# Stochastic quantization of two-dimensional
$P(\Phi)$ Quantum Field Theory

Paweł Duch, Wojciech Dybalski and Azam Jahandideh
Faculty of Mathematics and Computer Science
Adam Mickiewicz University in Poznań
ul. Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland
{pawel.duch, wojciech.dybalski<EMAIL_ADDRESS>
###### Abstract

We give a simple and self-contained construction of of the $P(\Phi)$ Euclidean
Quantum Field Theory in the plane and verify the Osterwalder-Schrader axioms:
translational and rotational invariance, reflection positivity and regularity.
In the intermediate steps of the construction we study measures on spheres. In
order to control the infinite volume limit we use the parabolic stochastic
quantization equation and the energy method. To prove the translational and
rotational invariance of the limit measure we take advantage of the fact that
the symmetry groups of the plane and the sphere have the same dimension.

###### Contents

  1. 1 Introduction
  2. 2 Ultraviolet limit
  3. 3 Stochastic quantization
  4. 4 Stereographic projection
  5. 5 A priori bound
  6. 6 Tightness
  7. 7 Integrability
  8. 8 Reflection positivity
  9. 9 Euclidean invariance
  10. A Function spaces
  11. B Mathematical preliminaries
  12. C Stochastic estimates

##  1 Introduction

We revisit the construction of the Euclidean two-dimensional $P(\Phi)$ quantum
field theory model also known as the $P(\Phi)_{2}$ model. The main new
contribution is a simple construction of the infinite volume measure of this
model using the stochastic quantization technique and the verification of all
Osterwalder-Schrader axioms with the exception of the clustering. By the
Osterwalder-Schrader reconstruction theorem [25] this yields the existence of
the theory in the Lorentzian signature that satisfies all the Wightman axioms
possibly with the exception of the uniqueness of the vacuum. Let us point out
that the proof of the invariance of the infinite volume measure under all of
the Euclidean transformations of the plane, which is one of the Osterwalder-
Schrader axioms, is quite non-trivial. In fact, finite-volume measures, which
are typically introduced in the intermediate steps of the construction, are
usually invariant only under a certain subset of the Euclidean
transformations. The novelty of the approach taken in the present work is to
study finite-volume $P(\Phi)_{2}$ measures defined on spheres in the
intermediate steps of the construction. Such measures are invariant under the
action of the three-dimensional Lie group of the rotations of the sphere (in
contrast, measures defined on a torus are only invariant under the action of
the two-dimensional Lie group). To prove the Euclidean invariance of the
infinite volume measure we crucially use the fact that the symmetry groups of
the plane and the sphere have the same dimension.

Fix $n\in 2\mathbb{N}_{+}$, $n\geq 4$, and a real polynomial

$P(\tau)=\sum_{m=0}^{n}a_{m}\tau^{m},\qquad\tau\in\mathbb{R},\qquad
a_{0},\ldots,a_{n-1}\in\mathbb{R},\quad a_{n}=1/n\,.$ (1.1)

Let $\mathbb{S}_{R}$ be a round two-dimensional sphere of radius
$R\in\mathbb{N}_{+}$ with the metric induced from $\mathbb{R}^{3}$. The
Laplace-Beltrami operator on $\mathbb{S}_{R}$ is denoted by $\Delta_{R}$ and
the canonical Riemannian volume form on $\mathbb{S}_{R}$ is denoted by
$\rho_{R}$. For $R\in\mathbb{N}_{+}$ a probability measure $\mu_{R}$ on
$\mathscr{D}^{\prime}(\mathbb{S}_{R})$ is defined by

$\mu_{R}(\mathrm{d}\phi):=\frac{1}{\mathcal{Z}_{R}}\exp\left(-\int_{\mathbb{S}_{R}}\lambda:\\!P(\phi(\mathrm{x}))\\!:\,\rho_{R}(\mathrm{d}\mathrm{x})\right)\nu_{R}(\mathrm{d}\phi),$
(1.2)

where $\lambda\in(0,\infty)$ is the coupling constant,
$\mathcal{Z}_{R}\in(0,\infty)$ is the normalization factor, $\nu_{R}$ is the
Gaussian measure on $\mathscr{D}^{\prime}(\mathbb{S}_{R})$ with covariance
$G_{R}:=(1-\Delta_{R})^{-1}$ and
$:{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}}:$ denotes the Wick
ordering. The measure $\mu_{R}$ is called the $P(\Phi)_{2}$ measure on
$\mathbb{S}_{R}$. In Sec. 2 we review the construction of this measure based
on the Nelson hypercontractivity argument [23]. By construction, $\mu_{R}$ is
invariant under rotations of $\mathbb{S}_{R}$. Let
$\jmath_{R}\,:\,\mathbb{R}^{2}\to\mathbb{S}_{R}$ be the parametrization of
$\mathbb{S}_{R}$ by the stereographic coordinates. By
$\jmath_{R}^{*}\sharp\mu_{R}$ we denote the measure on
$\mathscr{S}^{\prime}(\mathbb{R}^{2})$ obtained by the push-forward of
$\mu_{R}$ by the pullback
$\jmath_{R}^{*}\,:\,\mathscr{D}^{\prime}(\mathbb{S}_{R})\to\mathscr{S}^{\prime}(\mathbb{R}^{2})$.
The main result of the paper is the following theorem.

######  Theorem 1.1.

The sequence of measures $(\jmath_{R}^{*}\sharp\mu_{R})_{R\in\mathbb{N}_{+}}$
on $\mathscr{S}^{\prime}(\mathbb{R}^{2})$ has a weakly convergent subsequence.
Every accumulation point $\mu$ of
$(\jmath_{R}^{*}\sharp\mu_{R})_{R\in\mathbb{N}_{+}}$ is invariant under the
Euclidean symmetries of the plane and reflection positive. Moreover, there
exists a ball $B\subset\mathscr{S}(\mathbb{R}^{2})$ with respect to some
Schwartz semi-norm centered at the origin such that for all $f\in B$ it holds

$\int\exp(\phi(f)^{n})\,\mu(\mathrm{d}\phi)\leq 2.$ (1.3)

######  Remark 1.2.

Any accumulation point $\mu$ of
$(\jmath_{R}^{*}\sharp\mu_{R})_{R\in\mathbb{N}_{+}}$ is called the
$P(\Phi)_{2}$ measure on the plane.

######  Remark 1.3.

The bound (1.3) implies that $\mu$ is non-Gaussian as Gaussian measures do not
integrate functions growing so fast. Moreover, the Osterwalder-Schrader
regularity axiom [25] is an immediate consequence of this bound.

######  Remark 1.4.

By the above theorem $\mu$ satisfies all the Osterwalder-Schrader axioms [25]
possibly with the exception of the cluster property (the decay of correlation
functions). It is known that the $P(\Phi)_{2}$ measure on the plane is unique
provided $\lambda\in(0,\infty)$ is sufficiently small [15]. In general
uniqueness does not hold and the model exhibits phase transitions [16, Ch.
16]. The cluster property is only expected to hold in pure phases. Our
construction of the $P(\Phi)_{2}$ measure does not need any smallness
assumption on $\lambda$. However, it does not give any information about the
uniqueness of the infinite volume limit or the decay of correlation functions.
In what follows we set $\lambda=1$.

###### Proof.

The existence of a weakly convergent subsequence of
$(\jmath_{R}^{*}\sharp\mu_{R})_{R\in\mathbb{N}_{+}}$ follows from tightness
and Prokhorov’s theorem. The proof of tightness is presented in Sec. 6 and
uses parabolic stochastic quantization combined with a PDE energy estimate.
The invariance of $\mu$ under the Euclidean symmetries is established in Sec.
9 and is based on the fact that for all $R\in\mathbb{N}_{+}$ the measure
$\mu_{R}$ is invariant under the group of rotations of $\mathbb{S}_{R}$. The
proof that $\mu$ is reflection positive is given in Sec. 8 and is based on the
fact that for all $R\in\mathbb{N}_{+}$ the measure $\mu_{R}$ is reflection
positive. The bound (1.3) is proved in Sec. 7 with the use of the Hairer-
Steele argument [20]. ∎

The $P(\Phi)_{2}$ model has been extensively studied in the literature and is
arguably the simplest example of an interacting QFT. The overview of various
approaches used to construct this model can be found in the books [17, 27] and
the review article [28]. Since the finite-volume $P(\Phi)_{2}$ measure is
absolutely continuous with respect to the free field measure the construction
of the model in finite volume is quite elementary and was given by Nelson in
[23, 10]. We also mention constructions of the $P(\Phi)_{2}$ models on de
Sitter spacetime, whose Euclidean counterpart is a sphere [14, 5, 21]. The
construction of the infinite volume $P(\Phi)_{2}$ model directly in the
Lorentzian signature including the verification of the Haag-Kastler axioms was
carried out in early 70’s by Glimm and Jaffe [16]. The construction was later
revisited in the Euclidean setting. For $\lambda>0$ sufficiently small a
complete construction of the Euclidean $P(\Phi)_{2}$ model and the
verification of all of the Osterwalder-Schrader axioms including exponential
decay of correlations was given in [15] (see also [17, 16]) using the cluster
expansion technique. Let us also mention an alternative technique based on the
correlation equalities that works for all $\lambda$ positive and polynomials
$P(\tau)=Q(\tau)-h\,\tau$ such that $Q$ is an even polynomial and
$h\in\mathbb{R}$, which was originally developed in [19] (see also [27, 17]).
We stress that the method of our paper works for all $\lambda$ positive and
all polynomials $P$ bounded from below. In order to control the infinite
volume limit we have to prove certain bounds for moments of the regularized
measures uniform in both the ultraviolet and infrared cutoffs. To this end, we
use the parabolic stochastic quantization and energy method. Let us note that
a similar approach has already been used, for example, in [1, 18, 2] to
construct the $\Phi^{4}_{3}$ model. The analysis of the above-mentioned
references can be trivially adapted to the case of the much simpler
$P(\Phi)_{2}$ model. Let us point out that in [18] the infinite volume measure
is constructed as a limit of a sequence of measures defined on tori of
increasing size. The symmetry group of the torus consists of translations,
reflections and rotations by a multiple of $\pi/2$ and it is easy to prove
that the infinite volume measure also has these symmetries. However, it is not
clear whether it is invariant under all rotations. In the construction of [2]
an infrared cutoff preserving the rotational invariance was used. The
rotational invariance of the infinite volume limit is then obvious. However,
the translational invariance is far from clear as it is explicitly broken by
the infrared cutoff. In [1] infinite volume limit was not investigated. In the
present work we study $P(\Phi)_{2}$ measures defined on spheres of increasing
radius. In order to show the invariance of the infinite volume $P(\Phi)_{2}$
measure under all Euclidean transformations we take advantage of the fact that
the symmetry groups of the plane and the sphere have the same dimension. We
remark that using the strategy of this paper it should also be possible to
construct the infinite volume $\Phi^{4}_{3}$ measure and prove its invariance
under all Euclidean transformations of $\mathbb{R}^{3}$ by appropriately
adapting the analysis of [18, 2].

The plan of the paper is as follows. In Sec. 2 we introduce the $P(\Phi)_{2}$
measure $\mu_{R,N}$ on the sphere $\mathbb{S}_{R}$ with a certain UV cutoff
$N\in\mathbb{N}_{+}$ in the frequency space and prove the convergence of
$\mu_{R,N}$ as $N\to\infty$ to the measure $\mu_{R}$ formally given by Eq.
(1.2). We also investigate a certain auxiliary measure $\mu_{R,N}^{g}$, which
coincides with $\mu_{R,N}$ when $g=0$. The auxiliary measure $\mu_{R,N}^{g}$
is used in Sec. 7 to prove the bound (1.3). In Sec. 3 we study the stochastic
quantization equation of the measure $\mu_{R,N}^{g}$. We also introduce a
related stochastic PDE obtained with the use of the so-called Da Prato-
Debussche trick that, in contrast to the former SPDE, is well defined in the
limit $N\to\infty$. In Sec. 5 we apply the energy technique to prove a certain
a priori bound for the latter SPDE. The a priori bound is uniform in both the
radius of the sphere $R$ as well as the UV cutoff $N$ and is the main
ingredient in the proof of the existence of the infinite volume limit of the
measures $\mu_{R}$, which is presented in Sec. 6. In order to make sense of
the infinite volume limit we have to first identify the measure $\mu_{R}$ on
$\mathscr{D}^{\prime}(\mathbb{S}_{R})$ with a certain measure on
$\mathscr{S}^{\prime}(\mathbb{R}^{2})$. To this end, we use the stereographic
projection of the sphere $\mathbb{S}_{R}$ onto the plane $\mathbb{R}^{2}$
whose definition is recalled in Sec. 4. Sec. 8 is devoted to the proof of the
reflection positivity. In Sec. 9 we show that an infinite volume $P(\Phi)_{2}$
measure is invariant under translations and rotations. The proof relies on the
invariance of the finite volume measure $\mu_{R}$ on
$\mathscr{D}^{\prime}(\mathbb{S}_{R})$ under all rotations of the sphere
$\mathbb{S}_{R}$ and some elementary properties of the stereographic
projection. More specifically, we use the observation that if the radius $R$
of the sphere is very big, then the Euclidean transformations of the plane are
well approximated by appropriately chosen rotations of the sphere. In Appendix
A we recall the definitions and collect useful facts about various function
spaces used in the paper. Appendix B contains some auxiliary results. In
Appendix C we prove uniform bounds for moments of norms of Wick polynomials of
regularized free fields.

##  2 Ultraviolet limit

In this section we recall the construction of the $P(\Phi)_{2}$ measure on
$\mathbb{S}_{R}$ based on the Nelson hypercontractivity estimate [23]. We
first introduce the measures $(\mu_{R,N})_{N\in\mathbb{N}_{+}}$ with the UV
regularization and show that the limit $\lim_{N\to\infty}\mu_{R,N}=\mu_{R}$
exists in the sense of weak convergence of measures. For
$R,N\in\mathbb{N}_{+}$ we define the bounded operators
$G_{R},K_{R,N}\,:\,L_{2}(\mathbb{S}_{R})\to L_{2}(\mathbb{S}_{R})$,

$G_{R}:=(1-\Delta_{R})^{-1},\qquad K_{R,N}:=(1-\Delta_{R}/N^{2})^{-1}$ (2.1)

and a probability measure on $\mathscr{D}^{\prime}(\mathbb{S}_{R})$

$\mu_{R,N}(\mathrm{d}\phi):=\frac{1}{\mathcal{Z}_{R,N}}\exp\left(-\int_{\mathbb{S}_{R}}P(\phi(\mathrm{x}),c_{R,N})\,\rho_{R}(\mathrm{d}\mathrm{x})\right)\nu_{R,N}(\mathrm{d}\phi),$
(2.2)

where $\nu_{R,N}$ is the Gaussian measure on
$\mathscr{D}^{\prime}(\mathbb{S}_{R})$ with covariance
$G_{R,N}:=K_{R,N}G_{R}K_{R,N}$,

$c_{R,N}:=\int_{\mathscr{D}^{\prime}(\mathbb{S}_{R})}\phi(\mathrm{x})^{2}\,\nu_{R,N}(\mathrm{d}\phi)=\mathrm{Tr}(K_{R,N}G_{R}K_{R,N})/4\pi
R^{2}$ (2.3)

is the so-called counterterm and

$P(\tau,c):=\sum_{m=0}^{n}a_{m}\,\sum_{k=0}^{\lfloor
m/2\rfloor}\frac{(-1)^{k}m!}{(m-2k)!k!2^{k}}c^{k}\tau^{m-2k},\qquad\tau,c\in\mathbb{R}.$
(2.4)

Note that by Lemma B.1 there exists $C\in(0,\infty)$ such that for all
$N,R\in\mathbb{N}_{+}$ it holds

$|c_{R,N}-1/2\pi\,\log N|\leq C.$ (2.5)

Observe also that $P(\phi(\mathrm{x}),c_{R,N})$ is obtained by Wick-ordering
$P(\phi(\mathrm{x}))$ with respect to the regularized measure $\nu_{R,N}$.
Accordingly, the sum over $k$ in the definition of $P(\tau,c)$ amounts for
$c\geq 0$ to $\tau\mapsto c^{m/2}H_{m}(\tau/c^{1/2})$, where $H_{m}$,
$m\in\mathbb{N}_{0}$, are the Hermite polynomials, cf. Appendix C.

######  Remark 2.1.

The radius of the sphere $R$ and the parameter $N$ play the role of the
infrared and ultraviolet cutoffs, respectively. To remove the cutoffs one has
to study the limits $R\to\infty$ and $N\to\infty$. In this section we
investigate the ultraviolet limit $N\to\infty$.

Actually, in order to establish the bound (1.3) we will study a more general
class of probability measures

$\mu_{R,N}^{g}(\mathrm{d}\phi):=\frac{1}{\mathcal{Z}^{g}_{R,N}}\exp\left(\phi(g)^{n}/n\right)\mu_{R,N}(\mathrm{d}\phi),$
(2.6)

with $g\in C^{\infty}(\mathbb{S}_{R})$ such that

$\|g\|^{n}_{L_{n/(n-1)}(\mathbb{S}_{R})}\leq
1/2,\qquad\|\Delta_{R}g\|^{n}_{L_{n/(n-1)}(\mathbb{S}_{R})}\leq 1/2.$ (2.7)

The usefulness of the measure $\mu_{R,N}^{g}$ comes from Lemma 7.2, which says
that in order to show the bound (1.3) it is sufficient to prove a certain
uniform bound for some finite moment of the measure $\mu_{R,N}^{g}$. In Lemma
2.2 we show that the measure $\mu_{R,N}^{g}$ is well defined. Proposition 2.8
implies in particular that for every $R\in\mathbb{N}_{+}$ the sequence of
measures $(\mu_{R,N}^{g})_{N\in\mathbb{N}_{+}}$ converges weakly to a measure
denoted by $\mu_{R}^{g}$. If $g=0$, then $\mu_{R}^{g}$ coincides with the
$P(\Phi)_{2}$ measure on $\mathbb{S}_{R}$, which is denoted by $\mu_{R}$.
Moreover, the measures $\mu_{R}^{g}$ and $\mu_{R}$ are related by a formula
analogous to (2.6).

