TRACKING ACTIVE TIME WINDOWS VIA NOVEL BOUNDARY-DECAY THEORY
13.2 Superalgebraic decay estimates of finite-time-history boundary densities This section extends the theoretical results of the previous section, establishing notThis section extends the theoretical results of the previous section, establishing not
only that the density 𝜓𝑘 is bounded in various Sobolev norms inR+ by the size of the density on some preceding finite subinterval of the time history as was shown in Theorem 4, but also through Theorem 5, that the Sobolev norms of the density on time intervals of the form [𝑇 ,∞), 𝑇 > 0, decay rapidly as 𝑇 → ∞. More precisely, given sufficient smoothness of the incident field, we show that the density
decays faster than any inverse power of𝑇. This result is crucial in view of the time- windowing methodology proposed in Chapter 2, since, in that context, there are many windows that contribute to the overall solution and whose computation is not desired over the entire space-time region of interest. In this context, the guarantees of Theorem 4 only provide for a uniform bound overR+, and the possible accumulation of errors from many such windows is problematic for rigorous guarantees of overall solution accuracy. In contrast, the Theorem 5 estimates on the interval[𝑇 ,∞),𝑇 > 0, that decay rapidly as𝑇 → ∞are of course greatly advantageous in connection with the algorithm presented in Section 12 (specifically, for Theorem 3), as they provide a termination criterion for computation of each one of the temporal densities 𝜓𝑘 arising from the temporal windowing procedure proposed in Chapter 2. Indeed, neglecting the contribution of many windows adds provably-small additional error, as can be seen by summing the geometric series present in the proof of Theorem 3.
Roughly speaking, Theorem 5 establishes that if the surface density is measured to be small for a certain period of time (approximately equal to the time it takes for a signal to traverse a distance equal to the diameter of the obstacle), then it not only remains small for all time thereafter, but, further, it decays superalgebraically fast starting from the small observed value. In contrast, previous works [57, 94–96]
claim exponential decay of the solution 𝑢 only relative to the total energy of the incident wave𝑏.
In a less algorithmic and a more theoretical direction, this analysis can be viewed in the context of the study of boundary integral density and local energy decay rates for the wave equation. The study of temporal decay rates of wave scattering of compactly-supported (finite energy) incident fields by bounded obstacles began in the𝑑 =3 case with reference [122] predicting exponential decay rates on compact subsets of the domainΩin the case of a spherical scatterer (though see also [65, Rem.
1]). These results were later extended to general “star-shaped” obstacles [95], with further generalization achieved in reference [96] to a wide class of “nontrapping”
obstacles (see also Remark 21). Most of the remaining results in this area can be found in references [16, 73, 74, 82, 83, 93, 94]—see also Remark 22 that discusses the results of references [73, 74], which together with reference [16] represent the only known local energy decay estimates for trapping structures.
There has been much less work concerning the decay of the densities themselves, although, in view of Theorems 1 through 3 in Section 12 of the present chapter and the preceding discussion concerning decay rates of the wave equation itself, such
results can be very useful with regards to both theory and numerical implemen- tation. In this connection, the author is only aware of reference [57], which uses techniques from classical scattering theory to show (only for nontrapping obstacles) that the integral equation operator𝐴𝜔 can be analytically continued to an invertible operator 𝐴𝜔,𝜎 on the uniform strip −𝜎 < Im(𝜔) ≤ 0 along the entire real 𝜔 axis, and it conjectures that the inverse operator 𝐴−1
𝜔,𝜎 satisfies in that strip the bound
𝐴−1 𝜔,𝜎
𝐿2(Γ)→𝐿2(Γ) ≤ 𝐶(𝜎) (1+ |𝜔|2)𝑞/2for some 𝑞. The claimed result on expo- nential decay in that reference rests entirely on the validity of this conjecture via a straightforward application of Cauchy’s theorem.
