A Function Spaces Definition A.1

We say that \(w\in C^\infty (}^d)\) is an admissible weight iff there exist \(b\in [0,\infty )\) and \(c\in (0,\infty )\) such that \(0<w(x)\le c\,w(y)\,(1+|x-y|)^b\) for all \(x,y\in }^d\) and for every \(a\in }_0^d\) there exists \(c_a\in (0,\infty )\) such that \(|\partial ^a w (x)|\le c_a\, w(x)\) for all \(x\in }^d\).

Definition A.2

Let w be an admissible weight, \(p\in [1,\infty ]\) and \(\alpha \in }\), \(n\in }_0\). By definition, \(L_p(}^d,w)\) is the Banach space with the norm

$$\begin \Vert f\Vert _}^d,w)}:=\Vert w f\Vert _}^d)}. \end$$

The weighted Bessel potential space \(L^\alpha _p(}^d,w)\) is the Banach space with the norm

$$\begin \Vert f\Vert _}^d,w)}:= \Vert (1-\Delta )^f\Vert _}^d,w)}. \end$$

We also set \(L^\alpha _p(}^d)=L^\alpha _p(}^d,1)\). The weighted Sobolev space \(W^n_p(}^d,w)\) is the Banach space with the norm

$$\begin \Vert f\Vert _}^d,w)} = \textstyle \sum _ a\in }^d,|a|\le n \end} \Vert \partial ^a f\Vert _}^d,w)}. \end$$

For \(R\in (0,\infty )\) the Bessel potential space \(L^\alpha _p(}_R)\) on the round sphere \(}_R\subset }^d\) of radius R is the Banach space with the norm

$$\begin \Vert f\Vert _}_R)}:= \Vert (1-\Delta _R)^f\Vert _}_R)}, \end$$

where \(L_p(}_R)\) is the \(L_p\) space on \(}_R\) with respect to the canonical measure \(\rho _R\) on \(}_R\).

Remark A.3

The following facts are standard: Let w be an admissible weight, \(p\in [1,\infty )\), \(\alpha \in }\) and \(n\in }_0\). The norms \(\Vert \bullet \Vert _}^d,w)}\) and \(\Vert w\bullet \Vert _}^d)}\) are equivalent. The Sobolev space \(W^n_p(}^d,w)\) coincides with the Bessel potential space \(L^n_p(}^d,w)\) with equivalent norms. The Bessel potential space \(L^\alpha _p(}^d,w)\) coincides with the Triebel–Lizorkin space \(F^\alpha _(}^d,w)\) with equivalent norms. Furthermore, the Bessel potential space \(L_p^(}^d,w)\) is continuously embedded in the Besov space \(B_^(}^d,w)\) and the Besov space \(B_^(}^d,w)\) is continuously embedded in the Bessel potential space \(L_\infty ^(}^d,w)\). These facts can be obtained, e.g., from [32, Theorem 6.5, Theorem 6.9] and [30, Sec. 2.5.7]. We note that [32, Theorem 6.5 (iii)] is useful to pass from \(\alpha =0\) to \(\alpha \in }\).

Remark A.4

For \(\alpha \in }\) and \(p,q\in [1,\infty ]\), we have the following generalized Hölder inequality

$$\begin |\langle f,g\rangle _}^d,w^)}| \le C\, \Vert f\Vert _}^d,w^)}\,\Vert g\Vert __q(}^d,w^)}, \qquad \frac+\frac=1, \end$$

where \(\langle \bullet ,\bullet \rangle _}^d,w^)}\) is the scalar product in \(L_2(}^d,w^)\) and the constant \(C\in (0,\infty )\) depends only on the weight w.