######  Lemma 2.2.

For all $R,N\in\mathbb{N}_{+}$ and $g\in C^{\infty}(\mathbb{S}_{R})$ such that
$\|g\|^{n}_{L_{n/(n-1)}(\mathbb{S}_{R})}\leq 1/2$ the measure $\mu_{R,N}^{g}$
is well-defined and both $\nu_{R,N}$ and $\mu_{R,N}^{g}$ are concentrated on
$L^{1}_{2}(\mathbb{S}_{R})\subset\mathscr{D}^{\prime}(\mathbb{S}_{R})$.

######  Remark 2.3.

We identify implicitly a function $\phi\in L_{1}(\mathbb{S}_{R})$ with a
distribution $\varphi\in\mathscr{D}^{\prime}(\mathbb{S}_{R})$ defined by
$\varphi(f)\equiv\langle\varphi,f\rangle:=\int_{\mathbb{S}_{R}}\phi(\mathrm{x})f(\mathrm{x})\,\rho_{R}(\mathrm{d}\mathrm{x})$.

###### Proof.

By Lemma C.6 the measure $\nu_{R,N}$ is concentrated on
$L^{1}_{2}(\mathbb{S}_{R})$. By the Sobolev embedding
$L^{1}_{2}(\mathbb{S}_{R})\subset L_{n}(\mathbb{S}_{R})$ stated in Lemma A.9,
the bound $\phi(g)^{n}/n\leq\|\phi\|^{n}_{L_{n}(\mathbb{S}_{R})}/2n$ and the
boundedness from below of the polynomial $\tau\mapsto
P(\tau,c_{R,N})-\tau^{n}/2n$ the function

$\mathcal{U}^{g}_{R,N}\,:\,L_{n}(\mathbb{S}_{R})\ni\phi\mapsto\exp\left(\frac{1}{n}\phi(g)^{n}-\int_{\mathbb{S}_{R}}P(\phi(\mathrm{x}),c_{R,N})\,\rho_{R}(\mathrm{d}\mathrm{x})\right)\in(0,\infty)$
(2.8)

is bounded and continuous. Moreover,
$\mathcal{Z}_{R,N}\mathcal{Z}_{R,N}^{g}=\|\mathcal{U}^{g}_{R,N}\|_{L_{1}(\mathscr{D}^{\prime}(\mathbb{S}_{R}),\nu_{R,N})}\geq
1$ by the Jensen inequality and Lemma C.1. This proves the claims about the
measure $\mu_{R,N}^{g}$. ∎

######  Definition 2.4.

We define $X_{R}$ to be the Gaussian random variable valued in
$\mathscr{D}^{\prime}(\mathbb{S}_{R})$ with mean zero and covariance $G_{R}$.
We set $X_{R,N}:=K_{R,N}X_{R}$,

$\begin{gathered}X_{R,N}^{:m:}(\mathrm{x}):=\sum_{k=0}^{\lfloor
m/2\rfloor}\frac{(-1)^{k}m!}{(m-2k)!k!2^{k}}(c_{R,N})^{k}X_{R,N}^{m-2k}(\mathrm{x}),\quad
X_{R,N}^{:m:}(h):=\int_{\mathbb{S}_{R}}X_{R,N}^{:m:}(\mathrm{x})\,h(\mathrm{x})\,\rho_{R}(\mathrm{d}\mathrm{x}),\\\
Y_{R,N}:=\sum_{m=0}^{n}a_{m}X_{R,N}^{:m:}(1_{\mathbb{S}_{R}}),\qquad
Y_{R,N}^{g}:=Y_{R,N}-X_{R,N}(g)^{n}/n,\end{gathered}$ (2.9)

where $h\in L_{\infty}(\mathbb{S}_{R})$ and $1_{B}$ denotes the characteristic
function of the set $B\subset\mathbb{S}_{R}$.

######  Remark 2.5.

By Lemma C.6 it holds $X_{R,N}\in L^{1}_{2}(\mathbb{S}_{R})\subset
L_{n}(\mathbb{S}_{R})$ almost surely. In particular, $Y^{g}_{R,N}$ is well-
defined. Moreover, for positive measurable $F$ we have

$\int
F(\phi)\,\mu^{g}_{R,N}(\mathrm{d}\phi)=\frac{\mathbb{E}F(X_{R,N})\exp(-Y^{g}_{R,N})}{\mathbb{E}\exp(-Y^{g}_{R,N})}.$
(2.10)

######  Lemma 2.6.

Let $X$ be a real-valued random variable such that $X\geq 0$. Suppose that the
function $F\,:\,[0,\infty)\to[0,\infty)$ is continuously differentiable and
such that $F(0)=0$ and $F^{\prime}\geq 0$. Then it holds

$\mathbb{E}F(X)=\int_{0}^{\infty}\mathbb{P}(X>t)\,F^{\prime}(t)\,\mathrm{d}t.$
(2.11)

######  Lemma 2.7.

There exists $A\in(0,\infty)$ depending only on the coefficients of the
polynomial $\tau\mapsto P(\tau)$ such that for all $\tau\in\mathbb{R}$ and
$c\in(1,\infty)$ it holds $P(\tau,c)\geq\tau^{n}/2n-A\,c^{n/2}$.

###### Proof.

By the Young inequality for all $m\in\\{0,1,\cdots,n-1\\}$,
$k\in\\{0,1,\cdots,\lfloor\frac{m}{2}\rfloor$}, $a\in\mathbb{R}$ and
$\delta\in(0,1)$ there exists $C\in(0,\infty)$ such that for all
$c\in(1,\infty)$ and $\tau\in\mathbb{R}$ it holds

$a\,\tau^{m-2k}c^{k}\geq-\delta\,\tau^{n}-C\,c^{\frac{n}{2}}.$ (2.12)

To conclude we apply the above bound to all terms of the polynomial
$P(\tau,c)$ but the term $\tau^{n}/n$ and choose $\delta\in(0,1)$ sufficiently
small. ∎

######  Proposition 2.8.

Let $R\in\mathbb{N}_{+}$ and $g\in C^{\infty}(\mathbb{S}_{R})$ satisfy the
bounds (2.7). There exist random variables
$X_{R}\in\mathscr{D}^{\prime}(\mathbb{S}_{R})$, see Def. 2.4, and
$Y_{R}^{g}:=Y_{R}-X_{R}(g)^{n}/n\in\mathbb{R}$ such that
$\mathbb{E}\exp(-Y_{R}^{g})<\infty$ and for all bounded and continuous
$F\,:\,\mathscr{D}^{\prime}(\mathbb{S}_{R})\to\mathbb{R}$ and $p\in(0,\infty)$
it holds

$\lim_{N\to\infty}\mathbb{E}F(X_{R,N})\exp(-p\,Y_{R,N}^{g})=\mathbb{E}F(X_{R})\exp(-p\,Y_{R}^{g}).$
(2.13)

###### Proof.

By Vitali’s theorem it suffices to establish that
$(F(X_{R,N})\exp(-p\,Y_{R,N}^{g}))_{N\in\mathbb{N}_{+}}$ converges in
probability to $F(X_{R})\exp(-p\,Y_{R}^{g})$ and is uniformly integrable. The
convergence in probability follows from Lemmas C.7 and C.8. To show uniform
integrability it is enough to demonstrate that
$(\mathbb{E}\exp(-p\,Y_{R,N}^{g}))_{N\in\mathbb{N}_{+}}$ is bounded for all
$p\in(0,\infty)$. By Lemma 2.6 we have

$\displaystyle\mathbb{E}\exp(-p\,Y_{R,N}^{g})$ $\displaystyle\leq
1+\mathbb{E}(\exp(-p\,(Y^{g}_{R,N}\wedge 0))-1)$ (2.14)
$\displaystyle=1+\int_{0}^{\infty}\mathbb{P}(-p\,(Y^{g}_{R,N}\wedge
0)>t)\,\exp(t)\,\mathrm{d}t$
$\displaystyle=1+\int_{0}^{\infty}\mathbb{P}(-p\,Y^{g}_{R,N}>t)\,\exp(t)\,\mathrm{d}t.$

By Lemma 2.7 for every $R\in\mathbb{N}_{+}$ there exists $A\in(0,\infty)$ such
that for all $M\in\mathbb{N}_{+}$ it holds $Y_{R,M}^{g}\geq-A\,c_{R,M}^{n/2}$.
Consequently, by adding the latter inequality to
$-Y^{g}_{R,N}>2A\,c_{R,M}^{n/2}$, for every $R\in\mathbb{N}_{+}$ there exist
$c,C\in(0,\infty)$ such that for all $N,M\in\mathbb{N}_{+}$ it holds

$\mathbb{P}(-Y^{g}_{R,N}>2A\,c_{R,M}^{n/2})\leq\mathbb{P}(|Y^{g}_{R,N}-Y^{g}_{R,M}|>A\,c_{R,M}^{n/2})\\\
\leq\exp(-c\,c_{R,M}\,A^{2/n}\,M^{1/n^{2}})\,\mathbb{E}\exp(c\,M^{1/n^{2}}\,|Y^{g}_{R,N}-Y^{g}_{R,M}|^{2/n})\leq
C\,\exp(-M^{1/n^{2}}),$ (2.15)

where the last bound follows from Lemmas C.7 and C.8, the Nelson
hypercontractivity estimate stated in Lemma C.2 and the estimate (2.5) for the
counterterm $c_{R,M}$. As a result, by the bound (2.5) for every
$R\in\mathbb{N}_{+}$ and $p\in(0,\infty)$ there exist $c,C\in(0,\infty)$ such
that for all $N\in\mathbb{N}_{+}$ and $t\in[0,\infty)$ it holds

$\mathbb{P}(-p\,Y^{g}_{R,N}>t)\leq C\,\exp(-\exp(c\,t^{2/n})).$ (2.16)

The above bound together with Eq. (2.14) imply that
$(F(X_{R,N})\exp(-p\,Y^{g}_{R,N}))_{N\in\mathbb{N}_{+}}$ is uniformly
integrable. This finishes the proof. ∎

##  3 Stochastic quantization

In order to show the existence of the infinite volume limit of the
$P(\Phi)_{2}$ model and prove the bound (1.3) we have to establish appropriate
bounds for moments of the regularized measure $\mu^{g}_{R,N}$ uniform in
$R,N\in\mathbb{N}_{+}$. To this end, we shall use the so-called parabolic
stochastic quantization technique. We study a certain stochastic process
evolving in fictitious time whose stationary distribution coincides with the
Euclidean QFT measure. The process satisfies a non-linear stochastic PDE that
is called the stochastic quantization equation. More specifically, to prove
desired uniform bounds we apply the energy method, which relies on testing the
equation by the solution itself and estimating the terms that are not
positive. Because of the UV problem the stochastic quantization equation of
the measure $\mu^{g}_{R,N}$, that is Eq. (3.2) below, becomes singular in the
limit $N\to\infty$. For this reason we cannot apply the energy method directly
to Eq. (3.2). We use the so-called Da Prato-Debussche trick that is based on
the observation that the most singular part of the solution
$\varPhi^{g}_{R,N}$ of Eq. (3.2) coincides with the solution $Z_{R,N}$ of the
stochastic quantization equation of the Gaussian measure $\nu_{R,N}$, that is
Eq. (3.1). It turns out that Eq. (3.10), which is satisfied by the process
$\varPsi^{g}_{R,N}:=\varPhi^{g}_{R,N}-Z_{R,N}$, is not singular in the limit
$N\to\infty$. The application of the energy method to Eq. (3.10) is the
subject of Sec. 5.

######  Definition 3.1.

For $R\in[1,\infty)$ we define
$(W_{R}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}}))_{t\in[0,\infty)}$
to be the cylindrical Wiener process on $L_{2}(\mathbb{S}_{R})$, see [9, p.
53]).

######  Definition 3.2.

For $R,N\in[1,\infty)$ we set
$Q_{R,N}:=(1-\Delta_{R})(1-\Delta_{R}/N^{2})^{2}$.

We study the following stochastic ODEs, which coincide with the stochastic
quantization equations of the measures $\nu_{R,N}$ and $\mu_{R,N}^{g}$,
respectively:

$\mathrm{d}Z_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})=\sqrt{2}\,\mathrm{d}W_{R}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})-Q_{R,N}Z_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}})\,\mathrm{d}t,$
(3.1)
$\mathrm{d}\varPhi_{R,N}^{g}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})=\sqrt{2}\,\mathrm{d}W_{R}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})-Q_{R,N}\varPhi_{R,N}^{g}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}})\,\mathrm{d}t\\\
-P^{\prime}(\varPhi_{R,N}^{g}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}),c_{R,N})\,\mathrm{d}t+(\varPhi_{R,N}^{g}(t,g))^{n-1}g\,\mathrm{d}t,$
(3.2)

where $P^{\prime}(\tau,c):=\partial_{\tau}P(\tau,c)$. The unique mild solution
$Z_{R,N}\in C([0,\infty),L^{1}_{2}(\mathbb{S}_{R}))$ of Eq. (3.1) with the
initial condition

$Z_{R,N}(0,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}})=z_{R,N}\in
L^{1}_{2}(\mathbb{S}_{R})$ (3.3)

is given by

$Z_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})=\mathrm{e}^{-tQ_{R,N}}z_{R,N}+\int_{0}^{t}\mathrm{e}^{-(t-s)Q_{R,N}}\,\sqrt{2}\,\mathrm{d}W_{R}(s,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}})$ (3.4)

for $t\in[0,\infty)$, see e.g. [9, Sec. 5.2]. By definition the mild solution
of Eq. (3.2) with the initial condition

$\varPhi_{R,N}^{g}(0,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}})=\phi_{R,N}^{g}\in
L^{1}_{2}(\mathbb{S}_{R})$ (3.5)

is the stochastic process $\varPhi_{R,N}^{g}\in
C([0,\infty),L^{1}_{2}(\mathbb{S}_{R}))$ such that for all $t\in[0,\infty)$ it
holds

$\varPhi_{R,N}^{g}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})=\mathrm{e}^{-tQ_{R,N}}\phi_{R,N}^{g}\\\
+\int_{0}^{t}\mathrm{e}^{-(t-s)Q_{R,N}}\,\big{(}\sqrt{2}\,\mathrm{d}W_{R}(s,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})-P^{\prime}(\varPhi_{R,N}^{g}(s,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}),c_{R,N})\mathrm{d}s+(\varPhi_{R,N}^{g}(s,g))^{n-1}g\,\mathrm{d}s\big{)}.$
(3.6)

The mild solution $\varPhi^{g}_{R,N}$ exists and is unique, cf. [9, Sect.
5.5].

######  Definition 3.3.

By definition the stochastic processes $Z_{R,N},\varPhi^{g}_{R,N}\in
C([0,\infty),L^{1}_{2}(\mathbb{S}_{R}))$ are the unique solutions of Eq. (3.4)
and Eq. (3.6), respectively, with random initial data $z_{R,N}$ and
$\phi^{g}_{R,N}$, respectively, such that $z_{R,N}$ and $\phi^{g}_{R,N}$ are
independent of
$(W_{R}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}}))_{t\in[0,\infty)}$
and $\mathrm{Law}(z_{R,N},\phi_{R,N}^{g})=\nu_{R,N}\times\mu^{g}_{R,N}$. We
also define the process

$\varPsi^{g}_{R,N}:=\varPhi^{g}_{R,N}-Z_{R,N}\in
C([0,\infty),L^{1}_{2}(\mathbb{S}_{R})).$ (3.7)

######  Remark 3.4.

The processes $Z_{R,N},\varPhi^{g}_{R,N},\varPsi^{g}_{R,N}\in
C([0,\infty),L^{1}_{2}(\mathbb{S}_{R}))$ are well-defined because the measures
$\nu_{R,N}$ and $\mu^{g}_{R,N}$ are concentrated on
$L^{1}_{2}(\mathbb{S}_{R})$.

The following lemma expresses the fact that the measures $\nu_{R,N}$ and
$\mu^{g}_{R,N}$ are invariant for Eq. (3.4) and Eq. (3.6), respectively.

######  Lemma 3.5.

For all $t\in[0,\infty)$ it holds

$\mathrm{Law}(Z_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}))=\nu_{R,N},\qquad\mathrm{Law}(\varPhi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}}))=\mu^{g}_{R,N}.$
(3.8)

###### Proof.

See [9, Sec. 8.6]. ∎

######  Lemma 3.6.

It holds

$\varPsi^{g}_{R,N}\in C([0,\infty),L^{1}_{2}(\mathbb{S}_{R}))\cap
C((0,\infty),L^{3}_{2}(\mathbb{S}_{R}))\cap
C^{1}((0,\infty),L^{-3}_{2}(\mathbb{S}_{R}))$ (3.9)

and the following equality

$\partial_{t}\varPsi^{g}_{R,N}=-Q_{R,N}\varPsi^{g}_{R,N}-P^{\prime}(\varPsi^{g}_{R,N}+Z_{R,N},c_{R,N})+((\varPsi^{g}_{R,N}+Z_{R,N})(t,g))^{n-1}g$
(3.10)

is satisfied in $C((0,\infty),L^{-3}_{2}(\mathbb{S}_{R}))$.