We proceed instead by using known results that establish𝑞-growth bounds of the form
𝐴−1
𝜔
𝐿2(Γ)→𝐿2(Γ) ≤ 𝐶(1+𝜔2)𝑞/2, 𝜔 ≥ 0,
forreal valuesof𝜔, under a variety of geometric contexts (cf. Remark 21), including
“parabolic” and “hyperbolic” trapping domains, nontrapping domains, etc. Utilizing these𝑞-growth bounds we establish, in Theorem 5, that, for smooth incident fields of compact temporal support, the scattering density (and, thus, the fields) decay superalgebraically fast (i.e. faster than any negative power of𝑡) as𝑡 grows. In fact, Theorem 5 provides a new avenue to the study of decay rates for wave scattering problems for new classes of obstacles which can be shown to satisfy the𝑞-growth condition given in Definition 2. In particular, we conclude that superalgebraically- fast decay takes place for all such obstacle types listed in Remark 21. As an example, in view of the result [37], which shows that parabolic trapping regions satisfy the𝑞- growth condition with𝑞=2 (or𝑞 =3—see also Remark 23), Theorem 5 shows that superalgebraically-fast decay takes place for parabolic trapping regions—for which no decay results were previously known. Examples are provided in Figure 4.1 of such trapping obstacles, for which no decay rates were hitherto established. Indeed, this is the first result showing decay rates for any kind of connected trapping obstacle (see Remark 23, and cf. the results of references [16, 73, 74] showing decay for classes of obstacle formed by certain finite unions of several obstacles, the only previously-established decay rates for a trapping obstacle of any kind). In contrast, our superalgebraic decay results are weaker than the exponential decay results presented for certain other obstacle types previously studied [73, 96], and they could be strengthened by extending the present results into the complex frequency domain, perhaps along the lines of [28, 29]. Indeed, it has been suggested [37, p. 855] that generalization from existing methods [47] could be utilized to pursue wave equation
Figure 4.1: Examples of connected trapping obstacles that satisfy a𝑞-growth con- dition (𝑞=3) and for which superalgebraically-fast wave equation time decay rates are established in this thesis. Left: Visualization of the obstacle given in Remark 23, and which serves to demonstrate the existence of connected trapping obstacles sat- isfying the𝑞-growth condition of Definition 2. Right: An elongated cavity trapping obstacle, with a vertical dimension of 12 units and rectangular dimensions of 4 units, that also satisfies a𝑞-growth condition (𝑞=3).
decay (and, indeed, possibly exponential decay).
We first state the main result, Theorem 5, and then outline the proof approach. It may be helpful to recall the definitions and assumptions made at the beginning of Section 13.1, which are assumed throughout this section; in particular note that per Equation (4.25) is 𝑘-dependent (though for notational ease this 𝑘-dependence is suppressed—see point 2 in Remark 20) and is supported in time in the bounded interval𝐼 defined in Equation (4.24).
Theorem 5. LetΓbe the boundary of an obstacle satisfying a𝑞-growth condition, and let 𝑛 and 𝑝 be given positive and nonnegative integers, respectively. Assume that the incident field 𝑏𝑘 satisfies 𝛾+𝑏𝑘 ∈ 𝐻𝑝+(𝑛+1) (𝑞+1)(R;𝐻1(Γ)) and 𝛾+𝜕n𝑏𝑘 ∈ 𝐻𝑝+𝑛(𝑞+1)+𝑞(R;𝐿2(Γ)). For arbitrary𝑇 >1, the density𝜓𝑘 can be bounded in the 𝐻𝑝( [𝑇 ,∞);𝐿2(Γ)) norm as
k𝜓𝑘k𝐻𝑝( [𝑇 ,∞)
;𝐿2(Γ)) ≤𝐶(Γ, 𝜏, 𝑝, 𝑛)𝑇1−𝑛k𝜓∗k𝐻𝑝+𝑛(𝑞+1)(𝐼;𝐿2(Γ)). (4.72) The choice of 𝑝 =1, additionally, allows the uniform estimate in time of
sup
𝑡 >𝑇
k𝜓𝑘(·, 𝑡) k𝐿2(Γ) ≤ 𝐶(Γ, 𝜏, 𝑛)𝑇1−𝑛k𝜓∗k𝐻𝑛(𝑞+1)+1(𝐼;𝐿2(Γ)). (4.73)
The proof of Theorem 5 proceeds by certain Fourier-shifting and convolution tech- niques and an overall approach based on integration-by-parts, and will require differ- entiation of the boundary integral density as a function of frequency𝜔. Estimating the resultant frequency-derivative terms in this approach necessitates establishing certain continuity results on the boundary integral density in frequency-domain, as well as new analysis on certain frequency-differentiated boundary integral operators.
Thus, Definition 5 introduces certain frequency-differentiated boundary integral op- erators, and, in Lemma 7, certain frequency-explicit norms for these operators are given. Lemma 9, in turn, guarantees that for smooth data, the frequency-domain integral equation solution is continuously differentiable as a function of frequency, and Lemma 10 provides pointwise bounds on such derivatives (for obstacles satis- fying a𝑞-growth condition). Finally, Lemma 11 is a technical lemma (of a similar character to some elements of the proof of Theorem 4), establishing bounds on the integrals of certain quantities that arise from the estimates in Lemma 10. The proof of Theorem 5 concludes the section.
