|$, where $|\mathfrak{m}|$ is
the scaled dimension of the complement of the singular hyperplane defined by
(3.18), there exists a unique distribution
$\mathcal{R}f\in\mathcal{C}^{{\alpha}\wedge\eta}_{\mathfrak{s}}$ such that
$(\mathcal{R}f)(\varphi)=(\tilde{\mathcal{R}}f)(\varphi)$ for any smooth test
function compactly supported away from $P$. If $f$ and $\bar{f}$ are given
with respect to two models $M$ and $\bar{M}$ then, for any compact
$\mathfrak{K}$, it holds that

$\|\mathcal{R}_{M}f-\mathcal{R}_{\bar{M}}\bar{f}\|_{\mathcal{C}^{{\alpha}\wedge\eta}(\mathfrak{K})}\lesssim{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|f;\bar{f}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\gamma,\eta;\bar{\mathfrak{K}}}+{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|M;\bar{M}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\gamma;\bar{\mathfrak{K}}},$

where the constant depends on semi-norms of $f,\,\bar{f}$ and $M,\,\bar{M}$ on
$\bar{\mathfrak{K}}$.

We provide the proof of Proposition 3.18 at the end of this section, let us
first make some preparatory observations. Recalling the decomposition
$\mathfrak{g}=\mathfrak{p}\oplus\mathfrak{p}^{c}$ defined at the start of the
subsection, we define the projections
$\pi_{c}:\mathfrak{g}\to\mathfrak{p}^{c}$ and
$\pi_{\mathfrak{p}}:\mathfrak{g}\to\mathfrak{p}$. Then using the decomposition
$X=X^{\mathfrak{p}}+X^{c}\in\mathfrak{p}\oplus\mathfrak{p}^{c}$, we define the
map

$\tilde{\Phi}:\mathfrak{g}\rightarrow\mathbb{G},\quad\tilde{\Phi}(X)=\exp(X^{\mathfrak{p}})\exp(X^{c}).$
(3.19)

Similarly to Remark 2.19 one sees that $\tilde{\Phi}$ is a global
diffeomorphism. We then define the map

$N_{P}:\mathbb{G}\to\mathbb{R}_{+},\qquad
x\mapsto\left|\exp\left(\pi_{c}\circ\tilde{\Phi}^{-1}(x)\right)\right|$ (3.20)

and observe the following properties.

  * •

For $x\in\mathbb{G}$ one has that $N_{P}(x)=0$ if and only if $x\in P\ .$ This
follows from the fact that $P=\exp(\mathfrak{p})$.

  * •

For $x\in\mathbb{G}$ and $\delta>0$ one has the identity

$N_{P}(\delta\cdot x)=\delta N_{P}(x)\ .$ (3.21)

Indeed, writing $x=\tilde{\Phi}(\tilde{X})$, we have

$N_{P}(\delta\cdot
x)=|\exp(\pi_{c}(\operatorname{D}_{\delta}\tilde{X}))|=|\exp(\operatorname{D}_{\delta}\pi_{c}(\tilde{X}))|=|\delta\cdot\exp(\pi_{c}(\tilde{X}))|=\delta
N_{P}(x)\ .$

  * •

For any $x\in\mathbb{G}$ and $y\in P$ one has

$N_{P}(yx)=N_{P}(x)\ .$ (3.22)

This follows from the observation that, writing $x=\tilde{\Phi}(\tilde{X})$
and $y=\tilde{\Phi}(Y)=\exp(Y)$ one finds that
$yx=\exp(\operatorname{H}(Y,\tilde{X}^{\mathfrak{p}}))\exp(\tilde{X}^{c})$
where $\operatorname{H}$ was defined in (2.1). Identity (3.22) then follows
from the fact that
$\operatorname{H}(Y,\tilde{X}^{\mathfrak{p}})\in\mathfrak{p}$.

  * •

The map $N_{P}$ is Lipschitz continuous on $\mathbb{G}$ and it follows from
Remark 2.4 that $N_{P}$ is smooth on ${\mathbb{G}\setminus P}$.