###### Proof.

We first note that

$\varPsi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})=\mathrm{e}^{-tQ_{R,N}}(\phi_{R,N}^{g}-z_{R,N})\\\
-\int_{0}^{t}\mathrm{e}^{-(t-s)Q_{R,N}}\big{(}P^{\prime}(\varPhi^{g}_{R,N}(s,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}),c_{R,N})-(\varPhi^{g}_{R,N}(s,g))^{n-1}g({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\big{)}\,\mathrm{d}s.$ (3.11)

It holds $\phi_{R,N}^{g}-z_{R,N}\in L_{2}^{1}(\mathbb{S}_{R})$ and
$P^{\prime}(\varPhi^{g}_{R,N},c_{R,N})\in C([0,\infty),L_{2}(\mathbb{S}_{R}))$
almost surely by Lemma A.9. The statement follows from the regularizing
properties of the semigroup $(\mathrm{e}^{-tQ_{R,N}})_{t\in[0,\infty)}$. ∎

######  Definition 3.7.

For $l\in\\{0,\ldots,n-2\\}$ and $m\in\\{l,\ldots,n-1\\}$ we define

$a_{m,l}:=-a_{m+1}\,(m+1)!/(m-l)!l!,$ (3.12)

where $(a_{m})_{m\in\\{1,\ldots,n\\}}$ are the coefficients of the polynomial
$P(\tau)$.

######  Definition 3.8.

By definition $Z_{R,N}^{:0:}:=1$ and for $m\in\\{1,\ldots,n-1\\}$

$Z_{R,N}^{:m:}:=\sum_{k=0}^{\lfloor
m/2\rfloor}\frac{(-1)^{k}m!}{(m-2k)!k!2^{k}}(c_{R,N})^{k}Z_{R,N}^{m-2k}.$
(3.13)

Note that it holds

$P^{\prime}(\varPsi^{g}_{R,N}+Z_{R,N},c_{R,N})=(\varPsi^{g}_{R,N})^{n-1}-\sum_{l=0}^{n-2}\sum_{m=l}^{n-1}a_{m,l}\,Z_{R,N}^{:m-l:}\,(\varPsi^{g}_{R,N})^{l},$
(3.14)

where $P^{\prime}(\tau,c):=\partial_{\tau}P(\tau,c)$. Consequently, Eq. (3.10)
can be rewritten in the form

$(\partial_{t}+Q_{R,N})\varPsi^{g}_{R,N}+(\varPsi^{g}_{R,N})^{n-1}\\\
=\sum_{l=0}^{n-2}\sum_{m=l}^{n-1}a_{m,l}\,Z_{R,N}^{:m-l:}\,(\varPsi^{g}_{R,N})^{l}+((\varPsi^{g}_{R,N}+Z_{R,N})({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}},g))^{n-1}g.$ (3.15)

##  4 Stereographic projection

######  Definition 4.1.

For $R\in[1,\infty)$ we define

$\mathbb{S}_{R}:=\\{\mathrm{x}=(\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3})\in\mathbb{R}^{3}\,|\,\mathrm{x}_{1}^{2}+\mathrm{x}_{2}^{2}+\mathrm{x}_{3}^{2}=R^{2}\\}.$
(4.1)

For $R\in[1,\infty)$ and $\mathrm{x},\mathrm{y}\in\mathbb{S}_{R}$ we denote by
$d_{R}(\mathrm{x},\mathrm{y})$ the length of the shortest curve in
$\mathbb{S}_{R}\subset\mathbb{R}^{3}$ connecting $\mathrm{x}$ and
$\mathrm{y}$. For $R\in[1,\infty)$ we denote by $\rho_{R}$ the rotationally
invariant measure on $\mathbb{S}_{R}$ normalized such that
$\rho_{R}(\mathbb{S}_{R})=4\pi R^{2}$. For $R\in[1,\infty)$ we denote by
$\Delta_{R}$ the Laplace-Beltrami operator on $\mathbb{S}_{R}$. We denote by
$\Delta$ the Laplace operator on $\mathbb{R}^{2}$.

######  Definition 4.2.

For $R\in[1,\infty)$ the map
$\jmath_{R}\,:\mathbb{R}^{2}\to\mathbb{S}_{R}\setminus\\{(0,0,R)\\}\subset\mathbb{R}^{3}$
is defined by

$\jmath_{R}(x_{1},x_{2}):=\mathrm{x}\equiv(\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3})=\frac{R\,(4Rx_{1},4Rx_{2},x_{1}^{2}+x_{2}^{2}-4R^{2})}{4R^{2}+x_{1}^{2}+x_{2}^{2}}.$
(4.2)

We call $(x_{1},x_{2})=x\in\mathbb{R}^{2}$ the stereographic coordinates of
$\mathrm{x}\in\mathbb{S}_{R}\setminus\\{(0,0,R)\\}$. We denote by
$\jmath_{R}^{*}$ the pullback by $\jmath_{R}$. We also set
$w_{R}(x):=16R^{4}/(4R^{2}+x_{1}^{2}+x_{2}^{2})^{2}$.

Figure 1: Stereographic projection

######  Remark 4.3.

If $f\in C(\mathbb{S}_{R})$, then $\jmath_{R}^{*}f=f\circ\jmath_{R}\in
C(\mathbb{R}^{2})$. Note that for $f\in
C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2})$ the function
$f\circ\jmath_{R}^{-1}\in
C_{\mathrm{c}}(\mathbb{S}_{R}\setminus\\{(0,0,R)\\})$ has unique smooth
extension to $\mathbb{S}_{R}$. If
$\phi\in\mathscr{D}^{\prime}(\mathbb{S}_{R})$, then
$\jmath_{R}^{*}\phi\in\mathscr{S}^{\prime}(\mathbb{R}^{2})$ is defined by
$\langle\jmath_{R}^{*}\phi,f\rangle:=\langle\phi,(w_{R}^{-1}f)\circ\jmath_{R}^{-1}\rangle$
for all $f\in C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2})$.

######  Remark 4.4.

Note that it holds

$\int_{\mathbb{S}_{R}}f(\mathrm{x})\,\rho_{R}(\mathrm{d}\mathrm{x})=\int_{\mathbb{R}^{2}}\jmath_{R}^{*}f(x)\,w_{R}(x)\,\mathrm{d}x,\qquad\jmath_{R}^{*}\Delta_{R}=w_{R}^{-1}\Delta\jmath_{R}^{*}\,.$
(4.3)

##  5 A priori bound

Note that Eq. (3.15) takes the following form in the stereographic coordinates

$(\partial_{t}+(1-w_{R}^{-1}(x)\Delta)(1-w_{R}^{-1}(x)\Delta/N^{2})^{2})\jmath_{R}^{*}\varPsi^{g}_{R,N}(t,x)\\\
=\sum_{l=0}^{n-2}\sum_{m=l}^{n-1}a_{m,l}\,\jmath_{R}^{*}Z_{R,N}^{:m-l:}(t,x)\,(\jmath_{R}^{*}\varPsi^{g}_{R,N}(t,x))^{l}\\\
+(\jmath_{R}^{*}(\varPsi^{g}_{R,N}+Z_{R,N})(t,w_{R}\jmath_{R}^{*}g))^{n-1}\jmath_{R}^{*}g(x)\,.$
(5.1)

In this section we prove an a priori bound by multiplying both sides of the
above equation by $v_{L}\jmath_{R}^{*}\varPsi^{g}_{R,N}$, where $v_{L}$ is a
suitable weight, and integrating over $\mathbb{R}^{2}$. The bound is stated in
the proposition below and is used in the next section to prove the existence
of the infinite volume limit.

######  Definition 5.1.

Let $v_{L}:=\frac{1}{4\pi L^{2}}w_{L}^{8}$, where $L\in[1,\infty)$ is fixed as
in Lemma 5.3.

######  Proposition 5.2.

There exist $\kappa\in(0,\infty)$, $C\in(0,\infty)$, $p\in[1,\infty)$ and a
ball $B\subset\mathscr{S}(\mathbb{R}^{2})$ with respect to some Schwartz semi-
norm centered at the origin such that for all $t\in(0,\infty)$ and
$R,N\in\mathbb{N}_{+}$, $R\geq L$, as well as all $g\in
C^{\infty}(\mathbb{S}_{R})$, $w_{R}\jmath_{R}^{*}g\in B$, it holds

$8\,\partial_{t}\|\jmath_{R}^{*}\varPsi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{2}_{L_{2}(\mathbb{R}^{2},v_{L}^{1/2})}+\|\jmath_{R}^{*}\varPsi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{n}_{L_{n}(\mathbb{R}^{2},v_{L}^{1/n})}\\\ \leq
C\sum_{k=0}^{n-1}\,\|\jmath_{R}^{*}Z_{R,N}^{:k:}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{p}_{L_{p}^{-\kappa}(\mathbb{R}^{2},v_{L}^{1/p})}\,.$ (5.2)

###### Proof.

After multiplying both sides Eq. (5.1) by
$v_{L}\jmath_{R}^{*}\varPsi^{g}_{R,N}$, integrating over space and applying
Lemma 5.3 and Remark 5.4 we obtain

$1/2~{}\partial_{t}\|\jmath_{R}^{*}\varPsi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{2}_{L_{2}(\mathbb{R}^{2},v_{L}^{1/2})}+1/2~{}\|\vec{\nabla}\jmath_{R}^{*}\varPsi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{2}_{L_{2}(\mathbb{R}^{2},v_{L}^{1/2})}\\\
+1/8~{}\|\jmath_{R}^{*}\varPsi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{n}_{L_{n}(\mathbb{R}^{2},v_{L}^{1/n})}\leq
R^{(1)}_{R,N}(t)+R^{(2)}_{R,N}(t),$ (5.3)

where

$\displaystyle R^{(1)}_{R,N}(t)$
$\displaystyle=\sum_{l=0}^{n-2}\sum_{m=l}^{n-1}\int_{\mathbb{R}^{2}}a_{m,l}\,v_{L}(x)\,\jmath_{R}^{*}Z_{R,N}^{:m-l:}(t,x)\,(\jmath_{R}^{*}\varPsi^{g}_{R,N}(t,x))^{l+1}\,\mathrm{d}x,$
(5.4) $\displaystyle R^{(2)}_{R,N}(t)$
$\displaystyle=(\jmath_{R}^{*}(\varPsi^{g}_{R,N}+Z_{R,N})(t,w_{R}\jmath_{R}^{*}g))^{n-1}\,(\jmath_{R}^{*}\varPsi^{g}_{R,N})(t,v_{L}\jmath_{R}^{*}g).$

By Lemma A.8 for every $\delta_{1}\in(0,1)$ there exists $C_{1}\in(0,\infty)$
such that

$|R_{R,N}^{(1)}(t)|\leq
C_{1}\sum_{k=0}^{n-1}\,\|\jmath_{R}^{*}Z_{R,N}^{:k:}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{p}_{L_{p}^{-\kappa}(\mathbb{R}^{2},v_{L}^{1/p})}\\\
+\delta_{1}\,\|\nabla\jmath_{R}^{*}\varPsi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{2}_{L_{2}(\mathbb{R}^{2},v_{L}^{1/2})}+\delta_{1}\,\|\jmath_{R}^{*}\varPsi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{n}_{L_{n}(\mathbb{R}^{2},v_{L}^{1/n})},\,$ (5.5)

where $k=0$ term of the sum above is a constant. Furthermore, by Hölder’s
inequality and elementary estimates there exists $C_{2}\in(0,\infty)$ such
that for all $\delta_{2}\in(0,1)$ it holds

$|R^{(2)}_{R,N}(t)|\leq
C_{2}\,\delta_{2}^{n}\,\|\jmath_{R}^{*}\varPsi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{n}_{L_{n}(\mathbb{R}^{2},v_{L}^{1/n})}+C_{2}\,\delta_{2}^{n}\,\|\jmath_{R}^{*}Z_{R,N}\|^{n}_{L_{n}^{-\kappa}(\mathbb{R}^{2},v_{L}^{1/n})}$
(5.6)

provided

$\|v_{L}^{(n-1)/n}\jmath_{R}^{*}g\|_{L_{n/(n-1)}(\mathbb{R}^{2})}\vee\|v_{L}^{-1/n}w_{R}\jmath_{R}^{*}g\|_{L_{n/(n-1)}(\mathbb{R}^{2})}\vee\|v_{L}^{-1/n}w_{R}\jmath_{R}^{*}g\|_{L^{\kappa}_{n/(n-1)}(\mathbb{R}^{2})}\leq\delta_{2}.$
(5.7)

We choose $\delta_{1},\delta_{2}$ such that $\delta_{1}\leq 1/2$ and
$\delta_{1}+C_{2}\,\delta_{2}^{n}\leq 1/16$. This finishes the proof. ∎

######  Lemma 5.3.

There exists $L\in[1,\infty)$ such that for all $R\in[L,\infty)$ it holds

  1. (A)

$\langle\Psi,v_{L}(-w^{-1}_{R}\Delta)\Psi\rangle_{L_{2}(\mathbb{R}^{2})}\geq
1/2\,\|\vec{\nabla}\Psi\|_{L_{2}(\mathbb{R}^{2},w^{-1/2}_{R}v_{L}^{1/2})}^{2}-1/8\,\|\Psi\|^{2}_{L_{2}(\mathbb{R}^{2},w_{R}^{-1/2}v_{L}^{1/2})}$,

  2. (B)

$\langle\Psi,v_{L}(-w^{-1}_{R}\Delta)^{2}\Psi\rangle_{L_{2}(\mathbb{R}^{2})}\geq
1/2\,\|\Delta\Psi\|_{L_{2}(\mathbb{R}^{2},w^{-1}_{R}v_{L}^{1/2})}^{2}-1/8\,\|\Psi\|^{2}_{L_{2}(\mathbb{R}^{2},w_{R}^{-1}v_{L}^{1/2})}$,

  3. (C)

$\langle\Psi,v_{L}(-w^{-1}_{R}\Delta)^{3}\Psi\rangle_{L_{2}(\mathbb{R}^{2})}\geq
1/2\,\|\vec{\nabla}\Delta\Psi\|_{L_{2}(\mathbb{R}^{2},w^{-3/2}_{R}v_{L}^{1/2})}^{2}-1/8\,\|\Psi\|^{2}_{L_{2}(\mathbb{R}^{2},w_{R}^{-3/2}v_{L}^{1/2})}$.

###### Proof.

There exists $C\in(0,\infty)$ such that for all $L\in[1,\infty)$,
$R\in[L,\infty)$ it holds

$|\vec{\nabla}w^{-1/2}_{R}|\leq
C\,w_{R}^{-1/2}/L,\qquad|\vec{\nabla}v^{1/2}_{L}|\leq C\,v_{L}^{1/2}/L.$ (5.8)

This gives readily (A) by integrating by parts in
$\|\vec{\nabla}\Psi\|_{L_{2}(\mathbb{R}^{2},w^{-1/2}_{R}v_{L}^{1/2})}^{2}$
applying the Leibniz rule and the Young inequality. Estimates (B) and (C) are
obtained analogously, with the help of the following auxiliary inequalities

$\|\vec{\nabla}\Psi\|_{L_{2}(\mathbb{R}^{2},w^{-1}_{R}v_{L}^{1/2})}^{2}\leq
2\|\Delta\Psi\|_{L_{2}(\mathbb{R}^{2},w^{-1}_{R}v_{L}^{1/2})}^{2}+2\|\Psi\|^{2}_{L_{2}(\mathbb{R}^{2},w_{R}^{-1}v_{L}^{1/2})}\,,$
(5.9)
$\|\vec{\nabla}\Psi\|_{L_{2}(\mathbb{R}^{2},w^{-3/2}_{R}v_{L}^{1/2})}^{2}+\|\Delta\Psi\|_{L_{2}(\mathbb{R}^{2},w^{-3/2}_{R}v_{L}^{1/2})}^{2}\leq
4\|\vec{\nabla}\Delta\Psi\|_{L_{2}(\mathbb{R}^{2},w^{-3/2}_{R}v_{L}^{1/2})}^{2}+4\|\Psi\|^{2}_{L_{2}(\mathbb{R}^{2},w_{R}^{-3/2}v_{L}^{1/2})}$
(5.10)

valid for sufficiently big $L\in[1,\infty)$ and all $R\in[L,\infty)$. The
latter inequalities are proven by the same token as (A). ∎

######  Remark 5.4.

For all $L\in[1,\infty)$, $R\in[L,\infty)$ and $p\in\\{1,2,3\\}$ it holds

$\|\Psi\|_{L_{2}(\mathbb{R}^{2},w_{R}^{-p/2}v_{L}^{1/2})}\leq\|w_{R}^{-p/2}v_{L}^{(n-2)/2n}\|_{L_{2n/(n-2)}(\mathbb{R}^{2})}\,\|\Psi\|_{L_{n}(\mathbb{R}^{2},v_{L}^{1/n})}\leq\|\Psi\|_{L_{n}(\mathbb{R}^{2},v_{L}^{1/n})}.$
(5.11)

##  6 Tightness

######  Proposition 6.1.

Let $\kappa\in(0,\infty)$. There exists a ball
$B\subset\mathscr{S}(\mathbb{R}^{2})$ with respect to some Schwartz semi-norm
centered at the origin and a constant $C\in(0,\infty)$ such that for all
$R\in\mathbb{N}_{+}$, $R\geq L$, $N\in\mathbb{N}_{+}$ and all $g\in
C^{\infty}(\mathbb{S}_{R})$, $w_{R}\jmath_{R}^{*}g\in B$, it holds

$\int\|\jmath^{*}_{R}\phi\|^{n}_{L_{n}^{-\kappa}(\mathbb{R}^{2},v_{L}^{1/n})}\,\mu^{g}_{R,N}(\mathrm{d}\phi)\leq
C.$ (6.1)

######  Remark 6.2.

By Proposition 2.8 with $F=1$ and Lemma C.9 we obtain

$\int\|\jmath_{R}^{*}\phi\|^{n}_{L_{n}^{-\kappa}(\mathbb{R}^{2},v_{L}^{1/n})}\,\mu_{R}^{g}(\mathrm{d}\phi)=\lim_{N\to\infty}\int\|\jmath_{R}^{*}\phi\|^{n}_{L_{n}^{-\kappa}(\mathbb{R}^{2},v_{L}^{1/n})}\,\mu_{R,N}^{g}(\mathrm{d}\phi)\leq
C\,.$ (6.2)

By Theorem A.5 (C) the embedding
${L^{-\kappa}_{n}(\mathbb{R}^{2},v_{L}^{1/n})}\to
L^{-2\kappa}_{n}(\mathbb{R}^{2},v_{L}^{2/n})$ is compact. As a result, by
Lemma B.4 the sequence of measures
$(\jmath_{R}^{*}\sharp\mu_{R})_{R\in\mathbb{N}_{+}}$ on
$L^{-2\kappa}_{n}(\mathbb{R}^{2},v_{L}^{2/n})$ is tight and by Prokhorov’s
theorem it has a weakly convergent subsequence.