Theorem A.5

Let w, v be admissible weights and

$$\begin -\infty<\alpha _2\le \alpha _1 <\infty , \qquad 1\le p_1\le p_2\le \infty . \end$$

(A) The embedding \(L^_(}^d,w)\rightarrow L^_(}^d,v)\) is continuous if

$$\begin p_2<\infty , \qquad \alpha _1-d/p_1\ge \alpha _2-d/p_2 \quad \text \quad \sup _}^d} v(x)/w(x) < \infty . \end$$

(B) The embedding \(L^_(}^d,w)\rightarrow L^_(}^d,v)\) is continuous if

$$\begin \alpha _1-d/p_1> \alpha _2 \quad \text \quad \sup _}^d} v(x)/w(x) < \infty . \end$$

(C) The embedding \(L^_(}^d,w)\rightarrow L^_(}^d,v)\) is compact if

$$\begin p_2<\infty , \qquad \alpha _1-d/p_1>\alpha _2-d/p_2 \quad \text \quad \lim _ v(x)/w(x) =0. \end$$

Proof

Parts (A) and (C) follow from [13, Sec. 4.2.3, Theorem] and the equivalence between \(L^\alpha _p(}^d,w)\) and \(F^\alpha _(}^d,w)\) mentioned in Remark A.3 above. Part (B) is covered by [13, Sec. 4.2.3, Remark] and the embeddings stated in Remark A.3. \(\square \)

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 \(f\in L^\alpha _(}^d,w^)\) and \(g\in L^\alpha _(}^d,w^)\)

$$\begin \Vert fg\Vert _}^d,w^)} \le C\,\Vert f\Vert _(}^d,w^)} \, \Vert g\Vert _(}^d,w^)}. \end$$

Proof

The statement follows from the equivalence of the norms \(\Vert \bullet \Vert _}^d,w)}\) and \(\Vert w\bullet \Vert _}^d)}\), the fractional Leibniz rule [23, Ch. 2] and Theorem A.5 (A). Alternatively, one can use [8, Lemma 5]. \(\square \)

Theorem A.7

Let w be an admissible weight, \(p_1,p_2\in [1,\infty )\), \(\alpha _1,\alpha _2\in }\), \(\theta \in (0,1)\) and

$$\begin \alpha =\theta \,\alpha _1+(1-\theta )\,\alpha _2, \qquad \qquad \frac = \frac + \frac. \end$$

There exists \(C\in (0,\infty )\) such that for all \(f\in L^_(}^d,w^)\cap L^_(}^d,w^)\) it holds

$$\begin \Vert f\Vert _}^d,w^)} \le C\,\Vert f\Vert ^\theta __(}^d,w^)}\, \Vert f\Vert ^__(}^d,w^)}. \end$$

Proof

The statement is a consequence of the equivalence of the Bessel potential spaces \(L^\alpha _p(}^d,w)\) with the Triebel–Lizorkin spaces \(F^\alpha _(}^d,w)\), mentioned in Remark A.3, and the Hölder inequality, cf. [8, Sec 3]. \(\square \)

Lemma A.8

Let \(w\in L_1(}^2)\) be an admissible weight, \(n\in \\), \(\delta \in (0,\infty )\) 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 \( m\in \ \)and \(\Psi \in L_2^1(}^2,w^) \cap L_n(}^2,w^)\), \(Z\in L^_p(}^2,w^)\) it holds

$$\begin |\langle Z,\Psi ^m \rangle _}^2,w^)}|\le & C\,\Vert Z\Vert __p(}^2,w^)}^p+ \delta \,\Vert \vec \nabla \Psi \Vert _}^2,w^)}^2\\ & + \delta \,\Vert \Psi \Vert _}^2,w^)}^n+\delta . \end$$

Proof

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

$$\begin |\langle Z,\Psi ^m \rangle _}^2,w^)}| \le C\,\Vert Z\Vert __(}^2,w^)}\,\Vert \Psi ^m\Vert _}^2,w^)}, \end$$

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

$$\begin \Vert \Psi ^m\Vert _}^d,w^)} \le C\, \Vert \Psi \Vert _(}^d,w^)}^m \end$$

and Theorem A.7 implies that

$$\begin \Vert \Psi \Vert _}^d,w^)} \le C\,\Vert \Psi \Vert _}^d,w^)}^\kappa \,\Vert \Psi \Vert _}^d,w^)}^ \end$$

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

$$\begin |\langle Z,\Psi ^m \rangle _}^d,w^)}| \le C\,\Vert Z\Vert __(}^d,w^)}\, \Vert \Psi \Vert _}^d,w^)}^ \,\Vert \Psi \Vert _}^d,w^)}^ \end$$