  * •

There exists a constant $C>0$ such that for all $x\in\mathbb{G}$

$CN_{P}(x)\leq d_{\mathbb{G}}(x,P)\ .$ (3.23)

In a neighbourhood of the origin this follows directly from the fact that
$N_{P}$ is Lipschitz continuous. Homogeneity of $N_{p}$ (given by (3.21)
above) and of the distance function $x\mapsto d_{\mathbb{G}}(x,P)$ then shows
that this constant is in fact uniform on all of $\mathbb{G}$.

  * •

For all $x\in\mathbb{G}$

$d_{\mathbb{G}}(x,P)\leq N_{P}(x)\ .$ (3.24)

Indeed, write $x=\tilde{\Phi}({X})=\exp(X^{\mathfrak{p}})\exp(X^{c})$ and note
that

$d(x,P)\leq|\exp(X^{\mathfrak{p}})^{-1}x|=|\exp(X^{c})|=N_{P}(x)\ .$

###### Proof of Proposition 3.18.

The proof is analogous to that of [39, Lem.6.9], the only element that does
not adapt ad verbatim, is the construction of the partition of unity
$\varphi_{x,n}$. We therefore present an alternative construction of a
partition of unity on $\mathbb{G}$, which satisfies all the required
conditions.

First let $\varphi:\mathbb{R}_{+}\rightarrow[0,1]$ be a smooth compactly,
supported function such that $\operatorname*{\text{supp}}(\varphi)=[1/2,2]$
and with the property that for all $r\in\mathbb{R}_{+}$,

$\sum_{n\in\mathbb{Z}}\varphi(2^{n}r)=1.$

Secondly, let $\mathfrak{Z}\subset P$ be a lattice (see Section 2.3) and let
$\tilde{\varphi}$ be smooth, compactly supported, such that

$\sum_{y\in\mathfrak{Z}}\tilde{\varphi}_{y}(x)=1$ (3.25)

for all $x$ in the $\frac{2}{C}$-fattening of $P\subset\mathbb{G}$, where $C$
is the constant in (3.23)and (3.24). For $y\in\mathfrak{Z}$ we then set

$\phi_{y}(x):=\varphi(CN_{P}(x))\tilde{\varphi}_{y}(x)\ ,$

where the constant $C$ is as in (3.23). To conclude, we then set for every
$n\geq 0$, $y\in\mathfrak{Z}$ and $x\in\mathbb{G}$,

$\phi_{n,y}(x):=\phi_{y}(2^{n}\cdot x).$

By using (3.21) and (3.22), it directly follows that
$\phi_{n,y}(x)=(\phi_{0,e})\left(y^{-1}(2^{n}\cdot x)\right)$. Since
$\phi_{1,e}$ has compact support and is such that for all
$x\in\operatorname*{\text{supp}}(\phi_{1,e})$, one has
$d_{\mathbb{G}}(x,P)\geq CN_{P}(x)\geq 1/2$, it only remains to check that
$\\{\phi_{n,y}\\}_{n\in\mathbb{Z},y\in\mathfrak{Z}}$ is in fact a partition of
unity. Indeed let $x\in\mathbb{G}$, then

$\sum_{n\in\mathbb{Z},y\in\mathfrak{Z}}\phi_{n,y}(x)=\sum_{n\in\mathbb{Z},y\in\mathfrak{Z}}\varphi(CN_{P}(2^{n}\cdot
x))\tilde{\varphi}_{y}(2^{n}\cdot
x)=\sum_{n\in\mathbb{Z}}\varphi(2^{n}CN_{P}(x)){\sum_{y\in\mathfrak{Z}}{\tilde{\varphi}_{y}(2^{n}\cdot
x)}}=1\ ,$

where in the second last equality we used that
$\sum_{y\in\mathfrak{Z}}\tilde{\varphi}_{y}(2^{n}\cdot x)=1$ whenever
$d(x,P)\leq\frac{1}{C}2^{-n+1}$ and in the last equality we used (3.25). The
remainder of the proof of [39, Lem. 6.9] then adapts ad verbatim by also also
making use of Proposition 3.13. ∎

######  Remark 3.19.

If the model $M$ is smooth, i.e. $\Pi_{x}\tau\in C^{\infty}(\mathbb{G})$ for
every $\tau\in\operatorname{T}$, one finds exactly as in [39, Rem. 3.15] that
for any modelled distribution $f\in\mathscr{D}^{\gamma}$ with $\gamma>0$ one
has the identity $\mathcal{R}_{M}f(x)=\left(\Pi_{x}f(x)\right)(x)$ (and in
particular $\mathcal{R}_{M}f$ is a continuous function).