###### Proof.

Recall from Lemma 3.5 that
$\mathrm{Law}(\varPhi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}}))=\mu_{R,N}$ for
all $t\in[0,\infty)$. Hence,

$\int\|\jmath^{*}_{R}\phi\|^{n}_{L_{n}^{-\kappa}(\mathbb{R}^{2},v_{L}^{1/n})}\,\mu^{g}_{R,N}(\mathrm{d}\phi)=\mathbb{E}\|\jmath_{R}^{*}\varPhi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{n}_{L_{n}^{-\kappa}(\mathbb{R}^{2},v_{L}^{1/n})}\,.$ (6.3)

Since
$\mathrm{Law}(X_{R,N})=\mathrm{Law}(Z_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}}))$ for all
$t\in[0,\infty)$ by Lemma C.9 and Proposition 5.2 we have

$8\,\partial_{t}\mathbb{E}\|\jmath_{R}^{*}\varPsi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{2}_{L_{2}(\mathbb{R}^{2},v_{L}^{1/2})}+\mathbb{E}\|\jmath_{R}^{*}\varPsi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{n}_{L_{n}(\mathbb{R}^{2},v_{L}^{1/n})}\leq C_{1}\,$ (6.4)

for some constant $C_{1}\in(0,\infty)$ independent of $g$, $R,N$ and $t$. The
above inequality implies that for all $T\in(0,\infty)$ it holds

$\frac{1}{T}\int_{0}^{T}\mathbb{E}\|\jmath_{R}^{*}\varPsi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{n}_{L_{n}(\mathbb{R}^{2},v_{L}^{1/n})}\mathrm{d}t\\\ \leq
C_{1}-\frac{8}{T}\mathbb{E}\|\jmath_{R}^{*}\Psi^{g}_{R,N}(T,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{2}_{L_{2}(\mathbb{R}^{2},v_{L}^{1/2})}+\frac{8}{T}\mathbb{E}\|\jmath_{R}^{*}\Psi^{g}_{R,N}(0,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{2}_{L_{2}(\mathbb{R}^{2},v_{L}^{1/2})}\leq
C_{1}+\frac{C_{R,N}}{T}\,,$ (6.5)

where

$C_{R,N}:=8\,\mathbb{E}\|\jmath_{R}^{*}\Psi^{g}_{R,N}(0,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{2}_{L_{2}(\mathbb{R}^{2},v_{L}^{1/2})}\leq
8\,\mathbb{E}\|\Psi^{g}_{R,N}(0,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{2}_{L_{2}(\mathbb{S}_{R})}<\infty$ (6.6)

for every $R,N\in\mathbb{N}_{+}$ and $R\geq L$. Using the fact that
$\jmath_{R}^{*}\varPhi^{g}_{R,N}$ and $\jmath_{R}^{*}Z_{R,N}$ are stationary
in time one deduces that

$\mathbb{E}\|\jmath_{R}^{*}\varPhi^{g}_{R,N}(0,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{n}_{L^{-\kappa}_{n}(\mathbb{R}^{2},v_{L}^{1/n})}=\frac{1}{T}\int_{0}^{T}\mathbb{E}\|\jmath_{R}^{*}\varPhi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{n}_{L^{-\kappa}_{n}(\mathbb{R}^{2},v_{L}^{1/n})}\mathrm{d}t\\\
\leq{c}\,\mathbb{E}\|\jmath_{R}^{*}Z_{R,N}(0,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{n}_{L^{-\kappa}_{n}(\mathbb{R}^{2},v_{L}^{1/n})}+\frac{c}{T}\int_{0}^{T}\mathbb{E}\|\jmath_{R}^{*}\Psi^{g}_{R,N}(t,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{n}_{L^{-\kappa}_{n}(\mathbb{R}^{2},v_{L}^{1/n})}\mathrm{d}t\,,$
(6.7)

where $c=2^{n-1}$. By Lemma C.9 there exists $C_{2}\in(0,\infty)$ such that
for all $R,N\in\mathbb{N}_{+}$ it holds

$\mathbb{E}\|\jmath_{R}^{*}Z_{R,N}(0,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{n}_{L^{-\kappa}_{n}(\mathbb{R}^{2},v_{L}^{1/n})}\leq
C_{2}.$ (6.8)

Combining the bounds proved above we obtain

$\mathbb{E}\|\jmath_{R}^{*}\varPhi^{g}_{R,N}(0,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{n}_{L^{-\kappa}_{n}(\mathbb{R}^{2},v_{L}^{1/n})}\leq
c\,C_{1}+c\,C_{2}+\frac{c\,C_{R,N}}{T}$ (6.9)

for all $T\in(0,\infty)$. Choosing $T=C_{R,N}$ concludes the proof. ∎

##  7 Integrability

######  Proposition 7.1.

There exists a ball $B\subset\mathscr{S}(\mathbb{R}^{2})$ with respect to some
Schwartz semi-norm centered at the origin such that for all $f\in B$ the bound
(1.3) holds true.

###### Proof.

It follows from properties of the stereographic coordinates that for all
$f\in\mathscr{S}(\mathbb{R}^{2})$ and $R\in\mathbb{N}_{+}$ there exists
$g_{R}\in C^{\infty}(\mathbb{S}_{R})$ such that $w_{R}\jmath_{R}^{*}g_{R}=f$.
Let $B$ be contained in the ball from the statement of Proposition 6.1 and
suppose that $f\in B$. Note that for arbitrary
$\phi\in\mathscr{D}^{\prime}(\mathbb{S}_{R})$ it holds

$\phi(g_{R})=(\jmath^{*}_{R}\phi)(w_{R}\jmath^{*}_{R}g_{R})=(\jmath^{*}_{R}\phi)(f).$
(7.1)

Then by Lemma 7.2 it holds

$\int\exp\left(\phi(g_{R})^{n}/n\right)\,\mu_{R,N}(\mathrm{d}\phi)\\\
\leq\exp\left(\frac{1}{n}\int\phi(g_{R})^{n}\,\mu^{g_{R}}_{R,N}(\mathrm{d}\phi)\right)\,.$
(7.2)

Note that the expression on the LHS is integrable by Lemma 2.2. The identity
(7.1), Hölder’s inequality and Proposition 6.1 yield

$\int\phi(g_{R})^{n}\,\mu_{R,N}^{g_{R}}(\mathrm{d}\phi)\leq\hat{C}\,\|v_{L}^{-1/n}w_{R}\jmath^{*}_{R}g_{R}\|^{n}_{L^{\kappa}_{n/(n-1)}(\mathbb{R}^{2})}\,\int\|\jmath_{R}^{*}\phi\|^{n}_{L_{n}^{-\kappa}(\mathbb{R}^{2},v_{L}^{1/n})}\,\mu_{R}^{g_{R}}(\mathrm{d}\phi)\\\
\leq\,C\,\,\|v_{L}^{-1/n}w_{R}\jmath^{*}_{R}g_{R}\|^{n}_{L^{\kappa}_{n/(n-1)}(\mathbb{R}^{2})}$
(7.3)

for some constants $\hat{C},C\in(0,\infty)$ independent of $R,N$ and $g_{R}$.
Choosing the ball $B$ so that
$\|v_{L}^{-1/n}f\|^{n}_{L^{\kappa}_{n/(n-1)}(\mathbb{R}^{2})}\leq n/2C$ for
all $f\in B$ by the above inequalities and Proposition 2.8 we obtain

$\int\exp\left((\jmath^{*}_{R}\phi)(f)^{n}/n\right)\,\mu_{R}(\mathrm{d}\phi)=\lim_{N\to\infty}\int\exp\left(\phi(g_{R})^{n}/n\right)\,\mu_{R,N}(\mathrm{d}\phi)\leq
2\,.$ (7.4)

This concludes the proof. ∎

######  Lemma 7.2.

[4, Lemma A.7] Let $(\Omega,\mathcal{F},\mu)$ be a probability space,
$F:\,\Omega\to\mathbb{R}$ be a measurable function such that $\exp(F)\in
L_{1}(\Omega,\mu)$ and

$\mu^{F}(\mathrm{d}\phi):=\frac{\exp(F(\phi))\,\mu(\mathrm{d}\phi)}{\int\exp(F(\phi))\,\mu(\mathrm{d}\phi)}\,.$
(7.5)

It holds

$\int\exp(F(\phi))\,\mu(d\phi)\leq\exp\left(\int
F(\phi)\,\mu^{F}(d\phi)\right)\,.$ (7.6)

##  8 Reflection positivity

In this section, in Proposition 8.5, we establish the reflection positivity of
every accumulation point of the sequence
$(\jmath_{R}^{*}\sharp\mu_{R})_{R\in\mathbb{N}_{+}}$ of measures on
$\mathscr{S}^{\prime}(\mathbb{R}^{2})$. To this end, we leverage the fact that
the finite volume measure $\mu_{R}$ on $\mathscr{D}^{\prime}(\mathbb{S}_{R})$
is reflection positive.

######  Definition 8.1.

For all $R,N\in[1,\infty)$ we set

$\mathbb{S}_{R,N}^{\pm}:=\\{(\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3})\in\mathbb{S}_{R}\,|\,\pm\mathrm{x}_{1}>1/N\\},\quad\mathbb{S}_{R,N}:=\mathbb{S}_{R,N}^{+}\cup\mathbb{S}_{R,N}^{-}\quad\mathbb{S}_{R}^{\pm}:=\cup_{N\in[1,\infty)}\mathbb{S}_{R,N}^{\pm}.$
(8.1)

######  Definition 8.2.

Let $R\in[1,\infty)$. A functional
$F\,:\mathscr{D}^{\prime}(\mathbb{S}_{R})\to\mathbb{C}$ is called cylindrical
iff there exists $k\in\mathbb{N}_{+}$, $G\in
C^{\infty}_{\mathrm{b}}(\mathbb{R}^{k})$ and $f_{l}\in
C^{\infty}_{\mathrm{c}}(\mathbb{S}_{R}):=C^{\infty}(\mathbb{S}_{R})$,
$l\in\\{1,\ldots,k\\}$, such that

$F(\phi)=G(\phi(f_{1}),\ldots,\phi(f_{k})).$ (8.2)

The algebra of cylindrical functions is denoted by $\mathcal{F}_{R}$. The
subalgebras of $\mathcal{F}_{R}$ consisting of functionals of the form (8.2)
with $\mathrm{supp}\,f_{l}\subset\mathbb{S}_{R}^{\pm}$,
$l\in\\{1,\ldots,k\\}$, or such that
$\mathrm{supp}\,f_{l}\subset\mathbb{S}_{R,N}^{\pm}$, $l\in\\{1,\ldots,k\\}$,
are denoted by $\mathcal{F}^{\pm}_{R}$ and $\mathcal{F}^{\pm}_{R,N}$,
respectively. The definitions of $\mathcal{F}$ and $\mathcal{F}^{\pm}$ are
analogous to the definitions of $\mathcal{F}_{R}$ and $\mathcal{F}_{R}^{\pm}$
with $\mathbb{S}_{R}$ and $\mathbb{S}_{R}^{\pm}$ replaced by $\mathbb{R}^{2}$
and the half-plane $\\{(x_{1},x_{2})\in\mathbb{R}^{2}|\pm x_{1}>0\\}$,
respectively.

######  Definition 8.3.

Let $R\in[1,\infty)$. For $f\in C^{\infty}(\mathbb{S}_{R})$ we define
$\Theta_{R}f\in C^{\infty}(\mathbb{S}_{R})$ by the formula
$(\Theta_{R}f)(\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3}):=f(-\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3})$.
For $\phi\in\mathscr{D}^{\prime}(\mathbb{S}_{R})$ we define
$\Theta_{R}\phi\in\mathscr{D}^{\prime}(\mathbb{S}_{R})$ by the formula
$\langle\Theta_{R}\phi,f\rangle:=\langle\phi,\Theta_{R}f\rangle$ for all $f\in
C^{\infty}(\mathbb{S}_{R})$. For $f\in C^{\infty}(\mathbb{R}^{2})$ we define
$\Theta f\in C^{\infty}(\mathbb{R}^{2})$ by the formula $(\Theta
f)(x_{1},x_{2}):=f(-x_{1},x_{2})$. For
$\phi\in\mathscr{S}^{\prime}(\mathbb{R}^{2})$ we define
$\Theta\phi\in\mathscr{S}^{\prime}(\mathbb{R}^{2})$ by the formula
$\langle\Theta\phi,f\rangle:=\langle\phi,\Theta f\rangle$ for all
$f\in\mathscr{S}(\mathbb{R}^{2})$.

######  Remark 8.4.

Note that $\jmath_{R}^{*}\Theta_{R}\phi=\Theta\jmath_{R}^{*}\phi$ for all
$\phi\in\mathscr{D}^{\prime}(\mathbb{S}_{R})$.

######  Proposition 8.5.

Let $\mu$ be a weak limit of a subsequence of the sequence of measures
$(\jmath_{R}^{*}\sharp\mu_{R})_{R\in\mathbb{N}_{+}}$ on
$\mathscr{S}^{\prime}(\mathbb{R}^{2})$. For all $F\in\mathcal{F}^{+}$ it holds
$\int\overline{F(\Theta\phi)}F(\phi)\,\mu(\mathrm{d}\phi)\geq 0$.

###### Proof.

It is enough to prove that

$\int\overline{F(\Theta\phi)}F(\phi)\,(\jmath_{R}^{*}\sharp\mu_{R})(\mathrm{d}\phi)=\int\overline{F(\Theta\jmath_{R}^{*}\phi)}F(\jmath_{R}^{*}\phi)\,\mu_{R}(\mathrm{d}\phi)=\int\overline{F(\jmath_{R}^{*}\Theta_{R}\phi)}F(\jmath_{R}^{*}\phi)\,\mu_{R}(\mathrm{d}\phi)\geq
0$ (8.3)

for all $R\in\mathbb{N}_{+}$ and $F\in\mathcal{F}^{+}$. By Def. 8.2 and Remark
4.3 for every $F\in\mathcal{F}^{+}$ it holds
$F\circ\jmath_{R}^{*}\in\mathcal{F}^{+}_{R}$. Hence, the last bound above
follows from the reflection positivity of the measure $\mu_{R}$. ∎

For completeness, we prove below the reflection positivity of the measure
$\mu_{R}$ on $\mathscr{D}^{\prime}(\mathbb{S}_{R})$, which is stated in Lemma
8.10 (D). Note that the UV cutoff in the definition of the measures
$\mu_{R,N}$, introduced in Sec. 2, breaks the reflection positivity, cf. [3].
For this reason, in this section we work with a different UV cutoff. We
introduce a free field $\hat{X}_{R,N}$ with a UV cutoff that preserves the
reflection positivity, see Lemma 8.10 (B), and show that the measure $\mu_{R}$
can be approximated, see Lemma 8.9 and Eq. (8.10), by measures with a UV
cutoff that are reflection positive, see Lemma 8.10 (C).