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

$$\begin |\langle Z,\Psi ^m \rangle _}^2,w^)}| \le C\,\Vert Z\Vert __(}^2,w^)}^\, +\delta \,\Vert \Psi \Vert _}^2,w^)}^2 +\delta \,\Vert \Psi \Vert _}^2,w^)}^n.\nonumber \\ \end$$

(A.1)

We observe that by Hölder’s inequality and the assumption \(w\in L_1(}^2)\) for all \(q,r\in [1,\infty )\) such that \(q\le r\) there exists \(C\in (0,\infty )\) such that \(\Vert \bullet \Vert _}^2,w^)}\le C\,\Vert \bullet \Vert _}^2,w^)}\). Hence, the bound (A.1) implies the statement of the lemma with \(1/p = (2-\kappa (n-1)(n-2))/2n\). \(\square \)

Lemma A.9

Let \(p\in [2,\infty )\) and \(\alpha =1-2/p\). Then, there exists \(C\in (0,\infty )\) such that \(\Vert f\Vert _}_R)}\le C\,\Vert f\Vert _}_R)}\) for all \(f\in L_2^\alpha (}_R)\) and all \(R\in [1,\infty )\).

Proof

See, e.g., [7, Theorem 6] or [33, Theorem II.2.7(ii)]. \(\square \)

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

$$\begin -C\le \sum _^\infty \frac - \log (N+1) \le C. \end$$

Proof

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

The absolute value of the above expression is bounded by

$$\begin & \int _0^\infty \left| \frac\right. \\ & \quad \left. - \frac\right| \textrml. \end$$

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

$$\begin }+ \int _0^\infty \left| \frac - \frac\right| \textrml \le C. \end$$

This finishes the proof. \(\square \)

Definition B.2

Let \(}\) be a topological space and let \((\mu _)_}_+}\) be a sequence of probability measures defined on \((},\textrm(}))\). The sequence \((\mu _)_}_+}\) is tight iff for every \(\epsilon >0\) there exists a compact set \(K_\epsilon \subset }\) such that \(\mu _n(K_\epsilon )\ge 1-\epsilon \) for all \(n\in }_+\). The sequence \((\mu _)_}_+}\) converges weakly if for every bounded \(F\in C(})\) the sequence of real numbers \((\mu _n(F))_}_+}\) converges.

Theorem B.3

(Prokhorov’s theorem). Let \(}\) be a separable metric space. A sequence of probability measures \((\mu _)_}_+}\) on \((},\textrm(}))\) is tight iff there exists a diverging sequence of natural numbers \((a_n)_}_+}\) such that the sequence \((\mu _)_}_+}\) converges weakly.

Lemma B.4

Let \(},}\) be separable normed spaces such that \(\imath :}\rightarrow }\) is a compact embedding and let \((\mu _)_}_+}\) be a sequence of probability measures on \((},\textrm(}))\). Assume that there exists \(M\in (0,\infty )\) such that \(\int _}}\Vert x\Vert _}}\,\mu _n(\textrmx)\le M\) for all \(n\in }_+\). Then, the sequence of measures \((\nu _n)_}_+}\) on \((},\textrm(}))\) defined by

$$\begin \nu _n(A):=\mu _n(\imath ^(A)), \qquad \quad n \in }_+, \quad A \in \textrm(}), \end$$

is tight.

Proof

Let \(\epsilon >0\), \(L_\epsilon := \}\,|\,\Vert x\Vert _}}\le M/\epsilon \}\) and \(K_\epsilon :=\overline)}\). Observe that \(K_\epsilon \subset }\) is compact. It holds

$$\begin 1-\nu _n(K_) \le 1-\mu _n(L_\epsilon ) = \mu _n(\Vert x\Vert _}}>M/\epsilon ) \le \epsilon /M~\int _}}\Vert x\Vert _}}\,\mu _n(\textrmx)\le \epsilon . \end$$

This finishes the proof. \(\square \)

C Stochastic Estimates

We recall from [25, Section 1.1.1] some basic definitions related to the Wiener chaos. Let \(}\) be a real, separable Hilbert space with scalar product \(\langle \,\cdot \,,\, \cdot \, \rangle _}}\). We say that a stochastic process \(X=\}\}\) defined in a complete probability space \((\Omega ,}, })\) is a Gaussian process on \(}\) if X is a centered Gaussian family of random variables such that \(}(X(h) X(g))=\langle h, g\rangle _}\) for \(h,g\in }\). Now let \(H_n, n\in }_0\), be the Hermite polynomials. We denote by \(}_n\) the closed linear subspace of \(L^2(\Omega ,})\) generated by random variables \(\}, \Vert h\Vert _}=1\}\) and call it the Wiener chaos of order n. The subspace \(\bigoplus _^ }_\ell \) is called the inhomogeneous Wiener chaos of order n.

