# Uniqueness of an inverse source problem in experimental aeroacoustics

This paper is concerned with the mathematical analysis of experimental methods for the estimation of the power of an uncorrelated, extended aeroacoustic source from measurements of correlations of pressure fluctuations. We formulate a continuous, infinite dimensional model describing these experimental techniques based on the convected Helmholtz equation in R^3 or R^2. As a main result we prove that an unknown, compactly supported source power function is uniquely determined by idealized, noise-free correlation measurements. Our framework further allows for a precise characterization of state-of-the-art source reconstruction methods and their interrelations.

## Authors

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## 1 Introduction

In experimental aeroacoustics one measures acoustic, randomly generated signals and aims at reconstructing the power of sources. In this paper we consider time-harmonic sound propagation in homogeneous flow fields which may be considered as simplified models of wind tunnel experiments. In that case acoustic pressure fluctuations may be caused by fluid-structure interactions or local turbulent structures inside the flow field which are then propagated towards a measurement array in the homogeneous main flow.

The experimental investigation of aeroacoustic sound sources began in the 1970-s. Back then, the standard measurement device was an elliptic mirror [21]. However, the application of microphone arrays soon found its way into the field. Active and passive microphone array methods are widely applied for the localization of sources of wave fields or the imaging of the propagation medium [34, 13]. The field of applications covers many branches of physics and engineering for example radar (see, e.g., [22]) or geophysics (see, e.g., [7]). The first fundamental work on microphone array imaging methods in aeroacoustics, published in 1976 by Billingsley & Kinns [5], deals with aeroacoustic sound sources of a turbulent jet. Since then, the data processing and evaluation techniques for microphone array data were constantly developed further and microphone arrays are nowadays the standard experimental measurement devices for aeroacoustic experiments. Some of the most common applications of microphone arrays for (aero-)acoustic purposes are aircraft measurements (fly-over [24] or inside a wind tunnel [43]), jet noise [41], wheel/rail noise of trains [4] and wind turbines [37]. There exists also a close connection to helioseismic holography, which back-propagates correlations of acoustic waves observed on the Sun’s near-side to its interior or far-side to study the structure and dynamics therein (see Lindsey & Brown [30, 31], Gizon et al. [14], and Section 4.3). This article considers only free field sound propagation since many microphone array methods rely on this assumption. Sound propagation models based on numerical simulation allow more complex geometrical setups. A source localization method using Finite Element simulation on two- and three-dimensional geometries is presented by Kaltenbacher, Kaltenbacher & Gombots in [26, 15].

In wind tunnel experiments and many other applications discussed above it is common practice to compute the correlation matrix of the measurement array in a preprocessing step and reconstruct source powers from these correlation data. The main aim of this work is to establish a uniqueness result for the inverse problem of reconstructing bounded and compactly supported source power functions from correlation data in a continuous setting.

Let us distinguish our problem from two related inverse source problems with rather different properties. We are concerned with spatially extended sources, in contrast to the acoustic localization of a small number of point sources. Such source localization problems occur for example in speaker localization or speech enhancement for hearing aid devices (see, e.g., [2, 11, 12, 32, 42, 46]). Methods for this scenario seek to localize a small number of sources inside a reverberant room. Often, the direction of arrival (DOA) of the acoustic sources is of special interest, since it allows to suppress signals from other directions (speech enhancement). One of the main difficulties for this class of problems is that the source signal is superposed with disturbance signals due to wall reflections.

Moreover, whereas we consider the identification of fully uncorrelated sources, for the identification of deterministic (i.e. completely correlated) sources it is well known and easy to see that such sources are not uniquely determined from distant measurements of acoustic waves since there exist so-called non-radiating sources. Nevertheless, significant progress has been achieved recently by Griesmaier & Sylvester in retrieving partial information from such sources, e.g. stably splitting well-separated sources, see [18, 19, 20]. Note that for deterministic sources, data consist of deterministic wave fields, whereas for random sources one may consider correlation data. The fact that correlation functions depend on more independent variables than the corresponding wave fields is a first formal indication that uniqueness results for uncorrelated sources may be possible.

Since we consider spatially extended sources, a continuous source representation is natural for our scenario. Furthermore, in wind tunnel experiments sources may be considered as non-deterministic since the sound field is generated and measured inside a flow field with turbulent structures. As already mentioned above, one often uses the correlations between the microphone signals as input for the reconstruction process. One of the basic reconstruction techniques, based on microphone correlation measurements are Beamforming methods [36]. To improve the spatial resolution of Beamforming outputs, post processing methods like DAMAS [9] and Clean-SC [40] have been proposed. Covariance Matrix Fitting [6, 47] is an inverse method that reconstructs source powers directly from the measured correlation matrix. We will review these methods from a continuous perspective given by the setting of our uniqueness result.

