Consistency of the Bayes method for the inverse scattering problem

In this work, we consider the inverse scattering problem of determining an unknown refractive index from the far-field measurements using the nonparametric Bayesian approach. We use a collection of large ‘samples’, which are noisy discrete measurements taking from the scattering amplitude. We will study the frequentist property of the posterior distribution as the sample size tends to infinity. Our aim is to establish the consistency of the posterior distribution with an explicit contraction rate in terms of the sample size. We will consider two different priors on the space of parameters. The proof relies on the stability estimates of the forward and inverse problems. Due to the ill-posedness of the inverse scattering problem, the contraction rate is of a logarithmic type. We also show that such contraction rate is optimal in the statistical minimax sense.


Introduction
In this work, we study the Bayes method for solving the inverse medium scattering problem.Our aim is to prove the consistency property of the posterior distribution.Let the refractive index n ⩾ 0 and 1 − n be a compactly supported function in R 3 with supp(1 − n) ⊂ D, where D is an open bounded smooth domain, and having suitable regularity, which will be specified later.Let u = u inc + u sca n satisfy ∆u + κ 2 nu = 0 in R 3 , (1. Assume that u inc is the plane incident field, i.e. u inc = e iκx•θ with θ ∈ S 2 , where S 2 = θ ∈ R 3 : |θ| = 1 .Then the scattered field u sca n satisfies where θ ′ = x/|x|, see for example, [Ser17,p 232].The inverse scattering problem is to determine the medium perturbation 1 − n from the knowledge of the scattering amplitude u ∞ n (θ ′ , θ) for all θ ′ , θ ∈ S 2 at one fixed energy κ 2 .
It was known that the scattering amplitude u ∞ n (θ ′ , θ) uniquely determines the near-field data of (1.1a) on ∂D, which in turn, determines the Dirichlet-to-Neumann map of (1.1a) provided κ 2 is not an Dirichlet eigenvalue of −∆ on D, for example, see [Nac88].Combining this fact and the uniqueness results proved in [SU87], one can show that the scattering amplitude u ∞ n (θ ′ , θ) for all θ ′ , θ ∈ S 2 uniquely determines n, at least when n is essentially bounded.For the stability, a log-type estimate was derived in [HH01].
In this paper, we would like to apply the Bayes approach to the inverse scattering problem.The study of this inverse method is motivated by the practical consideration.In practice, it is impossible to obtain the full knowledge of the scattering amplitude.Instead, the following experiment is more realistic.We first randomly choose an incident direction and send out the corresponding incident field towards the probe region.Then we measure the scattering amplitude at another randomly chosen direction.The experiment can be repeated as many times as we wish.The goal now is to make an inference of the refractive index based on such observation data.To describe the method more precisely, we first introduce the measurement model.Let µ be the uniform distribution on S 2 × S 2 , i.e. µ = dω/|S 2 | 2 , where dω is the product measure on S 2 × S 2 , that is, ´S2 ×S 2 dω = |S 2 | 2 .We also write µ = dξ and hence ´S2 ×S 2 dξ = 1.Consider the iid random variables X i ∼ µ, i = 1, 2, . . ., N with N ∈ N. In other words, is a sequence of independent samples of µ on S 2 × S 2 .Denote where (θ ′ , θ) is a realization of X i .The observation of the scattering amplitude G(n)(X i ) is polluted by the measurement noise which is assumed to be a Gaussian random variable.Since G(n)(X i ) is a complex-valued function, we treat it as a R 2 -valued function.For convenience, we slightly abuse the notation by writing Consequently, the statistical model of the scattering problem is given as where σ > 0 is the noise level, I 2 is the 2 × 2 unit matrix.We also assume that W (N) and X (N) := {X i } N i =1 are independent.The theme of this paper is to consider the inference of n from the observational data (Y (N) , X (N) ) with Y (N) = {Y i } N i =1 by the Bayes method.In particular, we are interested in the asymptotic behavior of the posterior distribution induced from a large class of Gaussian process priors on n as N → ∞.The aim is to establish the statistical consistency theory of recovering n in (1.1c) with an explicit convergence rate as the number of measurements N increases, i.e. the contraction rate of the posterior distribution to the 'ground truth' n 0 when the observation data is indeed generated by n 0 .Gaussian process priors are often used in applications in which efficient numerical simulations can be carried out based on modern MCMC algorithms such as the pCN (preconditioned Crank-Nicholson) method [CRSW13].