In our case, \(\Omega =}'(}_R)\), \(}=\textrm(\Omega )\), \(}=\nu _R\) is the Gaussian measure with covariance \(G_R\) and \(}\!=\!L^_2(}_R)\). Observe that \(X^_\!=\!c^_ H_m(X_/c_^)\). The choice of the counterterm in (2.1) 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 2.7, we recall that convergence in \(L_2(\Omega , })\) 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 \(}(X)=}(Y)=0\) and \(}(X^2)=}(Y^2)=1\). Then, for all n, m we have

$$\begin }(H_n(X)H_m(Y))=\delta _ n! (}(XY))^n. \end$$

Proof

See [25, Lemma 1.1.1]. \(\square \)

Lemma C.2

(Nelson’s estimate). For every random variable X in an inhomogeneous Wiener chaos of order \(n\in }_+\), cf. [25], and every \(p\in [2,\infty )\) it holds

$$\begin }\big [ |X|^\big ]^} \le \sqrt(p - 1)^}\, }\big [ X^\big ]^}, \qquad }\exp \Bigg (\frac}}\big [ X^\big ]^}}\Bigg ) <\infty . \end$$

Proof

The first bound follows from the Nelson hypercontractivity of the Ornstein–Uhlenbeck operator (see, e.g., [25, Theorem 1.4.1] or [24]). The second bound is an immediate consequence of the first one. \(\square \)

Definition C.3

For an operator \(H:\,L_2(}_R)\rightarrow L_2(}_R)\), we denote by \(H(\bullet ,\bullet )\) its integral kernel (if it exists) such that \((Hf)(\textrm)=\int _}_R} H(})\,f(})\,\rho _R(})\). Similarly, for an operator \(H:\,L_2(}^2)\rightarrow L_2(}^2)\) we denote by \(H(\bullet ,\bullet )\) its integral kernel (if it exists) such that \((Hf)(x)=\int _}^2} H(x,y)\,f(y)\,\textrmy\).

Lemma C.4

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

$$\begin |}_| \le C\,\frac, \qquad |1-}_|\le C\,\frac, \end$$

where \(\hat_\) is introduced in Definition 8.6.

Remark C.5

Recall that \(K_=(1-\Delta _R/N^2)^\), \(G_R=(1-\Delta _R)^\) and the counterterms \(c_\), \(}_\) were introduced in Eq. (2.1) and Definition 8.8. Note that the operators \(G_R,K_,}_\) commute. Using the above lemma, we obtain

$$\begin |}_^2-K_^2|\le & |}_-K_|\,|}_+K_| \\\le & 2\,C\,(C+1)\,\frac. \end$$

Consequently, it holds

$$\begin |}_-c_|\le & \textrm(|}_^2-K_^2| G_R)/4\pi R^2 \\\le & 2C(C+1)\, \big [\textrm((1-\Delta _R/N^2)^ (1-\Delta _R)^) \\ & - \textrm((1-\Delta _R/N^2)^ (1-\Delta _R)^)\\ & + \textrm((1-\Delta _R/N^2)^ (1-\Delta _R)^)/N^2\big ]/(4\pi R^2). \end$$

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

Proof

Note that \(}_=\sum _^\infty (2\,l+1)\textrm(}_}_)\,}_\), where \(}_:\,L_2(}_R)\rightarrow L_2(}_R)\) is defined such that \((2l+1)}_\) 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 \(R,N\in }_+\) and \(l\in }_0\) it holds

$$\begin \begin&(1+l(l+1)/R^2N^2)\,|\textrm(}_}_)| \le C, \\&|\textrm((1-}_)}_)| \le C\,(1+l(l+1))/R^2N^2. \end \end$$

(C.1)