######  Definition 8.6.

Fix $h\in C^{\infty}(\mathbb{R})$ such that $\mathrm{supp}\,h\subset(-1,1)$,
$h=1$ on $[-1/2,1/2]$ and
$2\pi\int_{0}^{\infty}h(\theta)\,\theta\,\mathrm{d}\theta=1$. For
$R,N\in\mathbb{N}_{+}$ the operator $\hat{K}_{R,N}:\,L_{2}(\mathbb{S}_{R})\to
L_{2}(\mathbb{S}_{R})$ is defined by its integral kernel

$\hat{K}_{R,N}(\mathrm{x},\mathrm{y}):=N^{2}h(N\,d_{R}(\mathrm{x},\mathrm{y})).$
(8.4)

######  Definition 8.7.

Let $\hat{c}_{R,N}:=\mathrm{Tr}(\hat{K}_{R,N}G_{R}\hat{K}_{R,N})/4\pi R^{2}$.
By definition $\hat{X}_{R,N}:=\hat{K}_{R,N}X_{R}$,

$\begin{gathered}\hat{X}_{R,N}^{:m:}:=\sum_{k=0}^{\lfloor
m/2\rfloor}\frac{(-1)^{k}m!}{(m-2k)!k!2^{k}}(\hat{c}_{R,N})^{k}\hat{X}_{R,N}^{m-2k},\qquad\hat{X}_{R,N}^{:m:}(h):=\int_{\mathbb{S}_{R}}\hat{X}_{R,N}^{:m:}(\mathrm{x})\,h(\mathrm{x})\,\rho_{R}(\mathrm{d}\mathrm{x}),\\\
\hat{Y}_{R,N}:=\sum_{m=0}^{n}a_{m}\hat{X}_{R,N}^{:m:}(1_{\mathbb{S}_{R}}),\qquad\tilde{Y}_{R,N}^{\pm}:=\sum_{m=0}^{n}a_{m}\hat{X}_{R,N}^{:m:}(1_{\mathbb{S}^{\pm}_{R,N}}),\qquad\tilde{Y}_{R,N}=\tilde{Y}^{+}_{R,N}+\tilde{Y}^{-}_{R,N},\end{gathered}$
(8.5)

where $h\in L_{\infty}(\mathbb{S}_{R})$ and $1_{B}$ denotes the characteristic
function of the set $B$.

######  Remark 8.8.

By Lemma C.6 it holds $\hat{X}_{R,N}\in L^{1}_{2}(\mathbb{S}_{R})\subset
L_{n}(\mathbb{S}_{R})$ almost surely. In particular, $\hat{Y}_{R,N}$,
$\tilde{Y}_{R,N}$ are well-defined. Moreover, there exists $C\in(0,\infty)$
such that for all $N,R\in\mathbb{N}_{+}$ it holds $|\hat{c}_{R,N}-1/2\pi\,\log
N|\leq C$ by the bound (2.5) and Remark C.5.

######  Lemma 8.9.

For all $R\in\mathbb{N}_{+}$ and all bounded and continuous
$F\,:\,\mathscr{D}^{\prime}(\mathbb{S}_{R})\to\mathbb{R}$ it holds

$\lim_{N\to\infty}\mathbb{E}F(\hat{X}_{R,N})\exp(-\tilde{Y}_{R,N})=\lim_{N\to\infty}\mathbb{E}F(X_{R,N})\exp(-Y_{R,N}).$
(8.6)

###### Proof.

The proof follows the strategy of the proof of Proposition 2.8. By Lemmas C.7
and C.8 the sequences
$(F(\hat{X}_{R,N})\exp(-\tilde{Y}_{R,N}))_{N\in\mathbb{N}_{+}}$ and
$(F(X_{R,N})\exp(-Y_{R,N}))_{N\in\mathbb{N}_{+}}$ converge in probability to
$F(X_{R})\exp(-Y_{R})$. To conclude we show that the above-mentioned sequences
are uniformly integrable by repeating verbatim the argument from the proof of
Proposition 2.8. ∎

######  Lemma 8.10.

The following statements hold true for all $R,N\in\mathbb{N}_{+}$:

  1. (A)

If $F\in\mathcal{F}_{R}^{+}$, then
$\mathbb{E}\overline{F(\Theta_{R}X_{R})}F(X_{R})\geq 0$.

  2. (B)

If $F\in\mathcal{F}_{R,N}^{+}$, then
$\mathbb{E}\overline{F(\Theta_{R}\hat{X}_{R,N})}F(\hat{X}_{R,N})\geq 0$.

  3. (C)

If $F\in\mathcal{F}_{R,N}^{+}$, then
$\mathbb{E}\overline{F(\Theta_{R}\hat{X}_{R,N})}F(\hat{X}_{R,N})\exp(-\tilde{Y}_{R,N})\geq
0$.

  4. (D)

For all $F\in\mathcal{F}^{+}_{R}$ it holds
$\int\overline{F(\Theta_{R}\phi)}F(\phi)\,\mu_{R}(\mathrm{d}\phi)\geq 0$.

###### Proof.

For the proof of Item (A) see [11, Theorem 2]. To prove Item (B), observe that

$\mathbb{E}\overline{F(\Theta_{R}\hat{X}_{R,N})}F(\hat{X}_{R,N})=\mathbb{E}\overline{F(\hat{K}_{R,N}(\Theta_{R}X_{R}))}F(\hat{K}_{R,N}X_{R})\,,$
(8.7)

where we have used the fact that
$\Theta_{R}\hat{K}_{R,N}X_{R}=\hat{K}_{R,N}\Theta_{R}X_{R}$. By the support
property of the kernel $\hat{K}_{R,N}(\mathrm{x},\mathrm{y})$ if
$F\in\mathcal{F}^{+}_{R,N}$, then the functional $\phi\mapsto
F(\hat{K}_{R,N}\phi)$ belongs to $\mathcal{F}_{R}^{+}$. Consequently, the
statement follows from Item (A). To prove Item (C), note that

$\mathbb{E}\overline{F(\Theta_{R}\hat{X}_{R,N})}F(\hat{X}_{R,N})\exp(-\tilde{Y}_{R,N})=\mathbb{E}\overline{F(\Theta_{R}\hat{X}_{R,N})\exp(-\tilde{Y}^{-}_{R,N})}F(\hat{X}_{R,N})\exp(-\tilde{Y}^{+}_{R,N})\,.$
(8.8)

Denote $H(\hat{X}_{R,N}):=F(\hat{X}_{R,N})\exp(-\tilde{Y}_{R,N}^{+})$. It
holds

$\mathbb{E}\overline{F(\Theta_{R}\hat{X}_{R,N})}F(\hat{X}_{R,N})\exp(-\tilde{Y}_{R,N})=\mathbb{E}\overline{H(\Theta_{R}\hat{X}_{R,N})}H(\hat{X}_{R,N})\,.$
(8.9)

The RHS of the above equality can be approximated by a similar expression with
$H$ replaced by some functional belonging to $\mathcal{F}_{R,N}^{+}$. As a
result, Item (C) follows from Item (B). Let us turn to the proof of Item (D).
First note that for any $F\in\mathcal{F}^{+}_{R}$ there exists
$M\in\mathbb{N}_{+}$ such that $F\in\mathcal{F}^{+}_{R,M}$. Hence, it suffices
to show that
$\int\overline{F(\Theta_{R}\phi)}F(\phi)\,\mu_{R}(\mathrm{d}\phi)\geq 0$ for
all $R,M\in\mathbb{N}_{+}$ and $F\in\mathcal{F}^{+}_{R,M}$. To establish this
claim we note that by Lemma 8.9

$\int\overline{F(\Theta_{R}\phi)}F(\phi)\,\mu_{R}(\mathrm{d}\phi)=\lim_{N\to\infty}\frac{\mathbb{E}\overline{F(\Theta_{R}\hat{X}_{R,N})}F(\hat{X}_{R,N})\exp(-\tilde{Y}_{R,N})}{\mathbb{E}\exp(-\tilde{Y}_{R,N})}$
(8.10)

and use Item (C) together with the fact that
$\mathcal{F}^{+}_{R,M}\subset\mathcal{F}^{+}_{R,N}$ for all $N\geq M$. ∎

##  9 Euclidean invariance

In this section we establish the invariance under the Euclidean
transformations of the plane of every accumulation point $\mu$ of the sequence
$(\jmath_{R}^{*}\sharp\mu_{R})_{R\in\mathbb{N}_{+}}$ of measures on
$\mathscr{S}^{\prime}(\mathbb{R}^{2})$. We use the fact that for every
$R\in\mathbb{N}_{+}$ the measure $\mu_{R}$ is invariant under the rotations of
the sphere $\mathbb{S}_{R}$. The proof of the rotational invariance of $\mu$
is straight-forward as the rotations $\mathcal{R}_{R,\alpha}$ of the sphere
$\mathbb{S}_{R}$ around the $\mathrm{x}_{3}$ axis are mapped under the
stereographic projection to the rotations $\mathcal{R}_{\alpha}$ of the plane
$\mathbb{R}^{2}$ around the origin. Hence, for every $R\in\mathbb{R}_{+}$ the
measure $\jmath_{R}^{*}\sharp\mu_{R}$ on
$\mathscr{S}^{\prime}(\mathbb{R}^{2})$ is invariant under the rotations around
the origin and the same is true for every accumulation point $\mu$. The proof
of the translational invariance of $\mu$ is more complicated. There is no
rotation of the sphere $\mathbb{S}_{R}$ that is mapped under the stereographic
projection to the translation $\mathcal{T}_{\alpha}$ of the plane
$\mathbb{R}^{2}$ in the $x_{1}$ direction. In particular, for every
$R\in\mathbb{R}_{+}$ the measure $\jmath_{R}^{*}\sharp\mu_{R}$ on
$\mathscr{S}^{\prime}(\mathbb{R}^{2})$ is not invariant under the
translations. In order to establish the translational invariance of an
accumulation point $\mu$ we first prove that the rotations
$\mathcal{T}_{R,\alpha}$ of the sphere $\mathbb{S}_{R}$ around the
$\mathrm{x}_{2}$ axis are mapped under the stereographic projection to certain
transformations $\mathcal{S}_{R,\alpha}$ of the plane $\mathbb{R}^{2}$ and
subsequently show that the transformations $\mathcal{S}_{R,\alpha}$ converge
to the translations $\mathcal{T}_{\alpha}$ of the plane $\mathbb{R}^{2}$ in
the $x_{1}$ direction as $R\to\infty$.

######  Definition 9.1.

For $\alpha\in\mathbb{R}$ the maps
$\mathcal{R}_{\alpha},\mathcal{T}_{\alpha}\,:\,\mathbb{R}^{2}\to\mathbb{R}^{2}$
are defied by

$\mathcal{R}_{\alpha}(x_{1},x_{2}):=(x_{1}\cos\alpha+x_{2}\sin\alpha,x_{1}\sin\alpha-
x_{2}\cos\alpha),\qquad\mathcal{T}_{\alpha}(x_{1},x_{2}):=(x_{1}+\alpha,x_{2}),$
(9.1)

For $R\in\mathbb{N}_{+}$, $\alpha\in\mathbb{R}$ the maps
$\mathcal{R}_{R,\alpha},\mathcal{T}_{R,\alpha}\,:\,\mathbb{S}_{R}\to\mathbb{S}_{R}$
are defined by

$\begin{gathered}\mathcal{R}_{R,\alpha}(\mathrm{x})=(\mathrm{x}_{1}\cos\alpha+\mathrm{x}_{2}\sin\alpha,\mathrm{x}_{1}\sin\alpha-\mathrm{x}_{2}\cos\alpha,\mathrm{x}_{3}),\\\
\mathcal{T}_{R,\alpha}(\mathrm{x})=(\mathrm{x}_{1}\cos(\alpha/R)-\mathrm{x}_{3}\sin(\alpha/R),\mathrm{x}_{2},\mathrm{x}_{1}\sin(\alpha/R)+\mathrm{x}_{3}\cos(\alpha/R)).\end{gathered}$
(9.2)

where
$\mathrm{x}=(\mathrm{x}_{1},\mathrm{x}_{2},\mathrm{x}_{3})\in\mathbb{S}_{R}$.
For $R\in\mathbb{N}_{+}$, $\alpha\in(-R,R)$ the map
$\mathcal{S}_{R,\alpha}\,:\,B_{R}\to\mathbb{R}^{2}$ is defined by

$\mathcal{S}_{R,\alpha}(x_{1},x_{2}):=\frac{2(R\sin(\alpha/R)(1-(x_{1}^{2}+x_{2}^{2})/4R^{2})+x_{1}\cos(\alpha/R),x_{2})}{1+\cos(\alpha/R)+(1-\cos(\alpha/R))(x_{1}^{2}+x_{2}^{2})/4R^{2}-x_{1}/R\sin(\alpha/R)},$
(9.3)

where $B_{R}:=\\{x\in\mathbb{R}^{2}\,|\,|x|<R\\}$.

######  Remark 9.2.

For all $R\in\mathbb{N}_{+}$ and $\alpha\in\mathbb{R}$ it holds
$\mathcal{R}_{R,\alpha}\jmath_{R}=\jmath_{R}\mathcal{R}_{\alpha}$. For all
$R\in\mathbb{N}_{+}$ and $\alpha\in(-R,R)$ it holds
$\mathcal{T}_{R,\alpha}\jmath_{R}=\jmath_{R}\mathcal{S}_{R,\alpha}$ on
$B_{R}$.

######  Definition 9.3.

For $\alpha\in\mathbb{R}$, $f\in C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2})$,
$\phi\in\mathscr{D}^{\prime}(\mathbb{R}^{2})$ we set
$\mathcal{R}_{\alpha}^{*}f:=f\circ\mathcal{R}_{\alpha}\in
C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2})$,
$\mathcal{T}_{\alpha}^{*}f:=f\circ\mathcal{T}_{\alpha}\in
C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2})$ and
$\mathcal{R}_{\alpha}^{*}\phi:=\phi\circ\mathcal{R}_{-\alpha}^{*}\in\mathscr{D}^{\prime}(\mathbb{R}^{2})$,
$\mathcal{T}_{\alpha}^{*}\phi:=\phi\circ\mathcal{T}_{-\alpha}^{*}\in\mathscr{D}^{\prime}(\mathbb{R}^{2})$.
For $\alpha\in\mathbb{R}$, $f\in C^{\infty}(\mathbb{S}_{R}^{2})$,
$\phi\in\mathscr{D}^{\prime}(\mathbb{S}_{R}^{2})$ we set
$\mathcal{R}_{R,\alpha}^{*}f:=f\circ\mathcal{R}_{R,\alpha}\in
C^{\infty}(\mathbb{S}_{R}^{2})$,
$\mathcal{T}_{R,\alpha}^{*}f:=f\circ\mathcal{T}_{R,\alpha}\in
C^{\infty}(\mathbb{S}_{R}^{2})$ and
$\mathcal{R}_{R,\alpha}^{*}\phi:=\phi\circ\mathcal{R}_{R,-\alpha}^{*}\in\mathscr{D}^{\prime}(\mathbb{S}_{R}^{2})$,
$\mathcal{T}_{R,\alpha}^{*}\phi:=\phi\circ\mathcal{T}_{R,-\alpha}^{*}\in\mathscr{D}^{\prime}(\mathbb{S}_{R}^{2})$.

######  Definition 9.4.

Let $R\in\mathbb{N}_{+}$, $\alpha\in(-R,R)$. For $f\in
C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2})$ we set

$\mathcal{S}_{R,\alpha}^{*}f:=f\circ\mathcal{S}_{R,\alpha}\in
C^{\infty}(B_{R}).$

For $\phi\in\mathscr{D}^{\prime}(\mathbb{R}^{2})$, $\mathrm{supp}\,\phi\subset
B_{R}$, we define
$\mathcal{S}_{R,\alpha}^{*}\phi\in\mathscr{D}^{\prime}(\mathbb{R}^{2})$ by

$\langle\mathcal{S}_{R,\alpha}^{*}\phi,f\rangle:=\langle\phi,\det(\mathrm{T}\mathcal{S}_{R,-\alpha})\mathcal{S}_{R,-\alpha}^{*}f\rangle$
(9.4)

for all $f\in C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2})$, where
$\det(\mathrm{T}\mathcal{S}_{R,-\alpha})$ denotes the Jacobian, i.e. the
determinant of the tangent map of $\mathcal{S}_{R,-\alpha}$.

######  Remark 9.5.

For all $\alpha\in\mathbb{R}$, $a\in\mathbb{N}_{0}^{2}$ and $M\in(0,\infty)$
there exists $C\in(0,\infty)$ such that for all $x\in B_{M}$ and
$R\in(|\alpha|\vee M,\infty)$ it holds

$\mathcal{T}_{R,\alpha}\jmath_{R}(x)=\jmath_{R}\mathcal{S}_{R,\alpha}(x)\qquad\textrm{and}\qquad|\partial^{a}\mathcal{S}_{R,\alpha}(x)-\partial^{a}\mathcal{T}_{\alpha}(x)|\leq
C/R.$ (9.5)

Noting that $\mathrm{T}\mathcal{T}_{-\alpha}=1$ we conclude that for all
$\alpha\in\mathbb{R}$ and $f\in C^{\infty}_{\mathrm{c}}(\mathbb{R}^{2})$ there
exists $C$ such that for all sufficiently large $R\in\mathbb{N}_{+}$ it holds

$\|\det(\mathrm{T}\mathcal{S}_{R,-\alpha})\mathcal{S}_{R,-\alpha}^{*}f-\mathcal{T}_{-\alpha}^{*}f\|_{L^{1}_{2}(\mathbb{R}^{2},v_{L}^{-1/2})}\leq
C\,/R.$ (9.6)

######  Remark 9.6.

Let us note that the algebra of cylindrical functionals $\mathcal{F}$
separates points in
$L^{-1}_{2}(\mathbb{R}^{2},v_{L}^{1/2})\subset\mathscr{D}^{\prime}(\mathbb{R}^{2})$.
Hence, if $\mu_{j}$, $j=1,2$, are Borel probability measures on
$L^{-1}_{2}(\mathbb{R}^{2},v_{L}^{1/2})$ such that $\mu_{1}(F)=\mu_{2}(F)$ for
all $F\in\mathcal{F}$, then $\mu_{1}=\mu_{2}$ by [13, Theorem 4.5(a), Ch. 3].