The plan of the remainder of this paper is as follows: The forward problem for time harmonic sound propagation of uncorrelated sources is introduced in Section 2 before presenting our main uniqueness result in Section 3. In Section 4 we generalize commonly used discrete reconstruction methods to our continuous framework, study their interrelations and compare their performance for an experimental data example. Finally, we end this paper with some conclusions.

## 2 The forward problem

• a bounded, open domain with (source region)
such that is connected and

• a bounded, open domain (measurement region).

The sound propagation model inside a homogeneous flow is given by the convected Helmholtz equation for a subsonic, constant flow field . Let denote the speed of sound and

the Mach vector. We consider a subsonic regime, i.e. we assume that

. For the sign convection for the time factor, the convected Helmholtz equation for a function and a source term reads as

 (k+im⋅∇)2p+Δp=−Q . (1)

The free field Green’s function for equation (1) in three dimensions is (cf. [35, Appendix A])

 (2)

with the Mach scaled distance

 |x−y|m=√((x−y)⋅m)2+β2|x−y|2 (3)

and . Note that (3) is a norm on which is induced by a scalar product. Recall that the free Green’s function for in two dimensions is

 g(x,y)=i4H(1)0(k|x−y|)

where denotes the Hankel function of the first kind of order (see [10, p.74]). Straightforward computations show that the free Green’s function for general with is given by

 g(x,y)=i4βexp(−ikβ2(x−y)⋅m)H(1)0(kβ2|x−y|m) . (4)

In a real experimental setup and evaluation process, the data and the reconstructed quantities are finite. We consider microphones in the measurement region at positions . We also discretize the source region by focus points and corresponding disjoint sets such that and . E.g., the sets may be chosen as Voronoi cells. The random source term is now approximated by a sum of scaled approximate delta distributions with random complex amplitudes , e.g. a piecewise constant function

 Q––(N)(y)=N∑n=1Π(N)nϕNn(y) , (5)

where is an -normalized indicator function of . A standard assumption is that the complex amplitudes have zero mean and are mutually uncorrelated:

###### Assumption 2.1 (Uncorrelated sources).

The random source amplitude vector satisfies

 E(Π(N))=0and% Cov(Π(N))=M(N)q––––––

with the source powers for some continuous source power function and .

#### Notation:

Mathematical objects that posses a discrete and an infinite dimensional version will be indicated by the same symbol for the entire article. For distinction the discrete versions will be marked by an underscore.

At any point , the source term generates the pressure signal

 p–(N)(x):=∫Ωg(x,y)Q––(N)(y)dy . (6)

Note that under Assumption 2.1. For the microphone signal vector equation (6) yields

 p(M,N)=G––(M,N)Π(N) ,

where the propagation matrix is defined by

 G––(M,N)mn=∫ΩNng(xm,y)dy|ΩNn|−1/2≈g(xm,y(N)n)|ΩNn|1/2 . (7)

In aeroacoustic array measurements, one estimates the covariance matrix (also cross correlation matrix or cross-spectral matrix) of the microphone signal vector , where the estimation process is usually carried out by Welch’s method [45]. Assumption 2.1 yields the following representation for the estimated cross correlation matrix

 C––obs≈Cov(p(M,N))=Cov(G––(M,N)Π(N))=G––(M,N)Mq––––(N)G––(M,N)∗=:C––(M,N)(q–) . (8)

If we let tend to infinity such that the fill distance in tends to , then under mild assumptions on the choice of (e.g. for Voronoi cells) and the distribution of

(e.g. for Gaussian distributions) the random functions

tend to a Gaussian random process on with covariance operator

 (Mqv)(y):=q(y)v(y)

in the sense that

converges in probability to

for any and . (For , the process

is Gaussian white noise.) Moreover, the limiting pressure signal

can be written in terms of the volume potential operator

 (Gv)(x):=∫Ωg(x,y)v(y)dy. (9)

Note that and that has the covariance function

 cq(x1,x2):=Cov(p(x1),p(x2)) =∫Ωg(x1,y)q(y)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯g(x2,y)dy ,

which under the assumptions above is the limit of as . This implies that the integral operator ,

 (C(q)φ)(x1):=∫Mcq(x1,x2)φ(x2)dx2

(the covariance operator of the acoustic pressure signal ) can be written as

 C(q)=GMqG∗ . (10)

We will consider the forward operator as an operator-valued linear mapping from to the space of Hilbert-Schmidt operators on

 C: L2(Ω)→HS(L2(M)),    q↦C(q) .