(To obtain the second bound in the statement of the lemma one combines both estimates in (C.1).) Recall that [4, Theorem 2.9] the integral kernel of \(}_\) is given by \( }_(\textrm,\textrm)=P_l(\textrm\cdot \textrm/R^2)/4\pi R^2, \) where \(P_l\) is the l-th Legendre polynomial. Hence, it holds

$$\begin \textrm(}_}_) = 2\pi \int _0^1 P_l(\cos (\theta /RN))\,RN\sin (\theta /RN)\,h(\theta )\,\textrm\theta . \end$$

Using the fact that \(RN\sin (\theta /RN)\le \theta \), \(|P_l(\cos \vartheta )|\le 1\) (cf. [4, Sec. 2.7.5]) and

$$\begin l(l+1) P_l(\cos \vartheta ) \sin \vartheta = -\partial _\vartheta ^2(\sin \vartheta \, P_l(\cos \vartheta )) + \partial _\vartheta (\cos \vartheta \, P_l(\cos \vartheta )) \end$$

(cf. [4, Sec. 2.7.2]) we show the first of the bounds (C.1). Next, using that \(2\pi \int \theta h(\theta ) \textrm\theta =1\), we obtain that

$$\begin \textrm((1-}_)}_) = 2\pi \int _0^1 (P_l(\cos (\theta /RN))\,RN\,\sin (\theta /RN)-\theta )\,h(\theta )\,\textrm\theta . \end$$

We note the estimates

$$\begin & 0\le 1-P_l(\cos (\vartheta )) \le l(l+1)\,(1-\cos (\vartheta ))/2\le l(l+1)\,\vartheta ^2/4, \quad \\ & 0\le 1 - \sin (\vartheta )/\vartheta \le \vartheta ^2/6, \end$$

where the second inequality follows from

$$\begin 1-P_l(u)=P_l(1)-P_l(u)=\int ^1_u \fracP_l(v)dv\le (1-u)\,l(l+1)/2 \end$$

(cf. [4, Sec. 2.7.5]). This shows the second bound in (C.1) and finishes the proof. \(\square \)

Lemma C.6

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

(A)

\(}\Vert X_\Vert __2(}_R)}^2 \le R^2\, C^2\),

(B)

\(}\Vert }_\Vert __2(}_R)}^2 \le R^2\, C^2\).

Proof

Recall that \(X_=K_X_R\) and \(K_=(1-\Delta _R/N^2)^\). Consequently,

$$\begin }\Vert X_\Vert ^2__(}_R)} = }\Vert (1-\Delta _R)^ (1-\Delta _R/N^2)^ X_R\Vert ^2_}_R)}. \end$$

By Fubini’s theorem and the fact that \(}X_R(\textrm)X_R(\textrm)=G_R(\textrm,\textrm)\), where \(G_R=(1-\Delta _R)^\), we obtain

$$\begin }\Vert X_\Vert ^2__(}_R)}= & \textrm\big ((1-\Delta _R/N^2)^\big ) \le N^4\,\textrm\big ((1-\Delta _R)^\big )\\= & \sum _^ \frac. \end$$

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

Lemma C.7

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

(A)

\(}\Vert X_R\Vert __2(}_R)}^2 \le R^2\,C^2\),

(B)

\(}\Vert X_R- X_\Vert _(}_R)}^2 \le R^2\,C^2\, N^\),

(C)

\(}\Vert X_R-}_\Vert _(}_R)}^2 \le R^2\,C^2\, N^\).

Proof

Item (A) follows from Item (B) and Lemma C.6 (A) since, clearly, \(\Vert X_\Vert __2(}_R)} \le \Vert X_\Vert __2(}_R)}\). To prove Item (B) note that

$$\begin & }\Vert X_R - X_\Vert ^2__(}_R)} = \textrm\big ((1-\Delta _R)^ (1- (1 -\Delta _R/N^2)^)^2 \big ) \\ & \quad \le N^\,\textrm\big ((1-\Delta _R)^\big ) = \sum _^ \frac\,(2l+1)}}. \end$$

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). \(\square \)

Lemma C.8

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

(A)

\(}Y_R^2\le C^2\),

(B)

\(}(Y_R-Y_)^2\le C^2\,N^\),

(C)

\(}(Y_R-}_)^2\le C^2\,N^\),

(D)

\(}(}_-}_)^2\le C^2\,N^\).