###  3.4 Convolution with Singular Kernels

In this section we describe how to lift the action of singular kernels onto
the regularity structure. While most arguments adapt from [39] some care has
to be taken due to the fact that convolutions are not commutative and we do
not have an explicit formula for Taylor expansions. This latter issue is
circumvented using Lemma 3.6 which allows us to reduce our analysis to similar
expressions as appear in [39]. The examples we have in mind are the singular
part of Greens functions of left invariant differential operators satisfying
the following assumption.

######  Assumption 3.20.

For $\beta\in(0,|\mathfrak{s}|)$, the kernel
$K:\mathbb{G}\setminus\\{e\\}\to\mathbb{R}$ can be decomposed as

$K(x)=\sum_{n\in\mathbb{N}}K_{n}(x)$ (3.26)

where the smooth functions $K_{n}$ are supported on $B_{2^{-n}}$ and

  * •

for each $I\in\mathbb{N}^{d}$ there exists a constant $C(I)>0$, uniform in
$n\in\mathbb{N}$ such that

$\sup_{x\in\mathbb{G}}|X^{I}K_{n}(x)|\leq
C(I)2^{(|\mathfrak{s}|-\beta+d(I))n},$

  * •

for any multi-indices $I,\,J\in\mathbb{N}^{d}$ there exists a constant
$C(I,J)>0$ uniform in $n\in\mathbb{N}$ such that,

$\left|\,\int_{\mathbb{G}}\eta^{I}(x)X^{J}K(x)dx\,\right|\leq C(I,J)2^{-\beta
n},$

  * •

there exists an integer $r$, such that

$\int_{\mathbb{G}}\eta^{I}(x)K(x)\,dx=0$

for all multi-indices $I\in\mathbb{N}^{d}$ with scaled degree $d(I)\leq r$.

######  Remark 3.21.

We note that all of the analysis in the remainder of this section also applies
to kernels of the form
$K:(\mathbb{G}\setminus\\{e\\})^{2}\rightarrow\mathbb{R}$ satisfying an
analogue of [39, Ass. 5.1 & Ass. 5.4] adapted to our setting. Although
Assumption 3.20 is somewhat less general we choose to work with it for two
reasons; firstly it is simpler to verify and secondly it highlights the role
that translation invariance plays in our applications. Lemma 4.4 which is an
amalgam of [39, Lem. 5.5 & Lem. 7.7] in our setting, shows that fundamental
solutions of left-translation invariant, homogeneous linear operators can
always be decomposed into a compactly support part satisfying Assumption 3.20
and a sufficiently well-behaved remainder.

######  Remark 3.22.

Although we work explicitly with the one-parameter kernels of Assumption 3.20
it will sometimes be convenient in the proofs below to define
$K(x,y):=K(y^{-1}x)$ and use the notation $K(f)(x):=\int
K(x,y)f(y)\mathop{}\\!\mathrm{d}y=f\ast K(x)$. Note that under this
convention, for any left-translation invariant vector field $X$, one has
$(XK)(f)=X(Kf)$.

From now on we shall exclusively work with regularity structures $\mathcal{T}$
models $M$ satisfying the following assumption.

######  Assumption 3.23.

For each $a\in\triangle$, the vector space $\operatorname{T}_{a}$ coincides
with the linear span of abstract monomials $\boldsymbol{\eta}^{I}$ with
$d(I)=a$ and the model $M\in\mathcal{M}_{\mathcal{T}}$ restricted to the
polynomial sector
$\bar{\operatorname{T}}=\bigoplus_{a\in\triangle}\operatorname{T}_{a}$, is the
canonical polynomial model.

We point out that the assumption $K$ annihilates polynomials causes no real
restriction on the type of kernels since the result [39, Lem. 5.5] adapts in a
straightforward manner to our setting, see also Lemma 4.4 below. We point out
that this assumption is convenient but not crucial for the theory, c.f. [44].
In the remainder of this subsection we show how the action of kernels of this
type are lifted onto the regularity structure and act on modelled
distributions. Given a $\gamma\in\mathbb{R}\setminus\triangle$ we write
$\mathcal{K}_{\gamma}$ for this lift; it corresponds to $K$ in the sense that
for $f\in\mathscr{D}^{\gamma}$

$\mathcal{R}\mathcal{K}_{\gamma}f=K(\mathcal{R}f)$ (3.27)

and in that it satisfies an appropriate version of the classical Schauder
estimates.