Recall that a compact operator in a Hilbert space

is called a Hilbert-Schmidt operator if the eigenvalues of

are summable. Equipped with the inner product the space is a Hilbert space [38, Theorem VI.22 (c)]. As for integral operators, the Hilbert-Schmidt norm is given by the -norm of the kernel function (see [38, Theorem VI.23]), we have

 ∥C(q)∥HS=∥cq∥L2(M×M). (11)

As is the true input data of the inverse problem, this shows that is a natural choice for the image space.

###### Proposition 2.2 (Mapping property of the forward operator).

For any the operator belongs to .

See A. ∎

###### Remark 2.3 (Non-uniqueness for correlated sources).

One may ask if it is also possible to identify a general non-diagonal source covariance operator describing correlated sources from corresponding correlation data

 C(S)=GSG∗. (12)

It turns out that this is not possible, at least not in this generality: As already mentioned in the introduction, the operator has a non-trivial kernel (i.e. deterministic sources in are not uniquely determined from their generated wave fields in ). As covariance operators are self-adjoint and positive semi-definite, we can decompose into with . But there exist positive semi-definite operators such that . For such operators we have , i.e. the forward operator in (12) is not injective.

## 3 Uniqueness result for the inverse problem

The main goal of this section is to prove a uniqueness result for the operator equation

 C(q)=C (13)

for and a bounded source power function . The proof relies on several auxiliary statements which will be presented beforehand. Some less interesting and more technical proofs are shifted to an appendix.

For the convected Helmholtz equation propagation directions will be elements of the unit sphere with respect to the Mach-norm given by

 Sd−1(m):={x∈Rd: |x|m=1} .

The next proposition describes the asymptotic behavior of the Green’s functions (2)/(4) at infinity.

###### Proposition 3.1 (Asymptotic behavior of Green’s function).

Let be a ball, with radius such that , then for and the following asymptotic representation holds true:

 g(x,y)=C(d)h(x)|x|−(d−1)2mexp(ikβ2(m−A^x)⋅y)+O(|x|−(d+1)2)as |x|→∞ (14)

with the auxiliary quantities

 C(2):=eiπ4√8πk, h(x):=exp(ikβ2(|x|m−x⋅m)), C(3):=14π, A:=mm⊤+β2I, ^x:=x|x|m.

The asymptotic formula (14) holds uniformly for all and all directions .

###### Proof.

See A. ∎

For the volume potential of a function

 w(x)=∫Ωv(y)g(x,y)dy , (15)

we immediately get a similar representation with the far field pattern

 w∞(^x)=∫Ωexp(ikβ2(m−A^x)⋅y)v(y)dy . (16)
###### Corollary 3.2 (Asymptotic representation of the volume potential).

With defined in (16) the volume potential in (15) has the asymptotic behavior

 w(x)=C(d)h(x)|x|−(d−1)2mw∞(^x)+O(|x|−(d+1)2)
###### Proof.

Inserting the asymptotic representation (14) from the last proposition into the definition of the volume potential yields the claim. ∎

###### Remark 3.3 (No-flow case).

The statements of Proposition 3.1 and Corollary 3.2 for the case of reduce to the well-known formulas

 g(x,y) = C(d)eik|x||x|−(d−1)2e−ik^x⋅y+O(|x|−(d+1)2), (17) w(x) = (18) w∞(^x) = ∫Ωe−ik^x⋅yv(y)dy

(see [10] for (17) and [16, 17] for (18)).

In order to characterize elements of the kernel of the volume potential operator in (9), we show that the volume potential in (15) is real analytic outside of the source region.

###### Proposition 3.4 (Analyticity of the volume potential).

The volume potential (15) is real analytic on .

###### Proof.

See A

Now we can identify a set of specific plane waves as a subset of the closure of the range of the adjoint volume potential operator .

###### Proposition 3.5 (Plane waves).

For the plane wave functions

 um,^x(y):=exp(ikβ2(A^x−m)⋅y) ,

the following inclusion holds true

 W:={um,^x :  ^x∈Sd−1(m)}⊂¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ran(G∗) .
###### Proof.