A certain “time-history” technique used for the proof of Theorem 5 relies critically on use of certain finite-time boundary integral relations, according to which, the solution at any time future to a certain time point, can be obtained exactly, in absence of additional illumination, from the solution values in a finite time interval prior the given time point. Upon Fourier transformation, this approach allows us to leverage existing 𝑞-growth bounds on the inverse frequency-domain integral operators. The proof of decay then results by integration by parts in the frequency domain. Importantly, the technical Lemma 11 makes essential use of the bounded temporal support of the right-hand-side function ℎ(r, 𝑡), established in Lemma 5, in order to obtain an upper bound on certain powers of the temporal variable𝑡 that arise from frequency-differentiation. This is one of the main enabling steps in the overall decay proof, in that it allows for a study based purely on operator-norm estimates for real frequencies 𝜔 to establish decay rates for scattering problems.
As indicated in [37] alternative complex-variable approaches would require use of resolvent bounds for complex values of𝜔, which are not available at this time.
Before proceeding it is necessary to first define and establish properties of frequency- differentiated boundary integral operators.
Definition 5. For𝑚 ∈N, with reference to Definition1, define the operators
𝜕𝑚
𝜔𝐴𝜔 =
𝜕𝑚
𝜔𝐾∗
𝜔−i𝜕𝑚
𝜔𝑆𝜔, for 0≤ 𝜔 < 𝜔
0
𝜕𝑚
𝜔𝐾∗
𝜔−i𝑚 𝜕𝑚−1
𝜔 𝑆𝜔−i𝜔𝜕𝑚
𝜔𝑆𝜔, for 𝜔 > 𝜔
0,
(4.74)
where
𝜕𝑚
𝜔𝐾∗
𝜔
𝜇 (r) B
∫
Γ
𝜕𝑚
𝜔
𝜕 𝐺𝜔(r,r0)
𝜕n(r)
𝜇(r0)d𝜎(r0), r∈Γ, (4.75) and
𝜕𝑚
𝜔𝑆𝜔 𝜇
(r) B
∫
Γ
𝜕𝑚
𝜔𝐺𝜔(r,r0)
𝜇(r0)d𝜎(r0), r∈Γ. (4.76) Lemma 7. The boundary integral operator 𝐴𝜔 defined in Definition 1 can be continuously differentiated with respect to the parameter 𝜔 ≥ 0 at every 𝜔 ≠ 𝜔
0. That is, for𝑚 ∈Nand letting𝜔 =𝜔∗+Δ𝜔,Δ𝜔 ∈R,
Δ𝜔→lim0
𝜕𝑚−1
𝜔 𝐴𝜔−𝜕𝑚−1
𝜔 𝐴𝜔
∗
Δ𝜔
=𝜕𝑚
𝜔𝐴𝜔
∗, (4.77)
where 𝜕𝑖
𝜔𝐴𝜔 is as defined in Equation (4.74) (and 𝜕0
𝜔 = 𝐼) and all operators are understood in the sense of𝐿2(Γ). The operator𝜕𝑚
𝜔𝐴𝜔,𝑚 ∈N, satisfies the operator norm bound
k𝜕𝑚
𝜔𝐴𝜔k𝐿2(Γ)→𝐿2(Γ) ≤ 𝐶
1+𝐶
2𝜔 (4.78)
for all nonnegative 𝜔 ≠ 𝜔
0, where the finite constants 𝐶𝑗 = 𝐶𝑗(Γ, 𝑚) > 0 are 𝜔-independent.
Proof of Lemma7. We must show for an arbitrary fixed𝜔∗ ≥ 0, that for any 𝑓 ∈ 𝐿2(Γ)and𝑚 ∈N, the limit
Δ𝜔→lim0
𝜕𝑚−1
𝜔 𝐴𝜔−𝜕𝑚−1
𝜔 𝐴𝜔
∗
Δ𝜔
−𝜕𝑚
𝜔𝐴𝜔
∗
𝑓
𝐿2(Γ)
=0.