The study of inverse problems in the Bayesian inversion framework has recently attracted much attention since Stuart's seminal article [Stu10] (also see [DS17]).The setting of the problem considered in this paper is closely related to the ones studied in [GN20,Kek22].In [GN20], Gaussian process priors were used in the Bayesian approach to study the recovery of the diffusion coefficient in the elliptic equation by measuring the solution at randomly chosen interior points with uniform distribution.It was shown that the posterior distribution concentrates around the true parameter at a rate N −λ for some λ > 0 as N → ∞, where N is the number of measurements (or sample size).Previously, (frequentist) consistency of Bayesian inversion in the elliptic PDE considered in [GN20] was derived in [Vol13].However, the contraction rates obtained in [Vol13] were only implicitly given.Based on the method in [GN20], similar results were proved in [Kek22] for the parabolic equation where the aim is to recover the absorption coefficient by the interior measurements of the solution.For further results on the Bayesian inverse problems in the non-linear settings, we refer the reader to other interesting papers [Abr19, NS17, NS19, MNP21, Nic20, NP21, NW20].On the other hand, for linear inverse problems, the statistical guarantees of nonparametric Bayesian methods with Gaussian priors have been extensively studied and well understood, see for example [ALS13, KLS16, KvdVvZ11, MNP19, Ray13] and references therein.
The ideas used in proving the consistency of the Bayesian inversion for the inverse scattering problem studied here are similar to those used in [GN20,Kek22] in which main ideas are from [MNP21].Unlike the polynomial contraction rates derived in [GN20,Kek22], the posterior distribution Π(•|Y (N) , X (N) ) of n|(Y (N) , X (N) ) contracts at the true refractive index n 0 as N → ∞ with a logarithmic rate.The logarithmic rate is due to the ill-posedness of the inverse scattering medium problem by the knowledge of the scattering amplitude at one fixed energy, see [HH01].
This paper is organized as follows.In section 2, we describe the statistical model arising from the scattering problem.We state main consistency theorems with contraction rates assuming re-scaled Gaussian processes priors and re-scaled Gaussian sieve priors.The proofs of theorems are given in section 3.In appendix A, we derive some estimates for the forward scattering problem, and in appendix B, we prove the optimality of the logarithmic stability estimate in the inverse medium scattering problem based on Mandache's idea.Similar instability estimate was also derived in [Isa13].To make the paper self contained, we present our own proof and slightly refine the estimate obtained in [Isa13].

Some function spaces and notations
Throughout this paper, we shall use the symbol ≲ and ≳ for inequalities holding up to a universal constant.For two real sequences (a N ) and (b N ), we say that ≃ if both a N ≲ b N and b N ≲ a N for all sufficiently large N.For a sequence of random variables Z N and a real sequence (a N ), we write Let D be a bounded smooth domain in R 3 , let s ⩾ 0 be an integer and we consider the Hilbert space For non-integer s, H s (D) is defined in terms of the interpolation, see [LM72].It is known that the restriction operator to D is a continuous linear map of We now define the space of parameters.For M 0 > 1 and s > 3 2 , let where the traces are defined in the sense of [LM72,theorem 8.3].For each n ∈ F s M0 we extend n ≡ 1 in R 3 \ D, still denoted by n.Then it is clear that supp(1 − n) ⊂ D. Note that for n ∈ F s M0 , we only put the restriction on the size of n, but not on the H s (D)-norm of n.As in [NvdGW20, GN20, AN19, Kek22], we will consider a re-parametrization of F s M0 .We consider the link function Φ satisfying One example to satisfy (i) and (ii) is the logistic function As pointed out in [NvdGW20, section 3], by utilizing a characterization of the space H s 0 (D) (see e.g.[LM72, theorem 11.5]), one can show that the parameter space can be realized as We end this subsection by emphasizing that our link function is different to those in [NvdGW20, GN20, AN19, Kek22], see (i).

An abstract statistical model
For each forward map G : F s M0 → L 2 (S 2 × S 2 ), we define the reparametrized forward map by and consider the following random design regression model Assume that G satisfies and for each for some constant S 2 > 0 and t ⩾ 1.