######  Definition 3.6.

Given a sector $V$, a map $\mathcal{I}:V\mapsto\bar{\operatorname{T}}$ is
called an abstract integration map of order $\beta>0$ if it satisfies the
following properties:

  1. 1.

For each $\alpha\in A,\
\mathcal{I}:V_{\alpha}\to\operatorname{T}_{\alpha+\beta}$, where
$\operatorname{T}_{\alpha+\beta}:=\\{0\\}$ for $\alpha+\beta\notin A$.

  2. 2.

It annihilates polynomials, that is $\mathcal{I}:\bar{\operatorname{T}}\cap
V\to\\{0\\}$.

  3. 3.

For each $\tau\in\operatorname{T},\ \Gamma\in\mathbb{G}:\ (\mathcal{I}\
\Gamma-\Gamma\mathcal{I})(\tau)\in\bar{\operatorname{T}}.$

Assume that the kernel $K$ satisfies Assumption 3.20 for some $\beta>0$. We
associate to $K$ the map $\mathcal{J}:\mathbb{R}^{d}\to
L(\operatorname{T},\bar{\operatorname{T}})$ which for every
$\tau\in\operatorname{T}_{\alpha}$ is given by

$\mathcal{J}(x)\tau:=\sum_{n}\boldsymbol{\operatorname{{P}}}^{\alpha+\beta}_{x}(\Pi_{x}\tau\ast
K_{n}),$ (3.28)

where the last sum is seen to converge absolutely by first observing that by
Assumption 3.20 for any $\tau\in\operatorname{T}_{\alpha}$

$|X^{I}(\Pi_{x}\tau\ast K_{n})(x)|=|\Pi_{x}\tau\ast
X^{I}K_{n}(x)|=|\Pi_{x}\tau(X^{I}_{1}K_{n}(x,\cdot))|\lesssim
2^{-n(\alpha+\beta-d(I))}\ ,$

and then using Lemma 3.4.

######  Definition 3.7.

Given a regularity structure $\mathcal{T}$ equipped with an Integration map
$\mathcal{I}$, a kernel $K$ and a model $M=(\Pi,\Gamma)$, we say the model $M$
realises $K$ for $\mathcal{I}$ if for each $\tau\in\operatorname{T}_{\alpha}$
and each $x\in\mathbb{G}$,

$\Pi_{x}\mathcal{I}\tau=K(\Pi_{x}\tau)-\Pi_{x}\mathcal{J}(x)\tau.$

Now we can define the lift of the kernel K, namely for
$f\in\mathscr{D}^{\gamma}(V)$ we set:

$\mathcal{K}_{\gamma}f(x):=\mathcal{I}f(x)+\mathcal{J}(x)f(x)+\mathcal{N}_{\gamma}f(x),$

where

$(\mathcal{N}_{\gamma}f)(x)=\sum_{n}\boldsymbol{\operatorname{{P}}}^{\gamma+\beta}_{x}\big{(}(\mathcal{R}f-\Pi_{x}f(x))\ast
K_{n}\big{)}$

where the last sum converges by the same argument as for $\mathcal{J}(x)\tau$.

######  Remark 3.24.

Given a kernel $K$ satisfying Assumption 3.20 a regularity structure and model
satisfying Assumption 3.23, it turns out that one can always extend the
regularity structure and model to be equipped with an integration map
$\mathcal{I}$ realizing the kernel $K$. This is the content of the extension
theorem found as [39, Thm. 5.14], which holds in our setting as well. While we
do not reproduce the whole proof since it is a straightforward adaptation of
the original one, we present the main steps below, see Lemmas 3.26, 3.27 and
3.28, so that the interested reader will easily be able to fill in the
remaining details.

The next theorem which is an analogue of [39, Thm. 5.12], confirms that
$\mathcal{K}_{\gamma}$ does indeed correspond to $K$ in the sense of (3.27)
and satisfies the desired Schauder estimates.