######  Proposition 9.7.

Let $\mu$ be a weak limit of a subsequence of the sequence of measures
$(\jmath_{R}^{*}\sharp\mu_{R})_{R\in\mathbb{N}_{+}}$ on
$\mathscr{S}^{\prime}(\mathbb{R}^{2})$. It holds
$\mu(F)=\mu(F\circ\mathcal{R}_{\alpha}^{*})$ and
$\mu(F)=\mu(F\circ\mathcal{T}_{\alpha}^{*})$ for all bounded and measurable
$F\,:\,\mathscr{S}^{\prime}(\mathbb{R}^{2})\to\mathbb{R}$ and all
$\alpha\in\mathbb{R}$.

###### Proof.

Suppose that the sequence of measures $\jmath_{R}^{*}\sharp\mu_{R}$ on
$\mathscr{S}^{\prime}(\mathbb{R}^{2})$ converges to $\mu$ along the
subsequence $(R_{M})_{M\in\mathbb{N}_{+}}$. By Remark 6.2 the measure $\mu$ is
concentrated on $L^{-1}_{2}(\mathbb{R}^{2},v_{L}^{1/2})$. Hence, by Remark 9.6
without loss of generality we can assume that $F\in\mathcal{F}$ is a
cylindrical functional. Note that by the rotational invariance of the measure
$\mu_{R}$ it holds

$\mu_{R}(F_{R})=\mu_{R}(F_{R}\circ\mathcal{R}_{R,\alpha}^{*}),\qquad\mu_{R}(F_{R})=\mu_{R}(F_{R}\circ\mathcal{T}_{R,\alpha}^{*})$
(9.7)

for every $F_{R}\in\mathcal{F}_{R}$ and $\alpha\in\mathbb{R}$. By Remark 9.2
we have
$\mathcal{R}^{*}_{\alpha}\circ\jmath_{R}^{*}=\jmath_{R}^{*}\circ\mathcal{R}^{*}_{R,\alpha}$.
Hence, for every $F\in\mathcal{F}$ we obtain

$\mu(F\circ\mathcal{R}^{*}_{\alpha}-F)=\lim_{M\to\infty}\jmath_{R_{M}}^{*}\sharp\mu_{R_{M}}(F\circ\mathcal{R}^{*}_{\alpha}-F)=\lim_{M\to\infty}\mu_{R_{M}}(F\circ\mathcal{R}^{*}_{\alpha}\circ\jmath_{R_{M}}^{*}-F\circ\jmath_{R_{M}}^{*})\\\
=\lim_{M\to\infty}\mu_{R_{M}}(F\circ\jmath_{R_{M}}^{*}\circ\mathcal{R}_{R_{M},\alpha}^{*}-F\circ\jmath_{R_{M}}^{*})=\lim_{M\to\infty}(\jmath_{R_{M}}^{*}\sharp\mu_{R_{M}}-\mu)(F)=0.$
(9.8)

This finishes the proof of the rotational invariance.

Let us turn to the proof of the translational invariance. Note that by Remark
9.2 for every $F\in\mathcal{F}$ and all sufficiently large
$R\in\mathbb{N}_{+}$ it holds

$\jmath_{R}^{*}\sharp\mu_{R}(F)=\mu_{R}(F\circ\jmath_{R}^{*})=\mu_{R}(F\circ\jmath_{R}^{*}\circ\mathcal{T}_{R,\alpha}^{*})=\mu_{R}(F\circ\mathcal{S}_{R,\alpha}^{*}\circ\jmath_{R}^{*})=\jmath_{R}^{*}\sharp\mu_{R}(F\circ\mathcal{S}_{R,\alpha}^{*}).$
(9.9)

Remark 9.5 implies that for every $\alpha\in\mathbb{R}$ and $F\in\mathcal{F}$
there exists $C\in(0,\infty)$ such that for all $\psi\in
L^{-1}_{2}(\mathbb{R}^{2},v_{L}^{1/2})$ and all sufficiently large
$R\in\mathbb{N}_{+}$ it holds

$|F(\mathcal{S}_{R,\alpha}^{*}\psi)-F(\mathcal{T}_{\alpha}^{*}\psi)|\leq(C/R)~{}\|\psi\|_{L^{-1}_{2}(\mathbb{R}^{2},v_{L}^{1/2})}.$
(9.10)

By Proposition 6.1 and Hölder’s inequality we obtain that

$\int_{\mathscr{D}^{\prime}(\mathbb{S}_{R})}\|\jmath_{R}^{*}\phi\|_{L^{-1}_{2}(\mathbb{R}^{2},v_{L}^{1/2})}\,\mu_{R}(\mathrm{d}\phi)$
(9.11)

is uniformly bounded in $R\in\mathbb{N}_{+}$. Hence, for all
$\alpha\in\mathbb{R}$ and $F\in\mathcal{F}$ it holds

$\lim_{R\to\infty}\jmath_{R}^{*}\sharp\mu_{R}(F\circ\mathcal{S}_{R,\alpha}^{*}-F\circ\mathcal{T}_{\alpha}^{*})=0.$
(9.12)

Consequently,

$\mu(F\circ\mathcal{T}^{*}_{\alpha}-F)=\lim_{M\to\infty}\jmath_{R_{M}}^{*}\sharp\mu_{R_{M}}(F\circ\mathcal{T}^{*}_{\alpha}-F)\\\
=\lim_{M\to\infty}\jmath_{R_{M}}^{*}\sharp\mu_{R_{M}}(F\circ\mathcal{S}_{R_{M},\alpha}^{*}-F)=\lim_{M\to\infty}(\jmath_{R_{M}}^{*}\sharp\mu_{R_{M}}-\mu)(F)=0.$
(9.13)

This finishes the proof. ∎

##  Appendix A Function spaces

######  Definition A.1.

We say that $w\in C^{\infty}(\mathbb{R}^{d})$ is an admissible weight iff
there exist $b\in[0,\infty)$ and $c\in(0,\infty)$ such that $0<w(x)\leq
c\,w(y)\,(1+|x-y|)^{b}$ for all $x,y\in\mathbb{R}^{d}$ and for every
$a\in\mathbb{N}_{0}^{d}$ there exists $c_{a}\in(0,\infty)$ such that
$|\partial^{a}w(x)|\leq c_{a}\,w(x)$ for all $x\in\mathbb{R}^{d}$.

######  Definition A.2.

Let $w$ be an admissible weight, $p\in[1,\infty]$ and $\alpha\in\mathbb{R}$,
$n\in\mathbb{N}_{0}$. By definition $L_{p}(\mathbb{R}^{d},w)$ is the Banach
space with the norm

$\|f\|_{L_{p}(\mathbb{R}^{d},w)}:=\|wf\|_{L_{p}(\mathbb{R}^{d})}.$ (A.1)

The weighted Bessel potential space $L^{\alpha}_{p}(\mathbb{R}^{d},w)$ is the
Banach space with the norm

$\|f\|_{L^{\alpha}_{p}(\mathbb{R}^{d},w)}:=\|(1-\Delta)^{\alpha/2}f\|_{L_{p}(\mathbb{R}^{d},w)}.$
(A.2)

We also set $L^{\alpha}_{p}(\mathbb{R}^{d})=L^{\alpha}_{p}(\mathbb{R}^{d},1)$.
The weighted Sobolev space $W^{n}_{p}(\mathbb{R}^{d},w)$ is the Banach space
with the norm

$\|f\|_{W^{n}_{p}(\mathbb{R}^{d},w)}=\textstyle\sum_{\begin{subarray}{c}a\in\mathbb{N}^{d},|a|\leq
n\end{subarray}}\|\partial^{a}f\|_{L_{p}(\mathbb{R}^{d},w)}.$ (A.3)

For $R\in(0,\infty)$ the Bessel potential space
$L^{\alpha}_{p}(\mathbb{S}_{R})$ on the round sphere
$\mathbb{S}_{R}\subset\mathbb{R}^{d}$ of radius $R$ is the Banach space with
the norm

$\|f\|_{L^{\alpha}_{p}(\mathbb{S}_{R})}:=\|(1-\Delta_{R})^{\alpha/2}f\|_{L_{p}(\mathbb{S}_{R})},$
(A.4)

where $L^{\alpha}_{p}(\mathbb{S}_{R})$ is the $L_{p}$ space on
$\mathbb{S}_{R}$ with respect to the canonical measure $\rho_{R}$ on
$\mathbb{S}_{R}$.

######  Remark A.3.

The following facts are standard: Let $w$ be an admissible weight,
$p\in[1,\infty)$, $\alpha\in\mathbb{R}$ and $n\in\mathbb{N}_{0}$. The norms
$\|{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}\|_{L^{\alpha}_{p}(\mathbb{R}^{d},w)}$ and
$\|w{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}\|_{L^{\alpha}_{p}(\mathbb{R}^{d})}$ are equivalent. The
Sobolev space $W^{n}_{p}(\mathbb{R}^{d},w)$ coincides with the Bessel
potential space $L^{n}_{p}(\mathbb{R}^{d},w)$ with equivalent norms. The
Bessel potential space $L^{\alpha}_{p}(\mathbb{R}^{d},w)$ coincides with the
Triebel-Lizorkin space $F^{\alpha}_{p,2}(\mathbb{R}^{d},w)$ with equivalent
norms. Furthermore, the Bessel potential space
$L_{p}^{\alpha}(\mathbb{R}^{d},w)$ is continuously embedded in the Besov space
$B_{p,\infty}^{\alpha}(\mathbb{R}^{d},w)$ and the Besov space
$B_{\infty,1}^{\alpha}(\mathbb{R}^{d},w)$ is continuously embedded in the
Bessel potential space $L_{\infty}^{\alpha}(\mathbb{R}^{d},w)$. These facts
can be obtained e.g. from [31, Theorem 6.5, Theorem 6.9] and [29, Sec. 2.5.7].
We note that [31, Theorem 6.5 (iii)] is useful to pass from $\alpha=0$ to
$\alpha\in\mathbb{R}$.

######  Remark A.4.

For $\alpha\in\mathbb{R}$ and $p,q\in[1,\infty]$ we have the following
generalized Hölder inequality

$|\langle f,g\rangle_{L_{2}(\mathbb{R}^{d},w^{1/2})}|\leq
C\,\|f\|_{L^{\alpha}_{p}(\mathbb{R}^{d},w^{1/p})}\,\|g\|_{L^{-\alpha}_{q}(\mathbb{R}^{d},w^{1/q})},\qquad\frac{1}{p}+\frac{1}{q}=1,$
(A.5)

where $\langle{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}\rangle_{L_{2}(\mathbb{R}^{d},w^{1/2})}$ is the scalar product
in $L_{2}(\mathbb{R}^{d},w^{1/2})$ and the constant $C\in(0,\infty)$ depends
only on the weight $w$.

######  Theorem A.5.

Let $w,v$ be admissible weights and

$-\infty<\alpha_{2}\leq\alpha_{1}<\infty,\qquad 1\leq p_{1}\leq
p_{2}\leq\infty.$ (A.6)

(A) The embedding $L^{\alpha_{1}}_{p_{1}}(\mathbb{R}^{d},w)\to
L^{\alpha_{2}}_{p_{2}}(\mathbb{R}^{d},v)$ is continuous if

$p_{2}<\infty,\qquad\alpha_{1}-d/p_{1}\geq\alpha_{2}-d/p_{2}\quad\textrm{and}\quad\sup_{x\in\mathbb{R}^{d}}v(x)/w(x)<\infty.$
(A.7)

(B) The embedding $L^{\alpha_{1}}_{p_{1}}(\mathbb{R}^{d},w)\to
L^{\alpha_{2}}_{\infty}(\mathbb{R}^{d},v)$ is continuous if

$\alpha_{1}-d/p_{1}>\alpha_{2}\quad\textrm{and}\quad\sup_{x\in\mathbb{R}^{d}}v(x)/w(x)<\infty.$
(A.8)

(C) The embedding $L^{\alpha_{1}}_{p_{1}}(\mathbb{R}^{d},w)\to
L^{\alpha_{2}}_{p_{2}}(\mathbb{R}^{d},v)$ is compact if

$p_{2}<\infty,\qquad\alpha_{1}-d/p_{1}>\alpha_{2}-d/p_{2}\quad\textrm{and}\quad\lim_{|x|\to\infty}v(x)/w(x)=0.$
(A.9)

###### Proof.

Parts (A) and (C) follow from [12, Sec. 4.2.3, Theorem] and the equivalence
between $L^{\alpha}_{p}(\mathbb{R}^{d},w)$ and
$F^{\alpha}_{p,2}(\mathbb{R}^{d},w)$ mentioned in Remark A.3 above. Part (B)
is covered by [12, Sec. 4.2.3, Remark] and the embeddings stated in Remark
A.3. ∎

######  Theorem A.6.

Let $w$ be an admissible weight, $\alpha\in[0,\infty)$ and
$p,p_{1},p_{2}\in[1,\infty)$ be such that $1/p=1/p_{1}+1/p_{2}$. Then there
exists $C\in(0,\infty)$ such that for all $f\in
L^{\alpha}_{p_{1}}(\mathbb{R}^{d},w^{1/p_{1}})$ and $g\in
L^{\alpha}_{p_{2}}(\mathbb{R}^{d},w^{1/p_{2}})$

$\|fg\|_{L^{\alpha}_{p}(\mathbb{R}^{d},w^{1/p})}\leq
C\,\|f\|_{L^{\alpha}_{p_{1}}(\mathbb{R}^{d},w^{1/p_{1}})}\,\|g\|_{L^{\alpha}_{p_{2}}(\mathbb{R}^{d},w^{1/p_{2}})}.$
(A.10)

###### Proof.

The statement follows from the equivalence of the norms
$\|{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}\|_{L^{\alpha}_{p}(\mathbb{R}^{d},w)}$ and
$\|w{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}\|_{L^{\alpha}_{p}(\mathbb{R}^{d})}$, the fractional Leibniz
rule [22, Ch. 2] and Theorem A.5 (A). Alternatively, one can use [7, Lemma 5].
∎

######  Theorem A.7.

Let $w$ be an admissible weight, $p_{1},p_{2}\in[1,\infty)$,
$\alpha_{1},\alpha_{2}\in\mathbb{R}$, $\theta\in(0,1)$ and

$\alpha=\theta\,\alpha_{1}+(1-\theta)\,\alpha_{2},\qquad\qquad\frac{1}{p}=\frac{\theta}{p_{1}}+\frac{1-\theta}{p_{2}}.$
(A.11)

There exists $C\in(0,\infty)$ such that for all $f\in
L^{\alpha_{1}}_{p_{1}}(\mathbb{R}^{d},w^{1/p_{1}})\cap
L^{\alpha_{2}}_{p_{2}}(\mathbb{R}^{d},w^{1/p_{2}})$ it holds

$\|f\|_{L^{\alpha}_{p}(\mathbb{R}^{d},w^{1/p})}\leq
C\,\|f\|^{\theta}_{L^{\alpha_{1}}_{p_{1}}(\mathbb{R}^{d},w^{1/p_{1}})}\,\|f\|^{1-\theta}_{L^{\alpha_{2}}_{p_{2}}(\mathbb{R}^{d},w^{1/p_{2}})}\,.$
(A.12)

###### Proof.

The statement is a consequence of the equivalence of the Bessel potential
spaces $L^{\alpha}_{p}(\mathbb{R}^{d},w)$ with the Triebel-Lizorkin spaces
$F^{\alpha}_{p,2}(\mathbb{R}^{d},w)$, mentioned in Remark A.3, and the Hölder
inequality, cf. [7, Sec 3]. ∎

######  Lemma A.8.

Let $w\in L_{1}(\mathbb{R}^{2})$ be an admissible weight,
$n\in\\{3,4,\ldots\\}$, $\delta\in(0,\infty)$ and $\kappa\in(0,2/(n-1)(n-2))$.
Then there exists $C\in(0,\infty)$ and $p\in[1,\infty)$ such that for all
$m\in\\{1,\ldots,n-1\\}$ and $\Psi\in L_{2}^{1}(\mathbb{R}^{2},w^{1/2})\cap
L_{n}(\mathbb{R}^{2},w^{1/n})$, $Z\in L^{-\kappa}_{p}(\mathbb{R}^{2},w^{1/p})$
it holds

$|\langle Z,\Psi^{m}\rangle_{L_{2}(\mathbb{R}^{2},w^{1/2})}|\leq
C\,\|Z\|_{L^{-\kappa}_{p}(\mathbb{R}^{2},w^{1/p})}^{p}+\delta\,\|\vec{\nabla}\Psi\|_{L_{2}(\mathbb{R}^{2},w^{1/2})}^{2}+\delta\,\|\Psi\|_{L_{n}(\mathbb{R}^{2},w^{1/n})}^{n}+\delta.$
(A.13)

###### Proof.

Let $1/r=(1-\kappa)/n+\kappa/2$, $1/q=m/r$ and $1/p^{\prime}=1-1/q$. By
Hölder’s inequality