Remark 2.1.We implicitly assume that taking pointwise value of G(F) on S 2 × S 2 is legitimate.For instance, in the inverse scattering problem here, G(F) (corresponding to the scattering amplitude u ∞ n ) is, in fact, analytic on S 2 × S 2 .The statistical model (2.2b) with conditions (2.2c) and (2.2d) falls into the general framework described in [GN20].We want to remark that the uniform boundedness of the forward map G (condition (2.2c)) in [GN20,NvdGW20] (elliptic boundary value problem) or in [Kek22] (parabolic initial-boundary value problem) is ensured by the positivity assumption of the coefficient and the bound S 1 is determined by the fixed boundary value or the fixed initial and boundary values.Due to these facts, the ranges of the link functions used in [GN20,NvdGW20] and [Kek22] do not required to have finite upper bounds.In the scattering problem, the boundedness requirement of the forward map (2.2c) cannot be guaranteed by the sign restriction of the potential.In this case, we choose a link with finite range (like Φ given above) to ensure (2.2c).
Remark 2.2.In view of (1.2) and (2.1), one notes that the statistical model (1.3) fits into the framework of (2.2b).For the inverse scattering problem studied here, if the forward map G(n) is defined by the far-field pattern (1.2) with refractive index n ∈ F s M0 with s > 3 2 , from (A.9) we see that (2.2c) satisfies with S 1 = S 1 (D, κ, M 0 ).From (A.14) we have where C = C(D, k, M 0 ).By using (ii) and [NvdGW20, lemma 29], we have Observe that the random vectors (Y i , X i ) are iid with laws P i F .It turns out the Radon-Nikodym derivative of P i F is given by . (2.3) By slightly abusing the notation, now we define Moreover, E F i , E F N denote the expectation operators in terms of the laws P F i , P F N , respectively.In the Bayesian approach, let Π be a Borel probability measure on the parameter space H s 0 (D) supported in the Banach space C(D).From the continuity property of (F, (y, ξ) where the log-likelihood function is written as Finally, we end this subsection by referring to the monograph [Nic23] for a nice introduction on the above preliminaries.

Statistical convergence rates
In this work we would like to show that the posterior distribution arising from certain priors concentrates near sufficiently regular ground truth Φ(F 0 ), and derive a bound on the rate of contraction, assuming that the observation data (Y (N) , X (N) ) are generated through the model (2.2a)-(2.2d) of law P N F0 .
Assumption 2.3.Let s > t + 3/2, t ⩾ 1, and H be a Hilbert space continuously embedded into H s 0 (D), and let Π ′ be a centered Gaussian Borel probability measure on the Banach space C 0 c (D) that is supported on a separable measurable linear subspace of C t c (D). Assume that the reproducing kernel Hilbert space (RKHS) of Π ′ equals to H.
Here, we refer to [GN21, definition 2.6.4] for the definition of the RKHS, and we refer to [GN20, example 25] for an example satisfies assumption 2.3.For each s given in assumption 2.3, let Π ′ be given in assumption 2.3 and F ′ ∼ Π ′ , we consider the rescaled prior (2.5) Again, Π N defines a centered Gaussian prior on C(D) and its RKHS H N is still H but with the norm for all F ∈ H.We now introduce the main device in our proof, concerning the posterior contraction with an explicit rate around F 0 , whose proof can be found in [GN20, theorem 14].
Theorem 2.4.Let (H, Π ′ ) satisfies assumption 2.3 with integer s, let Π N ≡ Π N [s] be the rescaled prior given in (2.5), let Π N (•|Y (N) , X (N) ) be the posterior distribution given in (2.4a) with Π = Π N .Assume that F 0 ∈ H and the observation (Y (N) , X (N) ) to be generated through model (2.2b)-(2.2d) of law P N F0 .If we denote δ N = N −(s+1)/(2s+5) , then for any K > 0, there exists a large L > 0, depending on σ, F 0 , K, s, t, D, S 1 , S 2 , such that (2.7a) In addition, there exists a large L ′ > 0, depending on σ, K, s, t, such that (2.7b) We now apply theorem 2.4 to the inverse scattering problem considered here.Relying on the contraction rate (2.7a) and the regularization property (2.7b) and taking account of the stability estimate of G −1 (see theorem A.1), we can show that the posterior distribution arising from the statistical inverse scattering model (1.3) contracts around n 0 in the L ∞ -risk using ideas from [MNP21].In light of the link function, we define the push-forward posterior on the refractive index n by By the push-forward map, we can rewrite (2.7a) and (2.7b) in terms of n.That is, we have that for N → ∞ that (2.8a) and where the second estimate can be derived from the estimates proved in [NvdGW20, lemma 29] (see also [GN20,(27)]).Here L depends on σ, n 0 , K, s, t, D, κ, M 0 and L ′ depends on σ, K, s, t, κ, M 0 .