Remark C.9

Recall that \(n\in 2}_+\), \(n\ge 4\), is the degree of the polynomial P and the random variables \(Y_\) and \(}_,}_\) are introduced in Definitions 2.3 and 8.8, respectively.

Proof

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

$$\begin }X^_(1_}_R})(X^_-X^_)(1_}_R})\le C^2\,(N\wedge M)^. \end$$

Let \(G_:=K_G_R K_\). By Lemma C.1

$$\begin & }X^_(1_}_R})(X^_-X^_)(1_}_R}) \\ & \quad = m! \int _}_R^2}\!\!(G_(})^m-G_(})^m)\,\rho _R(})\rho _R(}). \end$$

Consequently, using Hölder’s inequality we obtain that for every \(m\in \\) there exists \(C\in (0,\infty )\) such that for all \(N,M\in }_+\) and \(\textrm\in }_R\) it holds

$$\begin & |}X^_(1_}_R})(X^_-X^_)(1_}_R})| \\ & \quad \le C\,\Vert (G_-G_)(\bullet ,\bullet )\Vert _}_R^2)} (\Vert G_(\bullet ,\bullet )\Vert _}_R^2)}^ + \Vert G_(\bullet ,\bullet )\Vert _}_R^2)}^) \\ & \quad \le }\,\Vert (G_-G_)(\bullet ,\bullet )\Vert _}_R^2)} (\Vert G_(\bullet ,\bullet )\Vert _}_R^2)}^ + \Vert G_(\bullet ,\bullet )\Vert _}_R^2)}^) \\ & \quad = }\,\Vert (G_-G_)(\textrm,\bullet )\Vert _}_R)} (\Vert G_(\textrm,\bullet )\Vert _}_R)}^ + \Vert G_(\textrm,\bullet )\Vert _}_R)}^), \end$$

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

$$\begin & \Vert (G_-G_)(\textrm,\bullet )\Vert ^2_}_R)} \le }\, \Vert (G_-G_)(\textrm,\bullet )\Vert ^2_(}_R)} \\ & \quad = (4\pi R^2)^\,}\,\big (\textrm\big [G_R^ K_^2 (K_-K_)^2\big ]\big ) \le C\,(N\wedge M)^. \end$$

The last estimate above follows from the bound

$$\begin & \textrm\big [G_R^ K_^2 (K_-K_)^2\big ] \\ & \quad \le \textrm\big [(1-\Delta _R)^ |(1-\Delta _R)/N^2+(1-\Delta _R)/M^2|^\big ] \\ & \quad \le 2\,(N\wedge M)^\,\textrm\big [(1-\Delta _R)^\big ] \end$$

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

$$\begin \Vert G_(\textrm,\bullet )\Vert ^2_}_R)}\le & }\, \Vert G_(\textrm,\bullet )\Vert ^2_(}_R)} \nonumber \\= & (4\pi R^2)^\,}\,\big (\textrm\big [G_R^ K_^4\big ]\big ) \le C \end$$

(C.2)

for some constants \(C,}\) independent of N and m. This proves (A) and (B). Thanks to Lemma C.4 the above estimates are also valid when \(X_\) is replaced with \(}_\) and \(G_\) is replaced with \(}_:=}_G_R}_\). Hence, (C) follows. To prove Item (D) note that for every \(m\in \\) there exists \(C\in (0,\infty )\) such that for all \(N\in }_+\) and \(\textrm\in }_R\) it holds

$$\begin & }}^_(1_}_R\setminus }_})}^_(1_}_R\setminus }_}) \le \Vert }_(\bullet ,\bullet )\Vert ^m_}_R\setminus }_\times }_R\setminus }_)} \\ & \quad \le \Vert }_(\bullet ,\bullet )\Vert ^m_}_R\setminus }_\times }_R)} \le C/N\, \Vert }_(\textrm,\bullet )\Vert ^m_}_R)}, \end$$

where in the last step we used the rotational invariance of \(}_\) and the fact that the volume of \(}_R\setminus }_\) is bounded by C/N. To conclude the proof of Item (D), we use an analog of the bound (C.2) with \(G_\) replaced by \(}_\) and Hölder inequality. \(\square \)