We will show that which is equivalent to the claim. Assume that , i.e. the volume potential vanishes on . Due to Proposition 3.4, is analytic and since it vanishes on the open set and is connected, it must vanish on all of . The representation formula of Corollary 3.2,

 w(x)=C(d)h(x)|x|−(d−1)2mw∞(^x)+O(|x|−(d+1)2)

implies that the far field pattern must also vanish identically. Together with the definition of the far field pattern we obtain

 0=w∞(^x)=∫Ω¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯um,^x(y)v(y)dy=⟨v,um,^x⟩L2(Ω),

which shows that . ∎

Finally, we are in the position to prove the uniqueness statement for bounded sources.

###### Theorem 3.6 (Uniqueness).

If such that , then

 q1=q2 .
###### Proof.

Due to linearity is suffices to show that implies . So let such that . Then

for all and hence

 ⟨qu1,u2⟩L2(Ω)=0    for all % u1,u2 ∈ ran(G∗) . (19)

Since , we can apply a density argument to show that property (19) holds also for elements of . By Proposition 3.5 we can choose for and plane waves of the form with . Together with (19) this implies

 (20)

Note that

 {^x2−^x1 : ^x1,^x2∈Sd−1(m)}={x∈Rd: |x|m≤2}

contains an open set with respect to and the set is also open as is a homeomorphism on . Extending by zero to the whole space (denoting the extension again by ) and using (20) we obtain

 ^q(ξ)=∫Rdq(y)e−iξ⋅ydy=0     for ξ∈V , (21)

i.e. the Fourier transform of

vanishes on the open set . Since has compact support, is real analytic [39, Section IX.3], and hence it must vanish everywhere. Since the Fourier transform is injective, we obtain that . ∎

## 4 A continuous perspective on common reconstruction methods

In this section we will analyze three common source reconstruction methods that are used for aeroacoustic measurement data, namely Covariance Matrix Fitting (CMF) (also known as spectral estimation method) [6, 47], Conventional Beamforming (CBF) [44, 25] and DAMAS [9]. All three methods can be generalized to the infinite dimensional framework, presented in the last two sections. For a broader overview on microphone array techniques for aeroacoustic purposes we refer to [29, 33].

### 4.1 The adjoint of the forward operator

Before we start with the specific source reconstruction approaches, we need to characterize the adjoint of the forward operator.

###### Proposition 4.1 (Adjoint forward operator).

The adjoint of the forward operator

 C:L2(Ω)→HS(L2(M)),      q↦C(q)

is given by

 C∗:HS(L2(M))→L2(Ω),      (C∗K)(y)=⟨K,Py⟩HS,

for and with the monopole operator defined by

 (Pyφ)(x1)=∫Mg(x1,y)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯g(x2,y)φ(x2)dx2

for and .

###### Proof.

See A

Note that if is given by its integral kernel , i.e. , then in view of the isometry (11) we have

 (22)

Recall the elements of the discrete measurement setup, presented in Section 2:  microphone a positions , focus points , propagation matrix (7) and source matrix .
The discrete forward operator is thus defined as

 C––(M,N): RN→CM×M,    q–↦G––(M,N)M(N)q––––––G––(M,N)∗ .

In the following we present each reconstruction method in a discrete version for an observed covariance matrix and an infinite dimensional version for an observed covariance operator .

### 4.2 Covariance Matrix Fitting

For infinite dimensional quantities we have the least squares problem

 ∥∥C(q)−Cobs∥∥2HS=min! , (23)

which is uniquely solvable for and exact data by Theorem 3.6. In the discrete version, the CMF problem is defined by the least squares problem

 ∥∥C––(q–)−C––obs∥∥2F=min! (24)

where denotes the Frobenius norm, the discrete analog of the Hilbert-Schmidt norm. In [6], the minimization problem (24) is solved for an experimental dataset of a wind tunnel experiment with an aircraft wing.

### 4.3 Conventional Beamforming

Conventional Beamforming is probably the most popular evaluation method for aeroacoustic measurement data since it yields a fast and robust estimator of the source power. Instead of solving an inverse problem for all source powers at once, CBF estimates the source power at each focus point separately. Such methods are often referred to as array imaging methods. For a broad overview on different imaging scenarios and their analysis we refer to [13].