Theorem 2.5.Let t ⩾ 2 and s > t + 3/2 be integers, and fix a real parameter M 0 > 1.We further assume that ε is any constant satisfying 0 It is clear that we can replace ∥n − n 0 ∥ L ∞ (D) by ∥n − n 0 ∥ L 2 (D) in (2.9).Unlike the polynomial rate proved in [GN20, theorem 5], we obtain a logarithmic contraction rate in (2.9), which is due to a log-type stability estimate of G −1 .To obtain an estimator of the unknown coefficient n, in view of the link function Φ, it is often convenient to derive an estimator of F. The posterior mean FN := E Π [F|Y (N) , X (N) ] of Π N (•|Y (N) , X (N) ), which can be approximated numerically by an MCMC algorithm, is the most natural choice of estimator.In light of theorem 2.5, we can also prove a contraction rate for the convergence FN to F 0 .
Theorem 2.6.Assume that the hypotheses of theorem 2.5 hold.Then, there exists a C = C(σ, n 0 , K, s, t, D, κ, ϵ, M 0 ) > 0 such that (2.10) Corollary 2.7.Assume that the hypotheses of theorem 2.5 hold.Then there exists a sufficiently large C′ = C′ (σ, n 0 , K, s, t, D, κ, ϵ, M 0 ) > 0 such that (2.11) The logarithmic contraction rates obtained in theorems 2.5, 2.6, and corollary 2.7 inherit from the log-type estimate of the inverse scattering problem.Nonetheless, in the next theorem, we will show that this contraction rate for the estimator n := Φ • FN is optimal is the statistical minimax sense, at least up to the exponent of ln N. We first define a parameter space.Let s > 3/2 be an integer, β > 0, and define Theorem 2.8.For integer s > 3/2, there exists β = β(s) > 0 such that for any δ > 5s 3 and ε ∈ (0, 1), we have for all N large enough, where the infimum is taken over all measurable functions ñ = ñ(Y, X) of the data (Y, X) ∼ P N n .
2.3.2.High-dimensional Gaussian sieve priors.From computational perspective, it is useful to consider sieve priors that are finite-dimensional approximations of the function space supporting the prior.Here we will use a randomly truncated Karhunen-Loéve type expansion in terms of Daubechies wavelets considered in [GN20, appendix B] or [GN21, chapter 4].Let Ψ ℓr : ℓ ⩾ −1, r ∈ Z 3 be the (3-dimensional) compactly supported Daubechies wavelets, which forms an orthonormal basis of L 2 (R 3 ).Let K be a compact subset in D and let

and consider the prior
where J ∈ N is the truncation level.In fact Π ′ J defines a centered Gaussian prior that is supported on the finite-dimensional space with RKHS norm satisfying [GN20, (B2)].As above, we consider the 're-scaled' prior Π N defined in (2.5) with F ′ ∼ Π ′ J .In analogy to theorem 2.4, we will derive a contraction rate for the statistical model (2.2b) with the prior Π N defined above.As in theorem 2.4, we obtain the same contraction rate in the L 2 -prediction risk of the regression function.
Theorem 2.9.Let t ⩾ 1, s > t + 3/2 be integers, and ) the posterior distribution arising from the noisy discrete measurements (Y (N) , X (N) ) of (2.2b).Let F 0 ∈ H s K (D) and δ N = N −(s+1)/(2s+5) .Then for any K > 0, there exists a large L > 0, depending on σ, F 0 , K, s, D, S 1 , S 2 , such that and for sufficiently large L ′ > 0, depending on σ, K, s, t, we have (2.15) The proof of theorem 2.9 only requires minor modification from the proof of theorem 2.4, and all necessary modifications are listed in [GN20, section 3.2].Therefore we omit the details.Having established theorem 2.9, similar to [GN20, proposition 7], theorems 2.5 and 2.6 can be directly extended to the case of Gaussian sieve priors.
Remark 2.10.As in [GN20], it is likely to extend the results for the Gaussian sieve priors with deterministic truncation level to randomly truncated ones, where the truncation level J itself is an appropriate random number.However, due to the log-type stability estimate of the inverse scattering problem, such extension is highly nontrivial.We will discuss the Bayes method with randomly truncated sieve priors for the inverse scattering problem in another paper.