$|\langle Z,\Psi^{m}\rangle_{L_{2}(\mathbb{R}^{2},w^{1/2})}|\leq
C\,\|Z\|_{L^{-\kappa}_{p^{\prime}}(\mathbb{R}^{2},w^{1/p^{\prime}})}\,\|\Psi^{m}\|_{L^{\kappa}_{q}(\mathbb{R}^{2},w^{1/q})},$
(A.14)

for some $C\in(0,\infty)$. Theorem A.6 implies that

$\|\Psi^{m}\|_{L^{\kappa}_{q}(\mathbb{R}^{d},w^{1/q})}\leq
C\,\|\Psi\|_{L^{\kappa}_{r}(\mathbb{R}^{d},w^{1/r})}^{m}$ (A.15)

and Theorem A.7 implies that

$\|\Psi\|_{L^{\kappa}_{r}(\mathbb{R}^{d},w^{1/r})}\leq
C\,\|\Psi\|_{L^{1}_{2}(\mathbb{R}^{d},w^{1/2})}^{\kappa}\,\|\Psi\|_{L_{n}(\mathbb{R}^{d},w^{1/n})}^{1-\kappa}$
(A.16)

for some $C\in(0,\infty)$. Combining the above bounds we obtain

$|\langle Z,\Psi^{n}\rangle_{L^{2}(\mathbb{R}^{d},w^{1/2})}|\leq
C\,\|Z\|_{L^{-\kappa}_{p^{\prime}}(\mathbb{R}^{d},w^{1/p^{\prime}})}\,\|\Psi\|_{L^{1}_{2}(\mathbb{R}^{d},w^{1/2})}^{m\kappa}\,\|\Psi\|_{L_{n}(\mathbb{R}^{d},w^{1/n})}^{m(1-\kappa)}$
(A.17)

for some $C\in(0,\infty)$. Hence, by Young’s inequality for every
$\delta\in(0,\infty)$ there is $C\in(0,\infty)$ such that

$|\langle Z,\Psi^{m}\rangle_{L_{2}(\mathbb{R}^{2},w^{1/2})}|\leq
C\,\|Z\|_{L^{-\kappa}_{p^{\prime}}(\mathbb{R}^{2},w^{1/p^{\prime}})}^{p^{\prime}}\,+\delta\,\|\Psi\|_{L^{1}_{2}(\mathbb{R}^{2},w^{1/2})}^{2}+\delta\,\|\Psi\|_{L_{n}(\mathbb{R}^{2},w^{1/n})}^{n}.$
(A.18)

We observe that by Hölder’s inequality and the assumption $w\in
L_{1}(\mathbb{R}^{2})$ for all $q,r\in[1,\infty)$ such that $q\leq r$ there
exists $C\in(0,\infty)$ such that
$\|{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}\|_{L_{q}(\mathbb{R}^{2},w^{1/q})}\leq
C\,\|{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}\|_{L_{r}(\mathbb{R}^{2},w^{1/r})}$. Hence, the bound (A.18)
implies the statement of the lemma with $1/p=(2-\kappa(n-1)(n-2))/2n$. ∎

######  Lemma A.9.

Let $p\in[2,\infty)$ and $\alpha=1-2/p$. Then there exists $C\in(0,\infty)$
such that $\|f\|_{L_{p}(\mathbb{S}_{R})}\leq
C\,\|f\|_{L_{2}^{\alpha}(\mathbb{S}_{R})}$ for all $f\in
L_{2}^{\alpha}(\mathbb{S}_{R})$ and all $R\in[1,\infty)$.

###### Proof.

See e.g. [6, Theorem 6] or [32, Theorem II.2.7(ii)]. ∎

##  Appendix B Mathematical preliminaries

######  Lemma B.1.

Let $\kappa\in(0,\infty)$. There exists $C\in(0,\infty)$ such that for all
$R,N\in[1,\infty)$ it holds

$-C\leq\sum_{l=0}^{\infty}\frac{(2l+1)}{2R^{2}\,(1+l(l+1)/R^{2})\,(1+l(l+1)/(NR)^{2})^{\kappa}}-\log(N+1)\leq
C.$ (B.1)

###### Proof.

Observe that the expression in the statement of the lemma coincides with

$\int_{0}^{\infty}\frac{(2\lfloor l\rfloor+1)\,\mathrm{d}l}{2R^{2}\,(1+\lfloor
l\rfloor(\lfloor l\rfloor+1)/R^{2})\,(1+\lfloor l\rfloor(\lfloor
l\rfloor+1)/(NR)^{2})^{\kappa}}-\int_{0}^{\infty}\frac{\mathrm{d}l}{(1+l)(1+(1+l)/N)}.$
(B.2)

The absolute value of the above expression is bounded by

$\int_{0}^{\infty}\left|\frac{(2\lfloor Rl\rfloor+1)/R}{2(1+\lfloor
Rl\rfloor(\lfloor Rl\rfloor+1)/R^{2})\,(1+\lfloor Rl\rfloor(\lfloor
Rl\rfloor+1)/(NR)^{2})^{\kappa}}-\frac{1}{(1+l)(1+(1+l)/N)}\right|\mathrm{d}l.$
(B.3)

Using $0\leq l-\lfloor Rl\rfloor/R\leq 1$ we show that there exists
$\hat{C}\in(0,\infty)$ such that the above expression is bounded by

$\hat{C}+\int_{0}^{\infty}\left|\frac{1}{(1+l)(1+l^{2}/N^{2})^{\kappa}}-\frac{1}{(1+l)(1+(1+l)/N)}\right|\mathrm{d}l\leq
C$ (B.4)

This finishes the proof. ∎

######  Definition B.2.

Let $\mathcal{X}$ be a topological space and let
$(\mu_{n})_{n\in\mathbb{N}_{+}}$ be a sequence of probability measures defined
on $(\mathcal{X},\mathrm{Borel}(\mathcal{X}))$. The sequence
$(\mu_{n})_{n\in\mathbb{N}_{+}}$ is tight iff for every $\epsilon>0$ there
exists a compact set $K_{\epsilon}\subset\mathcal{X}$ such that
$\mu_{n}(K_{\epsilon})\geq 1-\epsilon$ for all $n\in\mathbb{N}_{+}$. The
sequence $(\mu_{n})_{n\in\mathbb{N}_{+}}$ converges weakly if for every
bounded $F\in C(\mathcal{X})$ the sequence of real numbers
$(\mu_{n}(F))_{n\in\mathbb{N}_{+}}$ converges.

######  Theorem B.3 (Prokhorov’s theorem).

Let $\mathcal{X}$ be a separable metric space. A sequence of probability
measures $(\mu_{n})_{n\in\mathbb{N}_{+}}$ on
$(\mathcal{X},\mathrm{Borel}(\mathcal{X}))$ is tight iff there exists a
diverging sequence of natural numbers $(a_{n})_{n\in\mathbb{N}_{+}}$ such that
the sequence $(\mu_{a_{n}})_{n\in\mathbb{N}_{+}}$ converges weakly.

######  Lemma B.4.

Let $\mathcal{X},\mathcal{Y}$ be separable normed spaces such that
$\imath:\mathcal{X}\to\mathcal{Y}$ is a compact embedding and let
$(\mu_{n})_{n\in\mathbb{N}_{+}}$ be a sequence of probability measures on
$(\mathcal{X},\mathrm{Borel}(\mathcal{X}))$. Assume that there exists
$M\in(0,\infty)$ such that
$\int_{\mathcal{X}}\|x\|_{\mathcal{X}}\,\mu_{n}(\mathrm{d}x)\leq M$ for all
$n\in\mathbb{N}_{+}$. Then the sequence of measures
$(\nu_{n})_{n\in\mathbb{N}_{+}}$ on
$(\mathcal{Y},\mathrm{Borel}(\mathcal{Y}))$ defined by

$\nu_{n}(A):=\mu_{n}(\imath^{-1}(A)),\qquad\quad n\in\mathbb{N}_{+},\quad
A\in\mathrm{Borel}(\mathcal{Y}),$ (B.5)

is tight.

###### Proof.

Let $\epsilon>0$,
$L_{\epsilon}:=\\{x\in\mathcal{X}\,|\,\|x\|_{\mathcal{X}}\leq M/\epsilon\\}$
and $K_{\epsilon}:=\overline{\imath(L_{\epsilon})}$. Observe that
$K_{\epsilon}\subset\mathcal{Y}$ is compact. It holds

$1-\nu_{n}(K_{\epsilon})\leq
1-\mu_{n}(L_{\epsilon})=\mu_{n}(\|x\|_{\mathcal{X}}>M/\epsilon)\leq\epsilon/M~{}\int_{\mathcal{X}}\|x\|_{\mathcal{X}}\,\mu_{n}(\mathrm{d}x)\leq\epsilon.$
(B.6)

This finishes the proof. ∎

##  Appendix C Stochastic estimates

We recall from [24, Section 1.1.1] some basic definitions related to the
Wiener chaos. Let $\mathfrak{h}$ be a real, separable Hilbert space with
scalar product $\langle\,\cdot\,,\,\cdot\,\rangle_{\mathfrak{h}}$. We say that
a stochastic process $X=\\{X(h)\,|\,h\in\mathfrak{h}\\}$ defined in a complete
probability space $(\Omega,\mathcal{F},\mathbb{P})$ is a Gaussian process on
$\mathfrak{h}$ if $X$ is a centered Gaussian family of random variables s.t.
$\mathbb{E}(X(h)X(g))=\langle h,g\rangle_{\mathfrak{h}}$ for
$h,g\in\mathfrak{h}$. Now let $H_{n},n\in\mathbb{N}_{0}$, be the Hermite
polynomials. We denote by $\mathcal{H}_{n}$ the closed linear subspace of
$L^{2}(\Omega,\mathbb{P})$ generated by random variables
$\\{H_{n}(X(h)),h\in\mathfrak{h},\|h\|_{\mathfrak{h}}=1\\}$ and call it the
Wiener chaos of order $n$. The subspace
$\bigoplus_{\ell=0}^{n}\mathcal{H}_{\ell}$ is called the inhomogeneous Wiener
chaos of order $n$.

In our case $\Omega=\mathscr{D}^{\prime}(\mathbb{S}_{R})$,
$\mathcal{F}=\mathrm{Borel}(\Omega)$, $\mathbb{P}=\nu_{R}$ is the Gaussian
measure with covariance $G_{R}$ and $\mathfrak{h}=L^{-1}_{2}(\mathbb{S}_{R})$.
Observe that $X^{:m:}_{R,N}=c^{m/2}_{R,N}H_{m}(X_{R,N}/c_{R,N}^{1/2})$. The
choice of the counterterm in (2.3) is dictated by the assumptions of Lemma
C.1.

To facilitate the application of Lemmas C.7, C.8 below in the proof of
Proposition 2.8, we recall that convergence in $L_{2}(\Omega,\mathbb{P})$
implies convergence in probability, and that the latter property is preserved
under composition with continuous functions.

######  Lemma C.1.

Let $X,Y$ be two random variables with joint Gaussian distribution such that
$\mathbb{E}(X)=\mathbb{E}(Y)=0$ and $\mathbb{E}(X^{2})=\mathbb{E}(Y^{2})=1$.
Then, for all $n,m$ we have

$\mathbb{E}(H_{n}(X)H_{m}(Y))=\delta_{n,m}n!(\mathbb{E}(XY))^{n}.$ (C.1)

###### Proof.

See [24, Lemma 1.1.1]. ∎

######  Lemma C.2 (Nelson’s estimate).

For every random variable $X$ in an inhomogeneous Wiener chaos of order
$n\in\mathbb{N}_{+}$, cf. [24], and every $p\in[2,\infty)$ it holds

$\mathbb{E}\big{[}|X|^{p}\big{]}^{\frac{1}{p}}\leq\sqrt{n}(p-1)^{\frac{n}{2}}\,\mathbb{E}\big{[}X^{2}\big{]}^{\frac{1}{2}},\qquad\mathbb{E}\exp\Bigg{(}\frac{n|X|^{2/n}}{6\mathbb{E}\big{[}X^{2}\big{]}^{\frac{1}{n}}}\Bigg{)}<\infty.$
(C.2)

###### Proof.

The first bound follows from the Nelson hypercontractivity of the Ornstein-
Uhlenbeck operator (see e.g. [24, Theorem 1.4.1] or [23]). The second bound is
an immediate consequence of the first one. ∎

######  Definition C.3.

For an operator $H\,:\,L_{2}(\mathbb{S}_{R})\to L_{2}(\mathbb{S}_{R})$ we
denote by $H({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}})$ its integral
kernel (if it exists) such that
$(Hf)(\mathrm{x})=\int_{\mathbb{S}_{R}}H(\mathrm{x},\mathrm{y})\,f(\mathrm{y})\,\rho_{R}(\mathrm{d}\mathrm{y})$.
Similarly, for an operator $H\,:\,L_{2}(\mathbb{R}^{2})\to
L_{2}(\mathbb{R}^{2})$ we denote by
$H({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}})$ its integral
kernel (if it exists) such that
$(Hf)(x)=\int_{\mathbb{R}^{2}}H(x,y)\,f(y)\,\mathrm{d}y$.

######  Lemma C.4.

There exists $C\in(0,\infty)$ such that for all $R,N\in\mathbb{N}_{+}$ it
holds

$|\hat{K}_{R,N}|\leq
C\,\frac{1}{1-\Delta_{R}/N^{2}},\qquad|1-\hat{K}_{R,N}|\leq
C\,\frac{(1-\Delta_{R})/N^{2}}{1-\Delta_{R}/N^{2}},$ (C.3)

where $\hat{K}_{R,N}$ was introduced in Def. 8.6.

######  Remark C.5.

Recall that $K_{R,N}=(1-\Delta_{R}/N^{2})^{-1}$, $G_{R}=(1-\Delta_{R})^{-1}$
and the counterterms $c_{R,N}$, $\hat{c}_{R,N}$ were introduced in Eq. (2.3)
and Def. 8.7. Note that the operators $G_{R},K_{R,N},\hat{K}_{R,N}$ commute.
Using the above lemma we obtain

$|\hat{K}_{R,N}^{2}-K_{R,N}^{2}|\leq|\hat{K}_{R,N}-K_{R,N}|\,|\hat{K}_{R,N}+K_{R,N}|\leq
2\,C\,(C+1)\,\frac{(1-\Delta_{R})/N^{2}}{(1-\Delta_{R}/N^{2})^{2}}.$ (C.4)

Consequently, it holds

$|\hat{c}_{R,N}-c_{R,N}|\leq\mathrm{Tr}(|\hat{K}_{R,N}^{2}-K_{R,N}^{2}|G_{R})/4\pi
R^{2}\leq
2C(C+1)\,\big{[}\mathrm{Tr}((1-\Delta_{R}/N^{2})^{-1}(1-\Delta_{R})^{-1})\\\
-\mathrm{Tr}((1-\Delta_{R}/N^{2})^{-2}(1-\Delta_{R})^{-1})+\mathrm{Tr}((1-\Delta_{R}/N^{2})^{-2}(1-\Delta_{R})^{-1})/N^{2}\big{]}/(4\pi
R^{2}).$ (C.5)

By Lemma B.1 the RHS of the last inequality above is bounded by a constant
independent of $R,N\in\mathbb{N}_{+}$.

###### Proof.

Note that
$\hat{K}_{R,N}=\sum_{l=0}^{\infty}(2l+1)\mathrm{Tr}(\hat{K}_{R,N}\mathcal{P}_{R,l})\,\mathcal{P}_{R,l}$,
where $\mathcal{P}_{R,l}\,:\,L_{2}(\mathbb{S}_{R})\to L_{2}(\mathbb{S}_{R})$
is defined such that $(2l+1)\mathcal{P}_{R,l}$ is the orthogonal projection
onto the eigenspace of the operator $-\Delta_{R}$ corresponding to the
eigenvalue $l(l+1)/R^{2}$. Consequently, by the triangle inequality for the
commuting self-adjoint operators it is enough to show that there exists
$C\in(0,\infty)$ such that for all $R,N\in\mathbb{N}_{+}$ and
$l\in\mathbb{N}_{0}$ it holds

$\begin{gathered}(1+l(l+1)/R^{2}N^{2})\,|\mathrm{Tr}(\hat{K}_{R,N}\mathcal{P}_{R,l})|\leq
C,\\\ |\mathrm{Tr}((1-\hat{K}_{R,N})\mathcal{P}_{R,l})|\leq
C\,(1+l(l+1))/R^{2}N^{2}.\end{gathered}$ (C.6)

(To obtain the second bound in the statement of the lemma one combines both
estimates in (C.6).) Recall that the integral kernel of $\mathcal{P}_{R,l}$ is
given by
$\mathcal{P}_{R,l}(\mathrm{x},\mathrm{y})=P_{l}(\mathrm{x}\cdot\mathrm{y}/R^{2})/4\pi
R^{2},$ where $P_{l}$ is the $l$-th Legendre polynomial. Hence, it holds

$\mathrm{Tr}(\hat{K}_{R,N}\mathcal{P}_{R,l})=2\pi\int_{0}^{1}P_{l}(\cos(\theta/RN))\,RN\sin(\theta/RN)\,h(\theta)\,\mathrm{d}\theta,$
(C.7)

Using the fact that $RN\sin(\theta/RN)\leq\theta$, $|P_{l}(\cos\vartheta)|\leq
1$ and

$l(l+1)P_{l}(\cos\vartheta)\sin\vartheta=-\partial_{\vartheta}^{2}(\sin\vartheta\,P_{l}(\cos\vartheta))+\partial_{\vartheta}(\cos\vartheta\,P_{l}(\cos\vartheta))$
(C.8)

we establish the first of the bounds (C.6). Next, using that $2\pi\int\theta
h(\theta)\mathrm{d}\theta=1$, we obtain that

$\mathrm{Tr}((1-\hat{K}_{R,N})\mathcal{P}_{R,l})=2\pi\int_{0}^{1}(P_{l}(\cos(\theta/RN))\,RN\,\sin(\theta/RN)-\theta)\,h(\theta)\,\mathrm{d}\theta.$
(C.9)

We note the estimates

$0\leq 1-P_{l}(\cos(\vartheta))\leq l(l+1)\,(1-\cos(\vartheta))/2\leq
l(l+1)\,\vartheta^{2}/4,\quad 0\leq
1-\sin(\vartheta)/\vartheta\leq\vartheta^{2}/6,$ (C.10)

where the second inequality follows from
$P_{l}(1)-P_{l}(u)=\int^{1}_{u}\frac{d}{dv}P_{l}(v)dv$ by expressing the
derivative as a linear combination of $P_{l^{\prime}},l^{\prime}\leq l$, and
using $|P_{l^{\prime}}(v)|\leq 1$. This shows the second bound in (C.6) and
finishes the proof. ∎

######  Lemma C.6.

For every $N\in\mathbb{N}_{+}$ there exists $C\in(0,\infty)$ such that for all
$R\in\mathbb{N}_{+}$ it holds

  1. (A)

$\mathbb{E}\|X_{R,N}\|_{L^{1}_{2}(\mathbb{S}_{R})}^{2}\leq R^{2}\,C^{2}$,

  2. (B)

$\mathbb{E}\|\hat{X}_{R,N}\|_{L^{1}_{2}(\mathbb{S}_{R})}^{2}\leq
R^{2}\,C^{2}$.