The Beamforming imaging functional is defined as

If we assume that the empirical estimate of the covariance operator is self-adjoint, this one-dimensional minimization problem has the solution

 I(y)=Re⎛⎝⟨Cobs,Py⟩HS∥Py∥2HS⎞⎠=⟨Cobs,Py⟩HS∥Py∥2HS=(C∗(Cobs))(y)∥Py∥2HS . (25)

Here the second equality follows from eq. (22). In helioseismology imaging functionals analogous to (25) (without the scaling factor ) appear as special types of holographic imaging functionals (see [14, 31]).

The Hilbert-Schmidt norm of the monopole operator is given by

 ∥Py∥2HS =∫M×M∣∣g(x1,y)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯g(x2,y)∣∣2d(x1,x2)=∥g(⋅,y)∥4L2(M)>0 .

For discrete data and a fixed focus point , the steering vector is defined as the pointwise evaluation of the Green’s function at all microphones

 g(y(N)n)=⎛⎜ ⎜ ⎜⎝g(x1,y(N)n)⋮g(xM,y(N)n)⎞⎟ ⎟ ⎟⎠ .

The discrete monopole operator is therefore called steering matrix. In analogy to Proposition 4.1, the discrete adjoint forward operator is

 C––∗: CM×M→RN,    (C––∗(K––))n=⟨K––,Py(N)n––––––⟩F ,

where denotes the Frobenius scalar product. Thus the discrete Beamforming functional is defined as

###### Remark 4.2 (Time domain array imaging).

CBF can be motivated by the time domain principle of delay and sum (DAS) [25], which is strongly related to Kirchhoff migration (see e.g. [8]). The basic idea of DAS is to shift all sensor time signals according to the time delay to a fixed focus point. Summing up the shifted signals, source signals originating at the focus point accumulate.

### 4.4 Damas

The idea of DAMAS (deconvolution approach for the mapping of acoustic sources) is to deblur the source information obtained by a CBF solution. It is defined by an integral equation of the first kind,

 I(y)=∫Ωψ(y,y′)q(y′)dy′ . (26)

In this article, DAMAS will always refer to the integral equation (26) and not to the iterative Gauß-Seidel method that was suggested in [9] in order to solve the discrete version of (26). The integral kernel is usually referred to as point-spread function (PSF) and defined as

 ψ(y,y′)=⟨Py,Py′⟩HS∥∥Py′∥∥2HS=(C∗(Py))(y′)∥∥Py′∥∥2HS .

For a shift invariant PSF, (26) reduces to a convolution integral, but for our scenario the PSF is not shift invariant. Nevertheless, deblurring methods like DAMAS are usually called deconvolution methods in the aeroacoustic community. The next statement relates the integral equation of DAMAS to the least squares problem of CMF.

###### Proposition 4.3 (Normal equation).

The DAMAS problem (26) is equivalent to the operator equation

 C∗C(q)=C∗Cobs (27)

which is the normal equation of the CMF problem (23).

###### Proof.

First of all we can multiply (26) by which yields the equivalent integral equation

 (C∗(Cobs))(y)=∫Ω⟨Py,Py′⟩HS q(y′)dy′ . (28)

For an orthonormal basis of , reformulating the right-hand side of (28) yields

 = ∫Ω∞∑j=1⟨Pyφj,Py′φj⟩L2(M) q(y′)dy′ = ∞∑j=1∫Ω⟨Pyφj,Py′φj⟩L2(M) q(y′)dy′ = . (29)

We obtain further

 […] = ∫Ω(∫Mg(x,y′)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯g(x′,y′)φj(x′)dx′)q(y′)dy′ (30) = ∫Mcq(x,x′)φj(x′)dx′=(C(q)φj)(x) .

Inserting (30) into (29) yields

 ∫Ω⟨Py,Py′⟩HS q(y′)dy′=⟨C(q),Py⟩HS=(C∗C(q))(y) .\qed
###### Corollary 4.4 (DAMAS uniqueness).

For exact data and the source space the solution of (26) is unique.

###### Proof.

By the uniqueness of (23) we obtain

 ker(C∗C)∩L∞(Ω)=ker(C)∩L∞(Ω)={0} .\qed

The original, discrete version of DAMAS is given by the linear system

 (31)

with the discrete point spread function

Similar to Proposition 4.3 the discrete CMF and DAMAS problem are related by the normal equation.

###### Corollary 4.5 (Discrete normal equation).

The problem (31) is equivalent to the linear system

 C––∗C––(q–)=C––∗C––obs (32)

which is the normal equation of (24).