Lemma C.10

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

$$\begin }\Vert \jmath _R^* X^_\Vert __p(}^2,v_L^)}^p \le C, \qquad \lim _}\Vert \jmath _R^* (X_R-X_)\Vert __p(}^2,v_L^)}^p =0. \end$$

Proof

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

$$\begin & }\Vert \jmath _R^* X^_\Vert __p(}^2,v_L^)}^p\\ & \qquad \le \Vert v_L w_L^\Vert _}^2)}\,\Vert }((1-\Delta )^\jmath _R^* X^_(\bullet ))^p\Vert _}^2,w_L^)} \\ & \qquad \le C\,\Vert }((1-\Delta )^\jmath _R^* X^_(\bullet ))^2\Vert _}^2,w_L^)}^, \end$$

where the last bound is a consequence of Lemma C.2. Recall that \(}X_\otimes X_ = G_(\bullet ,\bullet )\), where \(G_=K_G_RK_\). By Lemma C.1

$$\begin }\jmath _R^*X^_\otimes \jmath _R^*X^_ = m!\, }_^m, \qquad }_:=(\jmath _R^*\otimes \jmath _R^*)G_(\bullet ,\bullet ). \end$$

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

$$\begin & }(1-\Delta )^\jmath _R^* X^_\otimes \, (1-\Delta )^\jmath _R^* X^_ \\ & \quad = m!\, \big ((1-\Delta )^ \otimes (1-\Delta )^\big )\, }_^m \in C(}^2\times }^2). \end$$

Since for \(F\in C(}^2\times }^2)\) it holds \(\sup _}^2} F(x,x)\le \sup _}^2}\sup _}^2} F(x,y)\) we obtain

$$\begin & \Vert }((1-\Delta )^\jmath _R^* X^_(\bullet ))^2\Vert _}^2,w_L^)} \\ & \quad \le m!\,\sup _y w_L^(y)\,\Vert ((1-\Delta )^ \otimes 1)\big (1 \otimes (1-\Delta )^) }_^m\big )(\bullet ,y)\Vert _}^2)} \\ & \quad = m!\,\sup _y w_L^(y)\,\Vert \big ( 1 \otimes (1-\Delta )^) }_^m\big )(\bullet ,y)\Vert __\infty (}^2)}. \end$$

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

$$\begin & \sup _}^2} w_L^(y)\,\Vert \big (1\otimes (1-\Delta )^) }_^m\big )(\bullet ,y)\Vert _}^2)} \\ & \quad = \sup _}^2} \Vert \big ( 1\otimes (1-\Delta )^) }_^m\big )(x,\bullet )\Vert _}^2,w_L^)} \\ & \quad = \sup _}^2} \Vert }_^m(x,\bullet )\Vert _(}^2,w_L^)} \end$$

up to a multiplicative constant C, which depends on m. The first equality above follows from the fact that for \( F\in C(}^2\times }^2)\) it holds \(\sup _}^2}\sup _}^2} F(x,y)=\sup _}^2}\sup _}^2} F(x,y)\). By Theorem A.5 (B), since \(q>2/\kappa \), the above expression is bounded by

$$\begin & \sup _}^2} \Vert }_^m(x,\bullet )\Vert _}^2,w_L^)} \\ & \quad \le \sup _}^2} \Vert }_^m(x,\bullet )\Vert _}^2,w_R^)} \\ & \quad = \sup _\in }_R} \Vert G_(\textrm,\bullet )^m\Vert _}_R)} \\ & \quad = \sup _\in }_R} \Vert G_(\textrm,\bullet )\Vert _(}_R)}^m \le C\sup _\in }_R} \Vert G_(\textrm,\bullet )\Vert __2(}_R)}^m \\ & \quad = C(4\pi R^2)^\,[\textrm(G_(1-\Delta _R)^G_)]^. \end$$

The first bound above is true because \(R\ge L\). The second bound is a consequence of the Sobolev embedding stated in Lemma A.9, since \(q\ge 2/m\). The first of the bounds from the statement of the lemma follows now from Lemma B.1 applied with \(N'=1\) and \(\kappa '=2/mq\). To prove the second of the bounds, we use exactly the same strategy as above with \(m=1\) and the operator \(G_\) replaced by \((1-K_)G_R(1-K_)\). \(\square \)