######  Theorem 3.25.

Let $\gamma\in\mathbb{R}\setminus\triangle$ and $\beta>0$ be such that
$\gamma+\beta\not\in\triangle$, let
$K:\mathbb{G}\setminus\\{e\\}\rightarrow\mathbb{R}$ be a kernel satisfying
Assumption 3.20 for $r\geq\gamma+\beta$, let
$\mathcal{T}=(\operatorname{T},\operatorname{G})$ be a regularity structure
and $M=(\Pi,\Gamma)$ be a model satisfying Assumption 3.23. Furthermore assume
that $\mathcal{T}$ is equipped with an abstract integration operator and $M$
realises $K$ for $\mathcal{I}$. Then for any sector $V$ of regularity
$\alpha\in A$ the operator $\mathcal{K}_{\gamma}$ is a continuous linear map
from $\mathscr{D}_{M}^{\gamma}(V)$ to $\mathscr{D}_{(\alpha+\beta)\wedge
0}^{\gamma+\beta}$ satisfying the identity

$\mathcal{R}\mathcal{K}_{\gamma}f=K(\mathcal{R}f),$

for all $f\in\mathscr{D}_{M}^{\gamma}(V)$. Furthermore, if we denote by
$M=(\Pi,\Gamma),\ \bar{M}=(\bar{\Pi},\bar{\Gamma})$ two models as above and by
$\mathcal{K}_{\gamma}$, respectively $\bar{\mathcal{K}}_{\gamma}$ the
associated operators, one has the bound

$\|\mathcal{K}_{\gamma}f;\bar{\mathcal{K}}_{\gamma}\bar{f}\|_{\gamma+\beta;\mathfrak{K}}\lesssim_{C}{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|f,\bar{f}\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}_{\gamma;\bar{\mathfrak{K}}}\\\
+\|\Pi-\bar{\Pi}\|_{\gamma;\bar{\mathfrak{K}}}+\|\Gamma-\bar{\Gamma}\|_{\gamma;\bar{\mathfrak{K}}}\
,$

where the implicit constant depends on the norms of $M,\bar{M}$ and
$f\in\mathscr{D}_{M}^{\gamma}(V)$ and
$\bar{f}\in\mathscr{D}^{\gamma}_{\bar{M}}(V)$.

The fact that the next lemma ([39, Lem. 5.16]) still holds in our setting is
the underlying reason that all proofs extend in a rather straight forward
manner from [39] and one does not require a more involved notion of abstract
integration map, which is for example the case on general Riemannian manifolds
c.f.[44].

######  Lemma 3.26.

In the setting of Theorem 3.25 one has the identity

$\Gamma_{x,y}(\mathcal{I}+\mathcal{J}(y))=(\mathcal{I}+\mathcal{J}(x))\Gamma_{x,y}$

for all $x,y\in\mathbb{G}$.

###### Proof.

The proof consists of unravelling the Definitions and using the fact that the
map $\Pi_{x}$ is injective when restricted to polynomials, exactly as in [39].
∎

We introduce the following quantity; for
$I\in\mathbb{N}^{d},\alpha>0,n\in\mathbb{N}$ and $x,y,z\in\mathbb{G}$, set

$K^{I,\alpha}_{n,xy}(z):=X^{I}_{1}K_{n}(y,z)-\tilde{\operatorname{P}}^{\alpha+\beta-d(I)}_{x}[X^{I}_{1}K_{n}(\cdot,z)](y)=X^{I}_{1}\big{(}K_{n}(\,\cdot\,,z)-\tilde{\operatorname{P}}^{\alpha+\beta}_{x}[K_{n}(\cdot,z)]\big{)}(y)$
(3.29)

where the second equality follows from Remark 2.12. Here we reiterate that we
are using the notation $K(x,y):=K(y^{-1}x)$ where $K$ satisfies Assumption
3.20. Taylor’s theorem and Remark 2.16 then yield that

$\displaystyle K^{I,\alpha}_{n,xy}(z)$
$\displaystyle=\sum_{|J|\leq[\alpha+\beta-d(I)]+1,d(J)\geq\alpha+\beta-d(I)}\int_{\mathbb{G}}X_{1}^{J}(X_{1}^{I}K_{n})(x\tilde{z},z)Q^{J}(x^{-1}y,d\tilde{z})\
.$ (3.30)