### 4.5 Regularization

Since the operator is infinitely smoothing, the inverse problem (13) is ill-posed. Therefore, at least for fine discretizations of the source intensity regularization is required to obtain stable reconstructions in the presence of noise. In case of CMF this leads to estimators of the form

 ˆqCMF,α∈argminq∈L∞(Ω)[∥∥C(q)−Cobs∥∥2\footnotesizeHS(L2(M))+αR(q)] (33)

and for the DAMAS problem

 ˆqDAMAS,α∈argminq∈L∞(Ω)[∥∥C∗C(q)−C∗Cobs∥∥2L2(Ω)+αR(q)] . (34)

where is a convex penalty term and is a regularization parameter. In [47], the authors present discrete versions of (33) and (34) using non-negativity constraints, box constraints on the sum of the source intensities and sparsity enforcing penalties.

### 4.6 Reconstructions from experimental data

To conclude this section we illustrate the application of the presented methods in an experimental setup. For a typical aeroacoustic experiment, a solid object (for example a model of an aircraft) is placed inside the velocity field of a wind tunnel. The fluid structure interactions generate an acoustic signal, which is measured by a microphone array. The raw time data is further processed to an estimator of the cross correlations . The reconstruction of the source powers is often called source map. Figure 2 shows an example of a source map for each method. The results for CMF and DAMAS are obtained by quadratic Tikhonov regularization with a non-negativity constraint.

## 5 Conclusions and outlook

Sound source reconstruction methods such as CBF, DAMAS or CMF are all based on the same discrete sound propagation model. Here we described these methods in an infinite dimensional setup which allowed to relate the reconstruction methods to each other via the adjoint forward operator and the normal equation.

Of course the free field propagation model used in this paper relies on strong simplifications such as the negligence of solid geometries (aeroacoustic model, wind tunnel walls) which are present in aeroacoustic experiments. Nevertheless, the discrete version of this simplified forward operator has been successfully applied in aeroacoustic testing for many decades and is still state-of-the-art for most applications in this field. The validation of the results for experimental data remains challenging since a ground truth for the source power function is usually not known in such cases.

As a main result we proved the injectivity of the forward operator for bounded sources. This gives rise to a number of further questions which may be addressed in future research. First, one may try to extend the technique of our uniqueness proof to more complicated geometries such as waveguides or the presence of known obstacles or inhomogeneous background media. A second natural direction of research concerns the extension of our uniqueness proof to a conditional stability result or even variational source conditions, which will most likely be of logarithmic type under natural smoothness assumptions. The latter would even yield error bounds for reconstruction methods such as regularized CMF, i.e. Tikhonov regularization (see [23]). Finally, the results of this paper may eventually lead to a better theoretical understanding and justification of helioseismic holography.

## Acknowledgments

We would like to thank Dan Yang for helpful discussions on helioseismic holography.

## Appendix A Proofs of auxiliary statements

###### Proof of Proposition 2.2 (Mapping property of the forward operator).

In view of (11) we have to show that . Since and are disjoint and bounded, the integral kernel

 κ(x1,x2,y)=g(x1,y)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯g(x2,y)

is continuous on and therefore . This implies

 ∥cq∥2L2(M×M) = ∫M×M∣∣∣∫Ωκ(x1,x2,y)q(y) dy∣∣∣2 d(x1,x2) ≤ = ∥q∥2L2(Ω)∥κ∥2L2(M×M×Ω) .

###### Proof of Proposition 3.1 (Asymptotic behavior of Green’s function).

We start with a detailed proof of the case . As the distance function , with has the partial derivatives

 ∂fx∂yi(y) = −(x−y)⋅mmi+β2(xi−yi)|x−y|m ∂2fx∂yjyi(y) = −((x−y)⋅m mj+β2(xj−yj))|x−y|3m ⋅((x−y)⋅m mi+β2(xi−yi))+mjmi+β2δij|x−y|m .

we obtain

 fx(0)=|x|m,∇fx(0)=−|x|−1m(x⋅m m+β2x) , (35)
 (Hfx(ty)y)⋅y=O(|x|−1)   t∈(0,1) . (36)

Here and in the rest of this proof stands for a complex-valued function of and the absolute value of which is bounded by with a constant independent of in the bounded set and sufficiently large. Inserting (35) and (36) into the second order Taylor formula yields

 fx(y) = |x|m−|x|−1m((x⋅m)(y⋅m)+β2(x⋅y))+O(|x|−1) (37) = |x|m−(A^x)⋅y+O(|x|−1) . (38)

Inserting (38) into the definition of the Green’s function we further compute

 g(x,y) =