Proofs of theorems
The main theme of this section is to prove theorem 2.5, 2.6 and 2.8.
Proof of theorem 2.5.For each M > 0 satisfies ∥1 − n∥ H t (D) ∨ ∥1 − n 0 ∥ H t (D) ⩽ M, one has the stability estimate of G −1 in theorem A.1: where . In view of (2.8a), (2.8b), and (3.1), for any K > 0, there exists large constants L, M ′ and L ′ (L ′ is determined by L and M ′ ) such that which conclude our result.
Having proved theorem 2.5, we then establish theorem 2.6 using the contraction rate in theorem 2.5 and the link function Φ.
Proof of theorem 2.6.We proof the theorem by modifying some ideas in [GN20, theorem 6].By the Jensen's inequality, it suffices to prove that there exists a large C > 0 such that For a large M > 0 to be chosen later, we write (3.2) ].By using Cauchy-Schwartz inequality, it is easy to see that (3.3) Let B > 0 be a constant to be determined later.By using (2.7b), one can choose a sufficiently large M = M(σ, B, s, t) > 0 such that We now define B N by By using [GN20, lemmas 16 and 23], we have ) and set the event By [GN21, lemma 7.3.2],we can show that By using the properties (3.6) of C N , we obtain which is bounded from above, using Markov's inequality and Fubini's theorem, by (3.7) By using Fernique's theorem (see e.g.[GN21, exercises 2.1.1,2.1.2and 2.1.5])one has The above discussions still valid if we replace M by a (possibly larger) M = M(σ, B, s, t, F 0 ) > 0 with ∥F 0 ∥ L ∞ ⩽ M. Since n = Φ • F and n 0 = Φ • F 0 , by (i), mean value theorem and inverse function theorems, there exists η lying between n 0 (x) and n(x) such that Therefore we see that and Let C > 0 be a constant to be determined later.From (3.9), we see that ⩽ C (ln N) We can modify the arguments as in Part I above by replacing the event {F : (resp.(2.9)) with C = C(σ, n 0 , K, s, t, D, κ, ϵ, M 0 ) > 0 given in theorem 2.5 to show that Finally, putting together (3.2), (3.8), and (3.10) yields theorem 2.6 with C = max{C, 1}.
Next, we prove the optimality of contraction rate in corollary 2.7 in the minimax sense.
Proof of theorem 2.8.We apply the method in the proof of the lower bound [AN19, theorem 2] to our case here.The idea is to find n 0 , n 1 ∈ Fs β (both are allowed to depend on N) such that, for some small ζ sufficiently small, (a) and [GN20, lemma 23] implies (3.12) then using the standard arguments as in [GN21, section 6.3.1](see also [Tsy09, chapter 2]), we conclude the theorem.
For the sake of completeness, here we present the details.From condition (a), we see that It follows from a general reduction principle that where the second infimum is over all tests ψ of (3.13).Similar to the proof of [GN21, theorem 6.3.2],we introduce the event Note that Let p 1 = P N n1 (ψ = 1), then It is clear that the infimum above is attained when 1 2 (p − P N n1 (Ω c )) = 1 − p and has the value Next, let us estimate Using the second Pinsker inequality [GN21, proposition 6.1.7b]and condition (b), we have We now choose ζ sufficiently small such that the last term above is bounded below by 1 − ε.
To verify condition (b), we use (3.12) to conclude that → 0 as N → ∞, since 3δ 5s > 1.Therefore, we can make ζ as small as we wish by taking N sufficiently large.
In other words, a 1 (•, •) is strictly coercive.Combining the Riesz representation theorem and the Lax-Milgram theorem, there exists an invertible operator A : Similarly, define the bounded linear operator B : * is bounded invertible provided the kernel of A + B is trivial, which follows the uniqueness of the scattered solution (by the combination of the Rellich lemma and the unique continuation property).Furthermore, we have the following estimate: where By the interior estimate [GT01, theorem 8.8], we further have which, by the Sobolev imbedding theorem, implies The scattering amplitude u ∞ n (θ ′ , θ) can be expressed explicitly by where u(y, θ) = u inc (y, θ) + u sca n (y, θ) is the total field with u inc (y, θ) = e iκy•θ , see [CCH23,(1.22)]or [CK19,(8.28)]or [Ser17, p 232].From (A.7) and (A.8), we have , applying the interior estimate and the Sobolev imbedding theorem again, we have that Next, assume that n 1 , n 2 satisfy (A.1).For each open set Ω in Euclidean space, we observe that then combining (A.5), (A.7), (A.10) and (A.11), yields uniformly in θ and C = C(D, k, M 0 ).Then it yields from (A.8) that where the inequality above is due to the integral form of Minkowski's inequality.Plugging (A.6), (A.10) and (A.12) into (A.13)gives where C = C(D, k, M 0 ).Next, we recall the following stability estimate for the determination of the potential from the scattering amplitude.

Appendix B. Optimality of the stability estimate
The purpose of this section is to show that the logarithmic estimate obtained in theorem A.1 is optimal by deriving an instability estimate.Similar instability estimate was already proved in [Isa13].To make the paper self contained, we present our own proof here and also slightly refine the estimate obtained in [Isa13].Throughout this section, we denote q(x) = n(x) − 1.Let integer s > 0 be a given regularizing parameter.Then there exists constants β = β(s, κ) > 0 and ϑ 0 = ϑ 0 (s, κ) > 0 such that: for each 0 < ϑ < ϑ 0 there exists non-negative q 1 , q 2 ∈ C ∞ (R 3 ) with supp (q j ) ⊂ B 1 2 satisfying the apriori bound ∥q j ∥ C s (R 3 ) ⩽ β and From the properties of the Hilbert-Schmidt norm [Con90, exercise IX.2.19(h)], one has Recall G(1 + q) is the far-field operator given in (1.2).
Our main strategy is to modify the ideas in [DCR03] (see also [KRS21] for more details about the mechanism).Given any ϑ > 0, s ⩾ 0 and β > 0, we consider the following set: , where B r denotes the ball or radius r centered at the origin.The following lemma verifies the assumption (a) of [DCR03, theorem 2.2], can be proved as in [Man01, lemma 2] (we omit the proof), see also [Isa13,KUW21,KW22,ZZ19].We refer [KT61] for a version in a more abstract form.
Lemma B.3.Fix d ∈ N and s ⩾ 0. There exists a constant6 µ = µ(d, s) > 0 such that the following statement holds for all β > 0 and for all ϑ ∈ (0, µβ): there exists a ϑ-discrete subset In addition, all elements in Z ϑ are in C ∞ (R d ).
Similar to [DCR03], the proof of theorem B.1 is quite delicate, which is not an obvious consequence of the abstract theorem in [DCR03, theorem 2.2].From the asymptotic expansion (1.1c), it is easy to see that |x||u sca 1+q (x, θ)| = O (1) as |x| → ∞ uniformly for all θ ′ = x/|x| ∈ S 2 .A crucial point is to bound |x||u sca 1+q (x, θ)| for all |x| ⩾ 2, by some constant which is independent of θ and q, like [DCR03, (4.21)].From now on, for simplicity, we restrict ourselves for the case when d = 3.We now prove the following lemma.
As in [HH01], we introduce the following index set . Accordingly, we write Lemma B.5.Let κ > 0 and let ∥q∥ L ∞ (R 3 ) ⩽ 1 with supp (q) ⊂ B 1/2 .Then there exist constants c abs = c abs (κ) > 0 and C abs = C abs (κ) > e such that |G (1 + q) jk mn | ⩽ C abs e −c abs max{m,n} for all (m, j, n, k) ∈ M. Proof.By [CK19, theorem 2.15], we can write where h (1) m is the spherical Hankel function of the first kind or order m.In fact, which converges uniformly in θ ′ ∈ S 2 , and hence uniformly in θ ′ ∈ S 2 .We now choose κr = 2 and reach From [KW22, (48)], we see that In view of the quantitative Stirling's formula [Rob55], we obtain Combining (B.4) and (B.6) implies Finally, by the first reciprocity relation [CK19, theorem 8.8], we have | for all (m, j, n, k) ∈ M, and the lemma is proved.
We also need a technical lemma.
Lemma B.6.Consider the normed space where we can estimate which implies the lemma.
We now construct a δ-net by following the procedures in [DCR03, lemma 2.3].
the law of a random variable Z.Let C t c (O) with t ⩾ 0 denote the Hölder space of order t with compact supports in the bounded smooth domain O.