As in [39] the motivation for defining these quantities comes from the
identity

$\displaystyle\Pi_{x}(\mathcal{I}\tau)(\phi)=K(\Pi_{x}\tau)(\phi)-\Pi_{x}J(x)\tau(\phi)$
$\displaystyle=\sum_{n}\int\big{(}\Pi_{x}\tau(K_{n}(y,\cdot))-\tilde{\operatorname{P}}_{x}^{\alpha+\beta}[K_{n}(\Pi_{x}\tau)](y)\big{)}\phi(y)\,dy$
$\displaystyle=\sum_{n}\int\Pi_{x}\tau\big{(}K_{n}(y,\cdot)-\tilde{\operatorname{P}}_{x,1}^{\alpha+\beta}[K_{n}(\cdot,\cdot)](y)\big{)}\phi(y)\,dy$
$\displaystyle=\sum_{n}\int\Pi_{x}\tau(K^{0,\alpha}_{n,xy})\phi(y)\,dy$

where we use the subscript for
$\tilde{\operatorname{P}}_{x,1}[K(\cdot,\cdot)](y)$ to clarify that one
expands in the first coordinate. The next Lemma collects the results of [39,
Lem. 5.18, Lemma 5.19], the proofs of which adapt ad verbatim.

######  Lemma 3.27.

Let $\alpha\in A$ and $\tau\in\operatorname{T}_{\alpha}$ and assume
$\alpha+\beta\notin\triangle$, then the following bound holds

$|(\Pi_{y}\tau)(K^{I,\alpha}_{n,xy})|\lesssim\|\Pi\|_{\alpha,B_{2}(x)}(1+\|\Gamma\|_{\alpha,B_{2}(x)})\sum_{\delta>0}2^{\delta
n}|y^{-1}x|^{\delta+\alpha+\beta-d(I)}\ ,$ (3.31)

where the sum over $\delta$ runs over a finite set of positive real numbers.
The same bound holds for $|(\Pi_{x}\tau)(K^{I,\alpha}_{n,xy})|$, as well as
the analogue bound on the difference of two models.

Furthermore one has the bound

$\sum_{n}\left|\int_{\mathbb{G}}(\Pi_{x}\tau)(K^{I,\alpha}_{n,xy})\phi_{x}^{\lambda}(y)\mathop{}\\!\mathrm{d}y\,\right|\lesssim\lambda^{\alpha+\beta}\|\Pi\|_{\alpha;B_{2}(x)}(1+\|\Gamma\|_{\alpha;B_{2}(x)})\
,$ (3.32)

as well as the analogous bound for the difference of two models, where all
these bounds hold uniformly over $x\in\mathbb{G}$, $\lambda\in(0,1]$ and
$\phi\in\mathfrak{B}_{r}$

As in [39] we introduce for $x,y\in\mathbb{G}$ the following operator

$\mathcal{J}_{x,y}:=\mathcal{J}(x)\Gamma_{x,y}-\Gamma_{x,y}\mathcal{J}(y)$
(3.33)

######  Lemma 3.28.

Under the assumptions of Theorem 3.25 one has for each $a\in\Delta$,
$\tau\in\operatorname{T}_{\alpha}$

$|\mathcal{J}_{x,y}\tau|_{a}\lesssim\|\Pi\|_{\alpha,B_{2}(x)}(1+\|\Gamma\|_{\alpha,B_{2}(x)})|y^{-1}x|^{\alpha+\beta-a}|\tau|$

uniformly in $x,y\in\mathbb{G}$ satisfying $x\in B_{1}(y)$. Again, the
analogous bound for the difference of two models holds as well.

###### Proof.

First we write $\mathcal{J}_{x,y}=\sum_{n}\mathcal{J}^{n}_{x,y}$ where each
$\mathcal{J}^{n}_{x,y}$ is defined by replacing $K$ by only one summand
$K_{n}$ in the definition of $\mathcal{J}_{x,y}$. Observe that since
$\mathcal{J}^{n}_{x,y}\tau\in\bar{\operatorname{T}}_{<\alpha+\beta}$ it
suffices by Lemma 3.6 to show that

$p^{n}_{\tau}(w):=\Pi_{e}(\mathcal{J}^{n}_{x,y}\tau)(w)\in\mathcal{P}_{\alpha+\beta}$

satisfies