###### Proof.

Recall that $X_{R,N}=K_{R,N}X_{R}$ and $K_{R,N}=(1-\Delta_{R}/N^{2})^{-1}$.
Consequently,

$\mathbb{E}\|X_{R,N}\|^{2}_{L^{1}_{2}(\mathbb{S}_{R})}=\mathbb{E}\|(1-\Delta_{R})^{1/2}(1-\Delta_{R}/N^{2})^{-1}X_{R}\|^{2}_{L_{2}(\mathbb{S}_{R})}.$
(C.11)

By Fubini’s theorem and the fact that
$\mathbb{E}X_{R}(\mathrm{x})X_{R}(\mathrm{y})=G_{R}(\mathrm{x},\mathrm{y})$,
where $G_{R}=(1-\Delta_{R})^{-1}$, we obtain

$\mathbb{E}\|X_{R,N}\|^{2}_{L^{1}_{2}(\mathbb{S}_{R})}=\mathrm{Tr}\big{(}(1-\Delta_{R}/N^{2})^{-2}\big{)}\leq
N^{4}\,\mathrm{Tr}\big{(}(1-\Delta_{R})^{-2}\big{)}=\sum_{l=0}^{\infty}\frac{N^{4}\,(2l+1)}{(1+l(l+1)/R^{2})^{2}}.$
(C.12)

Now, Item (A) follows from Lemma B.1. Thanks to Lemma C.4 the proof of Item
(B) is the same. ∎

######  Lemma C.7.

For every $\kappa\in(0,\infty)$, $\delta\in[0,2]$ there exists
$C\in(0,\infty)$ such that for all $R,N\in\mathbb{N}_{+}$ it holds

  1. (A)

$\mathbb{E}\|X_{R}\|_{L^{-\kappa}_{2}(\mathbb{S}_{R})}^{2}\leq R^{2}\,C^{2}$,

  2. (B)

$\mathbb{E}\|X_{R}-X_{R,N}\|_{L_{2}^{-\kappa-\delta}(\mathbb{S}_{R})}^{2}\leq
R^{2}\,C^{2}\,N^{-2\delta}$,

  3. (C)

$\mathbb{E}\|X_{R}-\hat{X}_{R,N}\|_{L_{2}^{-\kappa-\delta}(\mathbb{S}_{R})}^{2}\leq
R^{2}\,C^{2}\,N^{-2\delta}$.

###### Proof.

Item (A) follows from Item (B) and Lemma C.6 (A) since, clearly,
$\|X_{R,N}\|_{L^{-\kappa}_{2}(\mathbb{S}_{R})}\leq\|X_{R,N}\|_{L^{1}_{2}(\mathbb{S}_{R})}$.
To prove Item (B) note that

$\mathbb{E}\|X_{R}-X_{R,N}\|^{2}_{L^{-\kappa-\delta}_{2}(\mathbb{S}_{R})}=\mathrm{Tr}\big{(}(1-\Delta_{R})^{-1-\kappa-\delta}(1-(1-\Delta_{R}/N^{2})^{-1})^{2}\big{)}\\\
\leq
N^{-2\delta}\,\mathrm{Tr}\big{(}(1-\Delta_{R})^{-1-\kappa}\big{)}=\sum_{l=0}^{\infty}\frac{N^{-2\delta}\,(2l+1)}{(1+l(l+1)/R^{2})^{1+\kappa}}.$
(C.13)

Now, Item (B) follows from Lemma B.1. Thanks to Lemma C.4 the proof of Item
(C) is the same as the proof of Item (B). ∎

######  Lemma C.8.

Let $R\in\mathbb{N}_{+}$. There exists a real-valued random variable $Y_{R}$
and $C\in(0,\infty)$ such that for all $N\in\mathbb{N}_{+}$ it holds

  1. (A)

$\mathbb{E}Y_{R}^{2}\leq C^{2}$,

  2. (B)

$\mathbb{E}(Y_{R}-Y_{R,N})^{2}\leq C^{2}\,N^{-1/n}$,

  3. (C)

$\mathbb{E}(Y_{R}-\hat{Y}_{R,N})^{2}\leq C^{2}\,N^{-1/n}$,

  4. (D)

$\mathbb{E}(\hat{Y}_{R,N}-\tilde{Y}_{R,N})^{2}\leq C^{2}\,N^{-1}$.

###### Proof.

To prove Items (A) and (B) it is enough to show that for every
$m\in\\{1,\ldots,n\\}$ there exists $C\in(0,\infty)$ such that for all
$N,M\in\mathbb{N}_{+}$ it holds

$\mathbb{E}X^{:m:}_{R,N}(1_{\mathbb{S}_{R}})(X^{:m:}_{R,N}-X^{:m:}_{R,M})(1_{\mathbb{S}_{R}})\leq
C^{2}\,(N\wedge M)^{-1}.$ (C.14)

Let $G_{R,N,M}:=K_{R,N}G_{R}K_{R,M}$. By Lemma C.1

$\mathbb{E}X^{:m:}_{R,N}(1_{\mathbb{S}_{R}})(X^{:m:}_{R,N}-X^{:m:}_{R,M})(1_{\mathbb{S}_{R}})\\\
=m!\int_{\mathbb{S}_{R}^{2}}\\!\\!(G_{R,N,N}(\mathrm{x},\mathrm{y})^{m}-G_{R,N,M}(\mathrm{x},\mathrm{y})^{m})\,\rho_{R}(\mathrm{d}\mathrm{x})\rho_{R}(\mathrm{d}\mathrm{y}).$
(C.15)

Consequently, using Hölder’s inequality we obtain that for every
$m\in\\{1,\ldots,n\\}$ there exists $C\in(0,\infty)$ such that for all
$N,M\in\mathbb{N}_{+}$ and $\mathrm{x}\in\mathbb{S}_{R}$ it holds

$|\mathbb{E}X^{:m:}_{R,N}(1_{\mathbb{S}_{R}})(X^{:m:}_{R,N}-X^{:m:}_{R,M})(1_{\mathbb{S}_{R}})|\\\
\leq
C\,\|(G_{R,N,N}-G_{R,N,M})({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|_{L_{m}(\mathbb{S}_{R}^{2})}(\|G_{R,N,N}({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|_{L_{m}(\mathbb{S}_{R}^{2})}^{m-1}+\|G_{R,N,M}({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|_{L_{m}(\mathbb{S}_{R}^{2})}^{m-1})\\\
\leq\hat{C}\,\|(G_{R,N,N}-G_{R,N,M})({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|_{L_{n}(\mathbb{S}_{R}^{2})}(\|G_{R,N,N}({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|_{L_{n}(\mathbb{S}_{R}^{2})}^{m-1}+\|G_{R,N,M}({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|_{L_{n}(\mathbb{S}_{R}^{2})}^{m-1})\\\
=\check{C}\,\|(G_{R,N,N}-G_{R,N,M})(\mathrm{x},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|_{L_{n}(\mathbb{S}_{R})}(\|G_{R,N,N}(\mathrm{x},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|_{L_{n}(\mathbb{S}_{R})}^{m-1}+\|G_{R,N,M}(\mathrm{x},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|_{L_{n}(\mathbb{S}_{R})}^{m-1}),$ (C.16)

where in the last step above we used the fact that $G_{R,N,N}$ is invariant
under rotations and $\hat{C}=(4\pi R^{2})^{(2-2m/n)}C$, $\check{C}=(4\pi
R^{2})^{m/n}\hat{C}$. By the Sobolev embedding stated in Lemma A.9 there exist
$\hat{C},C\in(0,\infty)$ such that for all $N\in\mathbb{N}_{+}$ it holds

$\|(G_{R,N,N}-G_{R,N,M})(\mathrm{x},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{2}_{L_{n}(\mathbb{S}_{R})}\leq\hat{C}\,\|(G_{R,N,N}-G_{R,N,M})(\mathrm{x},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{2}_{L_{2}^{(n-2)/n}(\mathbb{S}_{R})}\\\ =(4\pi
R^{2})^{-1}\,\hat{C}\,\big{(}\mathrm{Tr}\big{[}G_{R}^{(n+2)/n}K_{R,N}^{2}(K_{R,N}-K_{R,M})^{2}\big{]}\big{)}\leq
C\,(N\wedge M)^{-2/n}.$ (C.17)

The last estimate above follows from the bound

$\mathrm{Tr}\big{[}G_{R}^{(n+2)/n}K_{R,N}^{2}(K_{R,N}-K_{R,M})^{2}\big{]}\\\
\leq\mathrm{Tr}\big{[}(1-\Delta_{R})^{-(n+2)/n}|(1-\Delta_{R})/N^{2}+(1-\Delta_{R})/M^{2}|^{1/n}\big{]}\\\
\leq 2\,(N\wedge
M)^{-2/n}\,\mathrm{Tr}\big{[}(1-\Delta_{R})^{-(n+1)/n}\big{]}$ (C.18)

and Lemma B.1. By an analogous reasoning we obtain

$\|G_{R,N,N}(\mathrm{x},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{2}_{L_{n}(\mathbb{S}_{R})}\leq\hat{C}\,\|G_{R,N,N}(\mathrm{x},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{2}_{L_{2}^{(n-2)/n}(\mathbb{S}_{R})}\\\ =(4\pi
R^{2})^{-1}\,\hat{C}\,\big{(}\mathrm{Tr}\big{[}G_{R}^{(n+2)/n}K_{R,N}^{4}\big{]}\big{)}\leq
C$ (C.19)

for some constants $C,\hat{C}$ independent of $N$ and $m$. This proves (A) and
(B). Thanks to Lemma C.4 the above estimates are also valid when $X_{R,N}$ is
replaced with $\hat{X}_{R,N}$ and $G_{R,N,M}$ is replaced with
$\hat{G}_{R,N,M}:=\hat{K}_{R,N}G_{R}\hat{K}_{R,M}$. Hence, (C) follows. To
prove Item (D) note that for every $m\in\\{1,\ldots,n\\}$ there exists
$C\in(0,\infty)$ such that for all $N\in\mathbb{N}_{+}$ and
$\mathrm{x}\in\mathbb{S}_{R}$ it holds

$\mathbb{E}\hat{X}^{:m:}_{R,N}(1_{\mathbb{S}_{R}\setminus\mathbb{S}_{R,N}})\hat{X}^{:m:}_{R,N}(1_{\mathbb{S}_{R}\setminus\mathbb{S}_{R,N}})\leq\|\hat{G}_{R,N,N}({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{m}_{L_{m}(\mathbb{S}_{R}\setminus\mathbb{S}_{R,N}\times\mathbb{S}_{R}\setminus\mathbb{S}_{R,N})}\\\
\leq\|\hat{G}_{R,N,N}({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{m}_{L_{m}(\mathbb{S}_{R}\setminus\mathbb{S}_{R,N}\times\mathbb{S}_{R})}\leq
C/N\,\|\hat{G}_{R,N,N}(\mathrm{x},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}})\|^{m}_{L_{m}(\mathbb{S}_{R})},$ (C.20)

where in the last step we used the rotational invariance of $\hat{G}_{R,N,N}$
and the fact that the volume of $\mathbb{S}_{R}\setminus\mathbb{S}_{R,N}$ is
bounded by $C/N$. To conclude the proof of Item (D) we use an analog of the
bound (C.19) with $G_{R,N,N}$ replaced by $\hat{G}_{R,N,N}$ and Hölder
inequality. ∎

######  Lemma C.9.

Let $m\in\mathbb{N}_{+}$, $p\in[1,\infty)$, $\kappa\in(0,\infty)$ and
$L\in[1,\infty)$. There exists $C\in(0,\infty)$ such that for all
$R,N\in\mathbb{N}_{+}$, $R\geq L$, it holds

$\mathbb{E}\|\jmath_{R}^{*}X^{:m:}_{R,N}\|_{L^{-\kappa}_{p}(\mathbb{R}^{2},v_{L}^{1/p})}^{p}\leq
C,\qquad\lim_{N\to\infty}\mathbb{E}\|\jmath_{R}^{*}(X_{R}-X_{R,N})\|_{L^{-\kappa}_{p}(\mathbb{R}^{2},v_{L}^{1/p})}^{p}=0.$
(C.21)

###### Proof.

By Jensen’s inequality it suffices to prove the statement for $p\in
2\mathbb{N}_{+}$. Let $q=(4/\kappa)\vee 4$. There exists $C\in(0,\infty)$
depending on $p$ and $\kappa$ such that for all $R,N\in\mathbb{N}_{+}$ it
holds

$\mathbb{E}\|\jmath_{R}^{*}X^{:m:}_{R,N}\|_{L^{-\kappa}_{p}(\mathbb{R}^{2},v_{L}^{1/p})}^{p}\leq\|v_{L}w_{L}^{-1/q}\|_{L_{1}(\mathbb{R}^{2})}\,\|\mathbb{E}((1-\Delta)^{-\kappa/2}\jmath_{R}^{*}X^{:m:}_{R,N}({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}))^{p}\|_{L_{\infty}(\mathbb{R}^{2},w_{L}^{1/q})}\\\ \leq
C\,\|\mathbb{E}((1-\Delta)^{-\kappa/2}\jmath_{R}^{*}X^{:m:}_{R,N}({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}))^{2}\|_{L_{\infty}(\mathbb{R}^{2},w_{L}^{1/q})}^{p/2}\,,$
(C.22)

where the last bound is a consequence of Lemma C.2. Recall that
$\mathbb{E}X_{R,N}\otimes
X_{R,N}=G_{R,N}({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}})$, where
$G_{R,N}=K_{R,N}G_{R}K_{R,N}$. By Lemma C.1

$\mathbb{E}\jmath_{R}^{*}X^{:m:}_{R,N}\otimes\jmath_{R}^{*}X^{:m:}_{R,N}=m!\,\tilde{G}_{R,N}^{m},\qquad\tilde{G}_{R,N}:=(\jmath_{R}^{*}\otimes\jmath_{R}^{*})G_{R,N}({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}},{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}})\,.$ (C.23)

Hence, by Fubini’s theorem and explicit formula for the kernel in terms of
spherical harmonics

$\mathbb{E}(1-\Delta)^{-\kappa/2}\jmath_{R}^{*}X^{:m:}_{R,N}\otimes\,(1-\Delta)^{-\kappa/2}\jmath_{R}^{*}X^{:m:}_{R,N}\\\
=m!\,\big{(}(1-\Delta)^{-\kappa/2}\otimes(1-\Delta)^{-\kappa/2}\big{)}\,\tilde{G}_{R,N}^{m}\in
C(\mathbb{R}^{2}\times\mathbb{R}^{2}).$ (C.24)

Since for $F\in C(\mathbb{R}^{2}\times\mathbb{R}^{2})$ it holds
$\sup_{x\in\mathbb{R}^{2}}F(x,x)\leq\sup_{y\in\mathbb{R}^{2}}\sup_{x\in\mathbb{R}^{2}}F(x,y)$
we obtain

$\|\mathbb{E}((1-\Delta)^{-\kappa/2}\jmath_{R}^{*}X^{:m:}_{R,N}({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}))^{2}\|_{L_{\infty}(\mathbb{R}^{2},w_{L}^{1/q})}\\\ \leq
m!\,\sup_{y}w_{L}^{1/q}(y)\,\|((1-\Delta)^{-\kappa/2}\otimes
1)\big{(}1\otimes(1-\Delta)^{-\kappa/2})\tilde{G}_{R,N}^{m}\big{)}({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}},y)\|_{L_{\infty}(\mathbb{R}^{2})}\\\
=m!\,\sup_{y}w_{L}^{1/q}(y)\,\|\big{(}1\otimes(1-\Delta)^{-\kappa/2})\tilde{G}_{R,N}^{m}\big{)}({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}},y)\|_{L^{-\kappa}_{\infty}(\mathbb{R}^{2})}.$ (C.25)

By Theorem A.5 (B) there exists $C\in(0,\infty)$ such that for all
$R,N\in\mathbb{N}_{+}$ the above expression is bounded by

$\sup_{y\in\mathbb{R}^{2}}w_{L}^{1/q}(y)\,\|\big{(}1\otimes(1-\Delta)^{-\kappa/2})\tilde{G}_{R,N}^{m}\big{)}({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}},y)\|_{L_{\infty}(\mathbb{R}^{2})}\\\
=\sup_{x\in\mathbb{R}^{2}}\|\big{(}1\otimes(1-\Delta)^{-\kappa/2})\tilde{G}_{R,N}^{m}\big{)}(x,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip