REFINED INSTABILITY ESTIMATES FOR SOME INVERSE PROBLEMS PU-ZHAO KOW AND JENN-NAN WANG Dedicated to the memory of Victor Isakov Abstract.

PU-ZHAO KOW AND JENN-NAN WANG Dedicated to the memory of Victor Isakov

Abstract. Many inverse problems are known to be ill-posed. The ill-posedness can be manifested by an instability estimate of exponential type, rst derived by Mandache [Man01].

In this work, based on Mandache's idea, we rene the instability estimates for two inverse problems, including the inverse inclusion problem and the inverse scattering problem. Our aim is to derive explicitly the dependence of the instability estimates on key parameters.

The rst result of this work is to show how the instability depends on the depth of the hidden inclusion and the conductivity of the background medium. This work can be regarded as a counterpart of the depth-dependent and conductivity-dependent stability esti- mate proved by Li, Wang, and Wang [LWW21], or pure dependent stability estimate proved by Nagayasu, Uhlmann, and Wang [NUW09]. We rigorously justify the intuition that the exponential instability becomes worse as the inclusion is hidden deeper inside a conductor or the conductivity is larger.

The second result is to justify the optimality of increasing stability in determining the near-eld of a radiating solution of the Helmholtz equation from the far-eld pattern. Isakov [Isa15] showed that the stability of this inverse problem increases as the frequency increases in the sense that the stability estimate changes from a logarithmic type to a Hölder type.

We prove in this work that the instability changes from an exponential type to a Hölder type as the frequency increases. This result is inspired by our recent work [KUW21].

1. Introduction

Many inverse problems are known to be ill-posed. Even the uniqueness holds in most cases, the continuous dependence of the unknown on the measurements is very weak. For some inverse problems, two estimates have been proved to quantify this ill-posedness. For example, in Calderón's problem, a logarithmic stability estimate was proved by Alessandrini [Ale88] and an exponential instability was derived by Mandache [Man01]. The estimate obtained in [Man01] guarantees that the logarithmic stability estimate in Calderón's problem is optimal. More rened stability estimates involving parameters of the equations, such as the frequency, the depth of the unknown, or the conductivity, etc. were derived for many cases, not just in inverse problems, but also in the unique continuation. Following Mandache's idea, exponential instability estimates containing the eect of the frequency in some inverse problems were proved in [ZZ19] (for the transport equation) and in [KUW21]

(for the Schrödinger equation). Inspired by the results in [KUW21], in this paper, we derive exponential instability estimates emphasizing on the eect of the parameters for two inverse problems, the inverse inclusion and the inverse scattering problems. We will explain our main results in detail below.

2020 Mathematics Subject Classication. 35J15; 35R25; 35R30.

Key words and phrases. inverse problems; instability; Calderón's problem; electrical impedance tomogra- phy; depth-dependent instability of exponential-type; Helmholtz equation; scattering theory; Rellich lemma;

increasing stability phenomena.

1

In a recent article [KRS21], Koch, Rüland, and Salo investigated the mechanisms that cause the instability for some linear and nonlinear inverse problems. The instability mechanisms were categorized by three smoothing properties  strong global smoothing, only weak global smoothing, and microlocal smoothing for the corresponding forward operators. They derived instability estimates in more general geometries and coecients. Here we are interested in how instability estimates depend on some key parameters. We achieve this by rening Mandache's approach and, therefore, work in the situation of symmetric geometries and constant coecients. In order to present the phenomena cleanly, we choose not to explore the possibility of extending the results to more general settings.

We dedicate this paper to the memory of Victor Isakov, who made numerous fundamental contributions in the development of inverse problems. His original research on the phenom- enon of increasing stability gives us a better understanding of the ill-posedness in inverse problems. This paper is largely inuenced by his results.

1.1. Depth-dependent and conductivity-dependent instability of the Electrical Impedance Tomography (EIT). We rst study the exponential instability of the EIT.

Dierent from early woks [DR03a,DR03b,Man01], here we would like to rene the previous estimates in which one can understand the inuence of other a priori factors of the conduc- tivity in instability. Precisely, we consider the inverse inclusion problem with the information of boundary data. We now describe the problem in more detailed. Let Ω ⊂ R² be a domain with smooth boundary and γ(x) > 0 (with a sucient regularity) represent the conductivity of Ω. Due to the conservation law, the electric potential u satises the conductivity equation

(1) ∇ · (γ(x)∇u) = 0 in Ω.

It is known that given any f ∈ H^1/2(∂Ω), there exists a unique solution u to (1) with u|∂Ω = f. The boundary data is given in the form of the Dirichlet-to-Neumann map (DN-map):

(2) Λ_γ(f ) := γ∂_νu

   ∂Ω

,

where ν is the unit exterior normal vector of ∂Ω. The information of the conductivity is encoded in Λγ and the EIT is to determine γ from the knowledge of Λγ.

This inverse problem was proposed by Calderón [Cal80] where he showed that the lin- earized DN-map at the constant conductivity is injective. The global uniqueness of the EIT was proved by Sylvester and Uhlmann [SU87] (for dimensions higher than two) and by Nach- man [Nach96] (for dimension two). The EIT is known to be ill-posed. A log-type stability estimate was rst established by Alessandrini [Ale88], while Mandache [Man01] conrmed that Alessandrini's result is optimal by showing that the problem of exponentially unstable.

In several practical situations, the conductivity coecient γ takes the following form:

γ(x) = γ₀(x) + γ₁(x)χ_D,

where χD is the characteristic function of the domain D. Here, D represents an inclusion in Ω having a dierent conductivity γ1. In [Isa88], Isakov showed that, if γ0(x) is known, then both γ1(x)and D can be uniquely determined by the DN-map (2). A log-type stability estimate was obtained in [AD05] for this inverse inclusion problem, i.e. determination of D from Λγ.

We now consider the inverse inclusion problem with γ0(x) = 1and γ1(x) = κ ̸= 1, that is,

(3) ∇ · ((1 + (κ − 1)χ_D)∇u) = 0 in Ω

an the DN-map ΛD is dened by

(4) Λ_D : H¹²(∂Ω) → H⁻¹²(∂Ω), Λ_D(u|_∂Ω) := ∂_νu|_∂Ω.

The exponential instability for the inverse inclusion problem described above was proved in [AD05]. However, the estimate obtained in [AD05] did not show that inuence of the depth of D on the instability. In [NUW09], they obtained a depth-dependent stability estimate by studying the linearized DN-map. Recently, the stability estimate of [NUW09] was extended to the multi-layer medium in [LWW21] where the eect of the conductivity of each layer on the stability was also discovered. To simplify the discussions, we consider the medium which has 3-layer structure (the ideas can be easily extended to multi-layer structure). Let Ω^′ be Lipschitz domains such that D ⊂ Ω^′ and Ω^′ ⊂ Ω. In this work, we study the inverse inclusion problem with

γ(x) = κ1χD + κ2χ_Ω′\D+ χ_Ω\Ω′,

where κi > 0 are dierent with κi ̸= 1 (i = 1, 2). We dene the following operator:

LDu := ∇ · ((κ1χD + κ2χ_Ω′\D+ χ_Ω\Ω′)∇u) in Ω.

Likewise, we can dene the DN-map ΛD by (4).

One of the main theme of this work is to investigate how the depth of the inclusion D and the conductivity κ2 aect the instability of the inverse problem. To formulate our problem precisely, we consider Ω = B1, Ω^′ = B³

4, and D = Br with 0 < r < ¹₄. We introduce a smooth function

ψ : ∂D → R

and the perturbed boundary ∂Ds of the inclusion Ds is described by the image of y = F_s(x) := x + sψ(x)ν_x(x), x ∈ ∂D.

Now the linearized DN-map of ΛDs at s = 0, denoted by dΛBr(ψ), is formally dened by

(5) dΛ_Br(ψ) := lim

s→0

1

s(Λ_Ds − Λ_D).

Indeed, dΛBr(ψ) : H¹²(∂B₁) → H⁻¹²(∂B₁)is a bounded linear operator, see [LWW21, Lemma 2.3]. A log-type stability estimate with dΛBr including the eect of the depth r of the inclusion Br and the conductivity κ2 was proved in [LWW21]. Precisely, under some apriori assumptions, the following estimate holds:

(6) ∥ψ∥_L²_(∂B1) ≤ C(κ₂+ 1) log(r⁻¹)| log ∥dΛ_Br(ψ)∥_∗|⁻¹, where

∥ • ∥∗ = ∥ • ∥

H¹²(∂B1)→H^{− 12}(∂B1).

Estimate (6) clearly indicates that the stability becomes worse as the depth of the inclusion increases, i.e. r becomes smaller, or the conductivity κ2 becomes larger. It was also showed in [NUW09] that, given any ϵ > 0, there exists no positive constant C^′ such that

∥ψ∥_L²_(∂Br)≤ C^′| log ∥dΛBr(ψ)∥∗|^−1−ϵ,

that is, the logarithmic stability (6) is optimal. The deterioration of the stability of re- constructing a deeply hidden inclusion by the DN-map was also observed numerically in [IINSU07, UW08, UWW09].

By combining the ideas of [LWW21] and [Man01], we proved the following depth-dependent and conductivity-dependent exponential instability for the linearized DN-map dΛBr:

Theorem 1.1. Fixing any 0 < r < ¹₄ and κ2 > 1 + κ₁. There exists a constant 0 < E < 1 such that, given any 0 < ϵ < E, there exists a function ψ ∈ C^∞(∂B_r) with

∥ψ∥_L^∞_(∂Br) ≥ ϵ such that

(7) ∥dΛ_Br(ψ)∥_∗ ≤ C 1

κ₂+ 1 exp(−| log r|²³ϵ^−3α1 ) for some constant C which is independent of κ2, r, ϵ.

Estimate (7) corresponds to the statement that the depth-dependent and conductivity- dependent stability obtained in [LWW21], as well as the depth-dependent stability obtained in [NUW09], are optimal from the instability perspective. We want to point out that, since dΛBr is a linear operator, a norm estimate was derived in [LWW21, Corollary 1], precisely, (8) ∥dΛ_Br(ψ)∥∗ ≤ C|κ₁− κ₂|

|κ₁+ κ₂| 1

κ₂+ 1r¹²∥ψ∥_L²_(∂Br)

for some constant C. The norm estimate (8) holds for all perturbations of the inclusion ψ.

It gives us only an upper bound of the size of dΛBr(ψ) in terms of ψ. The merit of (7) is that it provides a fact that the size of dΛBr(ψ) could be much smaller (exponentially small) in terms of some perturbation ψ. The derivation of (7) is more delicate that that of (8).

1.2. Instability estimate for the determination of the near-eld from the far-eld.

We now study the instability phenomenon of determining the near-eld of a radiating solution to the Helmholtz equation from the far-eld pattern. The uniqueness follows easily from Rellich's lemma. Likewise, this inverse problem is also ill-posed. Nonetheless, it was proved by Isakov [Isa15] that the stability of this inverse problem increases as the frequency increases.

In this work, we want to verify this increasing stability phenomenon from the viewpoint of instability estimate and hence shows that the result obtained in [Isa15] is optimal. The increasing stability phenomena were rigorously proved in other situations [DI07,DI10,HI04, IK11, ILX20, INUW14, Isa07, Isa11, Isa15, KU19, LLU19, NUW13], not only for inverse problems, but also for the unique continuation property.

Given any f ∈ H¹²(∂B₁), there exists a unique u ∈ Hloc¹ (R³ \ B₁) solving the following exterior problem:

(9)







(∆ + κ²)u = 0 in R³\ B₁,

u = f on ∂B1,

u satises Sommerfeld radiation condition at |x| → ∞, and the following estimate holds

(10) ∥u∥_H1(BR\B1) ≤ C(R, κ)∥f ∥

H¹²(∂B1),

see, for example, [KG08, Theorem 1.1] (see also [Ne01, Theorem 2.6.2] for rened inequality of (10) and [BP08, Theorem 3.3] for elastic waves). It is well-known that u satises the following asymptotic expansion [CK19, KG08, Tay96]:

u(x) = e^iκr

r u^∞(ˆx) + O(r⁻²) as r = |x| → ∞

uniformly for all ˆx = x/|x| ∈ S², where u^∞(ˆx) is called the far-eld pattern. We use u^∞(f ) to indicate the dependence of u^∞ on f.

It follows from Rellich's lemma that u^∞(f ) uniquely determine u in R³\ B₁ and therefore the boundary data f is uniquely also recovered, i.e., the mapping f → u^∞(f ) is injec- tive. We now want to remark on the stability estimate of determining f from u^∞(f ). Let

 Y_n^m n ≥ 0, |m| ≤ n

be the spherical harmonics, which forms a complete orthonormal basis in L²(S²). Therefore, we can write

u^∞ =X

n≥0

X

|m|≤n

u^∞_nmY_n^m. Dene

ϵ²₁ :=

⌊√ κ⌋

X

n=0

X

|m|≤n

|u^∞_nm|² and ϵ²2 :=

∞

X

n=⌊√ κ⌋+1

X

|m|≤n

|u^∞_nm|² Under some a priori assumptions, it was shown in [Isa15, Theorem 1.1] that

∥f ∥²_L2(∂B1)≤ 2e²

π ϵ²₁+ 2e²

π ϵ₂+ M₁ κ + | log ϵ₂|, (11a)

∥f ∥²_L2(∂B1)≤ 2e² π ϵ²₁+

r 2 πκeM₁ϵ

1 2

2 + M₁² κ + | log ϵ2| (11b)

for some constant M1 > 0. The estimates (11a) and (11b) indicate that the logarithmic part (κ + | log ϵ₂|)⁻¹ decreases as κ increases, and both estimates change from a logarithmic type to a Hölder type. In other words, Isakov's work [Isa15] can be regarded as a quantitative version of Rellich's lemma. Moreover, using (10), one can see that Neumann data can be easily recovered from Dirichlet data, and the recovery process is stable.

In this work, we will study the counterpart of the increasing stability by investigating how the exponential instability is aected by the frequency. Inspired by the work [ZZ19] and our recent preprint [KUW21], we prove the following theorem.

Theorem 1.2. Fixing any frequency κ > 0, and let ˜κ := (^κ₂)^exp(κ). There exists a positive constant E such that for any 0 < ϵ < E, there exists a function f ∈ C^∞(∂B₁) satisfying

∥f ∥_L^∞_(∂B1) ≥ ϵ and

(12) ∥u^∞(f )∥

H^{− 52}(S²) ≤ C

 exp



− max{˜κ, 1}

3 ϵ^−1α



+ min{1, ˜κ}ϵ^α1

 for some constant C which is independent of κ and ϵ.

Estimate (12) shows that the instability changes from an exponential type to a Hölder type when κ increases, and vice versa. Such transition of instability was also established for an inverse problem in the stationary radiative transport equation in [ZZ19] and in the Schrödinger equation in [KUW21]. In addition, this result shows that Isakov's result in [Isa15] is optimal.

Our proof relies a well-known expression (42) of u in terms of spherical harmonics Yn^m. The crucial step is the identity, which connects the Bessel function with Lommel polynomials, given in (44). This gives an explicit lower bound of the spherical Hankel functions, see Lemma 4.1. It is also interesting to mention that, using some rened properties of Bessel functions, John [John60] constructed an example showing a logarithmic stability uniformly

in κ in the continuation of solution to the Helmholtz equation from the unit disk into its complement in the plane.

1.3. Organization of the paper. We will follow the general procedure introduced in [Man01]. We rst discuss the construction of an ϵ-discrete set in some function space in Section2. Using this ϵ-discrete set, we will prove Theorem 1.1 and Theorem1.2 in Section3 and Section 4, respectively.

2. Construction of an ϵ-discrete set

Let d ≥ 1. We now want to construct an ϵ-discrete set (a.k.a. ϵ-distinguishable set) for some neighborhood which is not too large. Here we recall that a set Z of a metric space (M, d) is called an ϵ-discrete set if d(z1, z₂) ≥ ϵ for all z1 ̸= z₂ ∈ Z. For each ϵ > 0, we dene

Nˆϵ(B¹

2) :=

n ψ ∈ C_c^∞(R^d) ψ is real-valued, supp (ψ) ⊂ B¹

2, ∥ψ∥_L^∞_(R^d₎≤ ϵ o . We now prove the following lemma.

Lemma 2.1. Given any α > 0, there exists µ = µ(d, α) > 0 such that the following statement holds for any auxiliary parameter β > 0: Given any 0 < ϵ < µβ, there is an ϵ-discrete set ˆZ of ( ˆNϵ(B¹

2), ∥ • ∥L^∞) with

(13) | ˆZ| ≥ exp

 1 2^d+1

 µβ ϵ

^d_α , where | ˆZ| denotes the cardinality of ˆZ.

Remark 1. When d ≥ 2, Lemma 2.1is a special case of [Man01, Lemma 2]. See also [KT61, Theorem XIV] for more abstract setting, or [DR03a, Proposition 3.1], [KUW21, Proposition 2.1], [ZZ19, Lemma 5.2].

Proof of Lemma 2.1. It remains to prove this theorem for d = 1. We x ψ0 ∈ C_c^∞(R¹) such that supp (ψ0) ⊂ B¹

2 = (−¹₂,¹₂) and ∥ψ0∥_L^∞_(R¹₎ = 1. We now dene µ := ∥ψ₀∥⁻¹_Cα(R¹) and N = µβ

ϵ

_α¹ . Since 0 < ϵ < µβ, then ^µβ_ϵ > 1. Hence,

(14) N > 1

2

 µβ ϵ

_α¹ . We divide B¹

2 = (−¹₂,¹₂) into N smaller intervals of length 1/N. Let y1, · · · , y_N be their centers. Dening

Z :=ˆ (

ψ ψ(x) = ϵ

N

X

j=1

σ_jψ₀(N (x − y_j)), σ_j ∈ {0, 1}

) . Note that each element ψ ∈ ˆZ is smooth with ∥ψ∥L^∞ ≤ ϵ and ˆZ ⊂ ˆN_ϵ(B¹

2). Moreover, we see that

∥ψ₁− ψ₂∥_L^∞ = ϵ for all ψ1 ̸= ψ₂ ∈ ˆZ.

Finally, we see that

(15) | ˆZ| = 2^N = exp(N log 2) ≥ exp N 2

 .

Combining (14) and (15), we obtain (13). □

Let r > 0 and let P : ∂Br → R^d∪ {∞} be the stereographic projection. Let us dene N_ϵ(∂B_r, P) :=n

ψ : ∂B_r → R ψ ◦ P⁻¹ ∈ ˆN_ϵ(B¹

2) o , Z := ψ : ∂Br → R ψ ◦ P⁻¹ ∈ ˆZ , and

N_ϵ(∂B_r) :=  ψ : ∂B_r → R ψ is smooth with ∥ψ∥L^∞(∂Br) ≤ ϵ .

It is clear that Z ⊂ Nϵ(∂B_r, P) ⊂ N_ϵ(∂B_r)and |Z| = | ˆZ|. Hence, we can rephrase Lemma2.1 as follows:

Proposition 1. Given any α > 0, there exists µ = µ(d, α) > 0 such that the following statement holds for any auxiliary parameter β > 0: Given any 0 < ϵ < µβ, there is an ϵ-discrete set Z of (Nϵ(∂B_r), ∥ • ∥_L^∞_(∂Br)) with

|Z| ≥ exp

 1 2^d+1

 µβ ϵ

^d_α , 3. Proof of Theorem 1.1 We prove Theorem 1.1 in this section.

3.1. General framework of matrix representation. For each ρ > 0 and γ ∈ R, we have

∥ψ∥²_L2(∂Bρ)= ρ 2π

X

k∈Z

Z 2π 0

ψ(ρ cos θ, ρ sin θ)e^−ikθdθ    

2

(16a) ,

∥ψ∥²_Hγ(∂Bρ)= ρ 2π

X

k∈Z

(1 + k²)^γ    

Z 2π 0

ψ(ρ cos θ, ρ sin θ)e^−ikθdθ    

2

(16b) ,

see [NUW09, (2.1)]. For each n ∈ Z, we dene ϕn: ∂B₁ → C ϕ_n(cos θ, sin θ) := 1

√2π(1 + n²)¹⁴e^inθ. Using (16b) with γ = ¹₂ and ρ = 1 gives

∥ϕ_n∥²

H¹²(∂B1) = 1 2π

X

k∈Z

(1 + k²)¹²    

Z 2π 0

√ 1

2π(1 + n²)¹⁴e^inθe^−ikθdθ    

2

= 1

2π(1 + n²)¹²    

√2π (1 + n²)¹⁴

2

= 1

2π(1 + n²)¹² 2π

(2 + n²)¹² = 1.

That is,  ϕn n ∈ Z

forms a complete orthonormal set in H¹²(∂B₁).

Let A : H¹²(∂B₁) → H⁻¹²(∂B₁) be any bounded linear operator. For any pair (n, m) ∈ Z × Z, we dene the complex number

a_nm := ⟨Aϕ_n, ϕ_m⟩,

where ⟨•, •⟩ is the H⁻¹²(∂B₁) × H¹²(∂B₁) duality pair. We consider the Banach space X, which consists tensors (anm)with

∥(a_nm)∥_X := 1 4 sup

n,m∈Z

(1 + max{|n|, |m|})²|a_nm| < ∞.

We have the following proposition.

Proposition 2. There exists an absolute constant Cabs > 0 such that

(17) ∥A∥∗ ≤ Cabs∥(anm)∥X.

In other words, tensor (anm)can be treated as the matrix representation of the bounded linear operator A.

Proof. Using the Hilbert-Schmidt norm, we have

∥A∥∗ ≤

 X

n,m∈Z

|a_nm|²

¹₂

≤ 4

 X

n,m∈Z

1

(1 + max{|n|, |m|})⁴

¹₂

∥(a_nm)∥_X. We now compute

X

n,m∈Z

1

(1 + max{|n|, |m|})⁴

≤

 X

n≥0,m≥0

+ X

n≥0,m≤0

+ X

n≤0,m≥0

+ X

n≤0,m≤0

 1

(1 + max{|n|, |m|})⁴

= 4 X

n≥0,m≥0

1

(1 + max{n, m})⁴

≤ 4

 X

n≥m≥0

+ X

m≥n≥0

 1

(1 + max{n, m})⁴

= 8 X

n≥m≥0

1

(1 + n)⁴ = 8

∞

X

n=0 n

X

m=0

1

(1 + n)⁴ = 8

∞

X

n=0

1

(1 + n)³ < ∞,

which proves (2). □

3.2. Estimating the matrix representation of the linearized DN-map. The task here is to estimate dΛ^nmBr(ψ) := ⟨dΛ_Br(ψ)(ϕ_n), ϕ_m⟩. Precisely, we want to prove the following proposition.

Proposition 3. Given any ϵ > 0 and 0 < r < ¹₄. If κ2 > 1 + κ1, then there exists an absolute constant C such that

(18) |dΛ^nm_B

r(ψ)| ≤ CR 1

κ₂+ 1ℓ¹²r^ℓ−1 for all ψ ∈ NR(∂B_r), where ℓ = max{|n|, |m|}.

Remark 2. Observe that dΛ^nmBr(ψ) = 0 when n = 0 or m = 0, since ΛDs(1) = 0for all s ≥ 0.

Hence, we have

∥(dΛ^nm_Br(ψ))∥_X = 1 4 sup

0̸=n,m∈Z

(1 + max{|n|, |m|})²|dΛ^nm_Br (ψ)|

≤ sup

0̸=n,m∈Z

max{|n|, |m|}²|dΛ^nm_Br(ψ)|.

(19)

Given any function g ∈ L²(∂B_r) and k ∈ Z, we dene the Fourier coecient of g as

g_k :=

Z 2π 0

g(r cos θ, r sin θ)e^−ikθdθ.

It is easy to see that

(20) |g_k| ≤ 2π∥g∥_L^∞_(∂Br).

For f ∈ L²(∂B1), we abuse the notation and dene

f_k :=

Z 2π 0

f (cos θ, sin θ)e^−ikθdθ.

We need the following lemma, which is a special case of [LWW21, (18)] (taking R = 1 in [LWW21, (18)]).

Lemma 3.1. For f ∈ H¹²(∂B₁), we have dΛ_Br(ψ)(f )    ∂B1

=X

a∈Z

λ_a(f )e^iaθ, where λ0(f ) = 0 and for all a ∈ N

λ_−a(f ) = κ₁− κ₂ π² r⁻¹T_a

∞

X

p=1

S_p



(κ₁+ κ₂)ψ_−a+pf_−p− (κ₂ − κ₁)ψ_−a−pf_p

 ,

λ_a(f ) = κ₁− κ₂ π² r⁻¹T_a

∞

X

p=1

S_p



(κ₁+ κ₂)ψ_a−pf_p− (κ₂− κ₁)ψ_a+pf−p

 ,

where

T_a := −a

(κ₂− κ₁)r^a+ (κ₁+ κ₂)r^−a S_p := 2p 3

4

−p

(κ₂− κ₁)(κ₂− 1)r^p 3 4

−p

− (κ₁+ κ₂)(κ₂+ 1)r^−p 3 4

p

+ 3 4

p

− (κ₂− κ₁)(κ₂+ 1)r^p 3 4

^−p

+ (κ₁+ κ₂)(κ₂− 1)r^−p 3 4

p⁻¹ . Remark 3. When κ2 = 1 and κ1 = κ, that is, the case of 2-layer medium, we have Sp = T_p, and hence Lemma 3.1 reduces to [NUW09, Lemma 2.2] with R = 1.

The following inequalities can be found in the proof of [LWW21, Lemma 2.3]:

|T_k| ≤ 2k κ₁+ κ₂r^k, (21a)

|Sn| ≤ 1 min{¹₂, c₀}

2n

(κ₁+ κ₂)(κ₂ + 1)rⁿ, (21b)

where

c₀ := inf

τ ∈N

1 −κ₁ − κ₂

κ₁+ κ₂r^2τ +κ₁ − κ₂ κ₁+ κ₂

κ₂− 1 κ₂+ 1r^2τ 3

4

^−2τ

− κ₂− 1 κ₂+ 1

 3 4

2τ    .

Since 0 < r < ¹₄ and κ2 ≥ 1 + κ1, it is easy to see that c0 ≥ ¹₅, and hence (21b) becomes

(21c) |S_n| ≤ 10n

(κ₁+ κ₂)(κ₂+ 1)rⁿ. Now, we are ready to prove Proposition 3.

Proof of Proposition 3. Using (16b), we can estimate

|dΛ^nm_Br(ψ)| ≤ ∥dΛ_Br(ψ)(ϕ_n)∥

H^{− 12}(∂B1)

= 1 2π

X

k∈Z

(1 + k²)⁻¹²    

Z 2π 0

 X

a∈Z

λ_a(ϕ_n)e^iaθ



e^−ikθdθ    

2¹₂

= 1 2π

X

k∈Z

(1 + k²)⁻¹²|2πλk(ϕn)|²

¹₂

=

 2πX

k∈Z

(1 + k²)⁻¹²|λ_k(ϕ_n)|²

¹₂ (22) .

Note that the Fourier coecient (ϕn)_p of ϕn can be explicitly calculated:

(ϕ_n)_p = Z 2π

0

ϕ_n(cos θ, sin θ)e^−ipθdθ

= Z 2π

0

√ 1

2π(1 + n²)¹⁴e^inθe^−ipθdθ =

√2π (1 + n²)¹⁴δ_np. (23)

Now we consider n > 0. For any R > 0 and ψ ∈ NR(∂B_r), we can see that for k > 0,

|λ−k(ϕ_n)| =    

κ₁− κ₂ π² r⁻¹T_k

∞

X

p=1

S_p



(κ₁+ κ₂)ψ−k+p(ϕ_n)−p− (κ₂− κ₁)ψ−k−p(ϕ_n)_p

   

=    

κ₁− κ₂

π² r⁻¹T_kS_n



(κ₂− κ₁)ψ−k−n

√2π (1 + n²)¹⁴

  

 (from (23))

= 2π|κ₁ − κ₂|²

π² r⁻¹|T_k||S_n||ψ−k−n| 1 (1 + n²)¹⁴

≤ 20(2π)³²|κ₁− κ₂|² π² Rr⁻¹

 k

κ₁+ κ₂r^k

 n

(κ₁+ κ₂)(κ₂ + 1)rⁿ

 1

(1 + n²)¹⁴ (using (20),(21a), and (21c))

= 20(2π)³² π²

κ₁− κ₂ κ₁+ κ₂    

2

R 1

κ₂+ 1r⁻¹ knr^k+n (1 + n²)¹⁴

≤ 20(2π)³² π² R 1

κ₂+ 1r⁻¹ knr^k+n (1 + n²)¹⁴ (24a)

and

|λ_k(ϕ_n)| =    

κ₁− κ₂ π² r⁻¹T_k

∞

X

p=1

S_p



(κ₁+ κ₂)ψ_k−p(ϕ_n)_p− (κ₂− κ₁)ψ_k+p(ϕ_n)−p

   

=    

κ1− κ2

π² r⁻¹T_kS_n



(κ₂+ κ₁)ψ_k−n

√2π (1 + n²)¹⁴

  

 (from (23))

= 2π|κ1− κ2||κ1+ κ2|

π² r⁻¹|T_k||S_n||ψ_k−n| 1 (1 + n²)¹⁴

≤ 20(2π)³²|κ₁− κ₂||κ₁+ κ₂|

π² Rr⁻¹

 k

κ1+ κ2

r^k

 n

(κ1+ κ2)(κ2+ 1)rⁿ

 1

(1 + n²)¹⁴ (using (20),(21a), and (21c))

= 20(2π)³² π²

κ₁− κ₂ κ₁+ κ₂    

R 1

κ₂+ 1r⁻¹ knr^k+n (1 + n²)¹⁴

≤ 20(2π)³² π² R 1

κ₂ + 1r⁻¹ knr^k+n (1 + n²)¹⁴ (24b)

From (24a) and (24b), if we dene

C :=˜ 20(2π)³² π² , we then have

(25) |λ_k(ϕ_n)| ≤ ˜CR 1

κ₂+ 1r⁻¹|k|nr^|k|+n

(1 + n²)¹⁴ for all k ∈ Z.

Combining (22) and (25), we obtain

|dΛ^nm_Br(ψ)| ≤√

2π ˜CR 1

κ₂+ 1r⁻¹ nrⁿ (1 + n²)¹⁴

 X

k∈Z

(1 + k²)⁻¹²k²r^2|k|

¹₂

≤√

2π ˜CR 1

κ₂+ 1r⁻¹ nrⁿ (1 + n²)¹⁴

 X

k∈Z

(1 + k²)⁻¹²k² 1 4

|k|¹₂

= CR 1 κ₂+ 1

n

(1 + n²)¹⁴rⁿ⁻¹ (26)

with

C =√ 2π ˜C

 X

k∈Z

(1 + k²)⁻¹²k² 1 4

|k|¹₂

< ∞.

For n < 0, we can obtain an inequality similar to (26), precisely,

|dΛ^nm_Br(ψ)| ≤ CR 1 κ₂+ 1

|n|

(1 + n²)¹⁴r^|n|−1.

Since dΛBr(ψ)is self-adjoint, i.e. (dΛ^nmBr(ψ)) is symmetric, we thus conclude that

|dΛ^nm_B

r(ψ)| ≤ CR 1 κ₂+ 1

ℓ

(1 + ℓ²)¹⁴r^ℓ−1,

which implies (18). □

3.3. Construction of a δ-net. Given any ψ ∈ NR(∂B_r), (18) implies that

(27) ℓ²|dΛ^nm_Br(ψ)| ≤ C_R 1

κ₂+ 1ℓ⁵²r^ℓ−1

with ℓ = max{|n|, |m|} and CR depending on R > 0. Here, it suces to take CR > 1. Here, from (19), it follows that

(28) ∥(dΛ^nm_Br(ψ))∥_X ≤ C_R 1 κ2+ 1sup

ℓ≥1

ℓ⁵²r^ℓ−1 < ∞.

In other words, we have

(29) (dΛ^nm_Br(N_R(∂B_r))) ⊂ X.

In view of (29), we want to construct a _κ2^δ₊₁-net Y for ((dΛ^nm_Br(N_R(∂B_r))), ∥ • ∥_X), which is not too large. Precisely, we aim to derive the following proposition.

Proposition 4. Let 0 < r < ¹₄, R > 0, and κ2 > 1 + κ1. Given any 0 < δ < 1, there exists a _κ2^δ₊₁-net Y for ((dΛ^nmBr(NR(∂Br))), ∥ • ∥X) such that

(30) log |Y | ≤ C| log r|⁻²log³ η_R δ



+ C log η_R δ

 , where C is a general constant, and ηR is a constant depending only on R.

Remark 4. A set Y if a metric space (M, d) is called a δ-net for Y1 ⊂ M if for any x ∈ Y1, there is a y ∈ Y such that d(x, y) ≤ δ.

Proof of Proposition 4. Step 1: Initialization. Let CR be the constant given in Proposi- tion 3. Given any 0 < δ < 1, let τ0 > 1 be the unique positive solution (not necessarily an integer) to

(31) τ

5 2

0 r^τ0−1 = δ CR

. If we dene ℓ^∗ = ⌊τ₀⌋ (note 1 ≤ ℓ^∗ ≤ τ₀), then (31) implies

(32) δ

C_R ≤ ℓ∗⁵²r^ℓ∗−1 ≤ ℓ∗⁵²

 1 4

^ℓ∗−1₂

r^ℓ∗−12 ≤ C^′r^ℓ∗−12 with C^′ = sup

τ ≥1

τ⁵² 1 4

^{τ −1}₂ . Taking the logarithm both sides of (32) gives

log

 δ C^′C_R



≤ log(r^ℓ∗−12 ) = ℓ∗− 1 2 log r, and thus

log C^′C_R δ



= log



− δ

C^′C_R



≥ −ℓ_∗ − 1

2 log r = ℓ_∗− 1

2 | log r|, which is equivalent to

(33) ℓ∗ ≤ 2

| log r|log C^′C_R δ

 + 1.

Furthermore, we can observe that for any integer ℓ > ℓ∗, i.e. ℓ > τ0, it holds

(34) ℓ⁵²r^ℓ−1 ≤ δ

C_R.

Step 2: Construction of a set. For each pair (n, m) ∈ Z × Z with 0 < ℓ = max{|n|, |m|} ≤ ℓ_∗, (27) implies

|dΛ^nm_Br(ψ)| ≤ C_R 1

κ₂+ 1C^′′ with C^′′= sup

ℓ≥1

ℓ¹² 1 4

ℓ−1

. We set

δ^′ := δ

√2ℓ²_∗(κ₂+ 1), Y^′ :=

n

a = a₁+ ia₂ ∈ δ^′Z + iδ^′Z |a₁|, |a₂| ≤ _ℓ2^CRC′′

∗(κ2+1)

o , and

Y :=



(bnm) if ℓ = max{|n|, |m|} ≤ ℓ∗, then bnm∈ Y^′, otherwise, bnm = 0

 .

Step 3: Verifying that Y is a _κ2^δ₊₁-net. Given any ψ ∈ NR(∂B_r), our goal is to construct a tensor (bnm) ∈ Y that is close to the tensor (dΛ^nmBr(ψ)). If 0 < ℓ = max{|n|, |m|} ≤ ℓ∗, we choose bnm ∈ Y^′ as the closest element to dΛ^nmBr(ψ). Then, we have

(35a) ℓ²|b_nm− dΛ^nm_Br(ψ)| ≤√

2ℓ²_∗δ^′ = δ κ₂+ 1.

Otherwise, if ℓ = max{|n|, |m|} > ℓ∗, we choose bnm = 0. For such choice of (bnm), with the help of (27) and (34), we conclude that

(35b) ℓ²|b_nm− dΛ^nm_Br(ψ)| = ℓ²|dΛ^nm_Br(ψ)| ≤ C_R 1

κ2+ 1ℓ⁵²r^ℓ−1 ≤ δ κ2+ 1.

Combining (19), (35a), and (35b), we conclude that

∥(b_nm− dΛ^nm_Br(ψ))∥_X ≤ δ κ2+ 1, which shows that Y is a _κ2^δ₊₁-net for ((dΛ^nm_Br(N_R(∂B_r))), ∥ • ∥_X).

Step 4: Calculating the cardinality of Y . We see that

(36) |Y^′| =

 1 + 2

 C_RC^′′

ℓ²_∗(κ₂+ 1)δ^′

2

≤



1 + 2√

2C_RC^′′

δ

2

.

Let Nℓ be the number of pairs (n, m) ∈ (Z \ {0}) × (Z \ {0}) with max{|n|, |m|} = ℓ. We want to estimate Nℓ. When n = ±ℓ, then m can be any no-zero integer between −ℓ and ℓ (i.e. there are 2ℓ choices). Switching the role of n and m, we hence obtain that Nℓ ≤ 8ℓ. Consequently, we can estimate

(37) N_∗ :=

ℓ∗

X

ℓ=1

N_ℓ ≤

ℓ∗

X

ℓ=1

8ℓ = 4ℓ_∗(ℓ_∗+ 1) and |Y | = |Y^′|^N∗. Combining (33), (36), and (37), we obtain

log |Y | = N∗log |Y^′| ≤ 16ℓ²_∗log



1 + 2√

2C_RC^′′

δ



≤ 16

 2

| log r|log C^′CR

δ

 + 1

2

log



1 + 2√

2CRC^′′

δ

 ,

which implies (30). □

Remark 5. Note that inf

0<δ<1| log r|⁻²log³ η_R δ



= | log r|⁻²log³(η_R) <

 log 1

4

−2

log³(η_R) := ˚E_R. Therefore, given any 0 < ϵ < ˚E_R^−α, there exists a unique 0 < δ < 1 such that

(38) ϵ^−1α = | log r|⁻²log³ η_R δ

 

equivalently, δ = ηRexp(−| log r|²³ϵ^−3α1

 . Therefore, (30) can be rewritten as follows:

(39) log |Y | ≤ C(ϵ^−α1 + | log r|²³ϵ^−3α1 ).

3.4. Proof of the main result.

Proof of Theorem 1.1. Fixing any auxiliary parameters R > 0 and α > 0. For each 0 < ϵ <

min{µβ, ˚E_R^−α, R, 1}, we can construct an ϵ-discrete Z as in Proposition 1with d = 1. Then, let δ be the number given in (38), and we can construct a _κ2^δ₊₁-net Y as in Proposition 4 such that (39) holds. Since 0 < ϵ < R, then Z ⊂ NR(∂Br). Therefore, Y is also a _κ2^δ₊₁-net of ((dΛ^nmBr(Z)), ∥ • ∥_X).

Now, we choose β = β(α, r, R) such that 1

8(µβ)^α1 > C| log r|²³ and µβ > ϵ.

Then it follows from (39), 0 < r < ¹₄, and 0 < ϵ < 1 that

log |Z| ≥ 1 4

 µβ ϵ

_α¹

> C(ϵ^−α1 + | log r|²³ϵ^−3α1 ) ≥ log |Y |.

Using the pigeonhole principle, there exist two dierent ψ1, ψ₂ ∈ Z such that

∥(dΛ^nm_Br(ψ₁) − y_nm)∥_X ≤ δ

κ₂+ 1 and ∥(dΛ^nmBr(ψ₂) − y_nm)∥_X ≤ δ κ₂ + 1 for some (ynm) ∈ Y. Letting ψ = ψ1 − ψ₂, we obtain that

∥(dΛ^nm_Br (ψ))∥_X ≤ 2δ

κ₂+ 1 = 1

κ₂+ 1C_Rexp(−| log r|²³ϵ^−3α1 ), which, with the help of Proposition 2, gives

(40) ∥dΛ_Br(ψ)∥∗ ≤ 1

κ₂ + 1C_Rexp(−| log r|²³ϵ^−3α1 ).

Finally, since Z is a ϵ-discrete set, ∥ψ∥L^∞(∂Br) ≥ ϵ and the proof is complete. □ Remark 6. One may choose

ϵ^−1α = log³ η_R δ

 

equivalently, δ = ηRexp(−ϵ^−3α1 )

 and take β suciently large such that

1

8(µβ)^α1 > C > C| log r|⁻² and µβ > ϵ.

Then it follows from (30) and 0 < ϵ < 1 that

log |Z| ≥ 1 4

 µβ ϵ

_α¹

> C(| log r|⁻²ϵ^−α1 + ϵ^−3α1 ) ≥ log |Y |.

Following the same argument as above, we then conclude that there exists ψ with ∥ψ∥L^∞ ≥ ϵ, but

(41) ∥dΛ_Br(ψ)∥∗ ≤ 1

κ₂+ 1C_Rexp(−ϵ^−3α1 ).

Comparing with (40), the estimate (41) is clearly not optimal.

4. Proof of Theorem 1.2

In this section, we want to prove Theorem 1.2. We will follow the lines in the proof of Theorem 1.1.

4.1. Spherical harmonics and series expansion. For each γ ∈ R, the Banach space H^γ(S²) can be equipped with the following equivalent norm:

∥A∥²_Hγ(S²) =X

n≥0

X

|m|≤n

(1 + n)^2γ|a_nm|² where A =X

n≥0

X

|m|≤n

a_nmY_n^m. The following proposition is simple but crucial in our work.

Proposition 5. Let s ∈ R, and we dene the following Banach space:

X_s :=

 (a_nm) ∥(a_nm)∥_Xs := sup

n≥0,|m|≤n

(1 + n)^32−s|a_nm| < ∞  . If A =X

n≥0

X

|m|≤n

a_nmY_n^m, then there exists an absolute constant Cabs > 0 such that

∥A∥_H^−s_(S²₎ ≤ Cabs∥(anm)∥Xs. Proof. Using direct computations, we have

∥A∥²_H−s(S²)=X

n≥0

X

|m|≤n

(1 + n)^−2s|anm|² =X

n≥0

X

|m|≤n

1

(1 + n)³(1 + n)^3−2s|anm|²

≤X

n≥0

X

|m|≤n

1

(1 + n)³∥(anm)∥²_Xs

=X

n≥0

2n + 1

(1 + n)³∥(anm)∥²_Xs, which proves the proposition with C_abs² =X

n≥0

2n + 1

(1 + n)³ < ∞. □

4.2. Some elementary computations. Recall from [CK19, Theorem 2.15 and 2.16] the following representation of u satisfying (9):

(42) u(x) =X

n≥0

X

|m|≤n



κiⁿ⁺¹u^∞_nmh⁽¹⁾_n (κr)



Y_n^m(ˆx) where u^∞ =X

n≥0

X

|m|≤n

u^∞_nmY_n^m(ˆx), In view of the boundary condition u = f on ∂B1, we have that

(43) f_nm = κiⁿ⁺¹u^∞_nmh⁽¹⁾_n (κ) with f(x) =X

n≥0

X

|m|≤n

f_nmY_n^m(ˆx),

where h⁽¹⁾n (t)is the spherical Hankel function. We can prove the following elementary lemma.

Lemma 4.1. Let κ > 0 and dene ˜κ := (^κ₂)^exp(κ). Then there exists a constant C > 0, which is independent of n and κ, such that

|h⁽¹⁾_n (κ)|⁻¹ ≤

(Cκ2⁻ⁿ if κ ≤ log(n), Cκ˜κ if κ ≥ log(n), for all n = 0, 1, 2, · · · .

Proof. From [Wat44, (4) (5), Sec. 9.61, p.297], it follows

|J_n+¹

2(κ)|²+ |J_−n−¹

2(κ)|² = 2

πκ(−1)ⁿR_2n,¹

2−n(κ)

= 2 πκ

n

X

m=0

(2κ)^2m−2n(2n − m)!(2n − 2m)!

((n − m)!)²m!

(44)

where Jν is the Bessel function of 1^stkind and Rn,ν is the Lommel polynomials. From [Wat44, (3), Sec. 3.61, p.74] or [JEL60, p.142], we have

N_n+¹

2(κ) = (−1)ⁿ⁺¹J_−n−¹

2(κ),