Assuming that the result holds for𝑚−1 and𝑚−2, we use induction on𝑚(the base case𝑚 =1 proceeds identically to below, except that terms with a (𝑚−1) factor are taken to be zero). Considering𝜔∗ > 𝜔
0,
Δlim𝜔→0
𝜕𝑚−1
𝜔 𝐴𝜔−𝜕𝑚−1
𝜔 𝐴𝜔
∗
Δ𝜔
−𝜕𝑚
𝜔𝐴𝜔
∗
𝑓
𝐿2(Γ)
= lim
Δ𝜔→0
∫
Γ
(𝜕𝑚−1
𝜔 𝐴𝜔𝑓) (r) − (𝜕𝑚−1 𝜔 𝐴𝜔
∗𝑓) (r) Δ𝜔
− (𝜕𝜔𝐴𝜔
∗𝑓) (r)
2
d𝜎(r)
Expanding the first integrand (the quotient) in this quantity reveals
Δlim𝜔→0
1 Δ𝜔
(𝜕𝑚−1
𝜔 𝐴𝜔𝑓) (r) − (𝜕𝑚−1
𝜔 𝐴𝜔
∗𝑓) (r)
=
Δlim𝜔→0
∫
Γ
1 Δ𝜔
𝜕𝑚−1
𝜔
𝜕 𝐺𝜔(r,r0)
𝜕n(r) −𝜕𝑚−1
𝜔
𝜕 𝐺𝜔
∗(r,r0)
𝜕n(r)
𝑓(r0)d𝜎(r0)
−i(𝑚−1)
∫
Γ
1 Δ𝜔
𝜕𝑚−2
𝜔 𝐺𝜔(r,r0) −𝜕𝑚−2
𝜔 𝐺𝜔
∗(r,r0)
𝑓(r0)d𝜎(r0)
−i 𝜔
Δ𝜔
∫
Γ
𝜕𝑚−1
𝜔 𝐺𝜔(r,r0)𝑓(r)d𝜎(r0)
−𝜔∗ Δ𝜔
∫
Γ
𝜕𝑚−1
𝜔 𝐺𝜔
∗(r,r0)𝑓(r)d𝜎(r0) , while the last two terms can be expressed as
Δlim𝜔→0
∫
Γ
𝜕𝑚−1
𝜔 𝐺𝜔(r,r0)𝑓(r0)d𝜎(r0) + lim
Δ𝜔→0
𝜔∗
∫
Γ
1 Δ𝜔
𝜕𝑚−1
𝜔 𝐺𝜔(r,r0) −𝜕𝑚−1
𝜔 𝐺𝜔
∗(r,r0)
𝑓(r0)d𝜎(r0). Using dominated convergence (the functions𝐺𝜔 are smooth with respect to𝜔, and thus every term in the preceding expressions can be bounded independently ofΔ𝜔), we obtain
Δlim𝜔→0
1 Δ𝜔
(𝜕𝑚−1
𝜔 𝐴𝜔𝑓) (r) − (𝜕𝑚−1
𝜔 𝐴𝜔
∗𝑓) (r)
=
∫
Γ
𝜕𝑚
𝜔
𝜕 𝐺𝜔
∗(r,r0)
𝜕n(r) −i(𝑚−1)𝜕𝑚−1
𝜔 𝐺𝜔
∗(r,r0)
𝑓(r)d𝜎(r0)
−i
∫
Γ
𝜕𝑚−1
𝜔 𝐺𝜔
∗(r,r0)𝑓(rd𝜎(r0) −i𝜔∗
∫
Γ
𝜕𝑚
𝜔𝐺𝜔(r,r0)𝑓(r)d𝜎(r0)
= (𝜕𝑚
𝜔𝐾∗
𝜔∗𝑓) (r) −i𝑚(𝜕𝑚−1
𝜔 𝑆𝜔
∗𝑓) (r) −i𝜔∗(𝜕𝑚
𝜔𝑆𝜔
∗𝑓) (r)
= (𝜕𝑚
𝜔𝐴𝜔
∗𝑓) (r). Thus as desired,
Δlim𝜔→0
𝜕𝑚−1
𝜔 𝐴𝜔−𝜕𝑚−1 𝜔 𝐴𝜔
∗
Δ𝜔
−𝜕𝑚
𝜔𝐴𝜔
∗
𝑓
𝐿2(Γ)
=0. The case 0≤ 𝜔∗ < 𝜔
0proceeds similarly, but is simpler due to the simpler form of definition of 𝐴𝜔 in that regime and is omitted for brevity.
We next show the frequency-explicit operator norm bound in (4.78). For 0 ≤ 𝜔 <
𝜔0, we have (first case in Equation (4.74))
𝜕𝑚
𝜔𝐴𝜔 =𝜕𝑚
𝜔𝐾∗
𝜔−i𝜕𝑚
𝜔𝑆𝜔,
and therefore, k𝜕𝑚
𝜔𝐴𝜔k𝐿2(Γ)→𝐿2(Γ) ≤ k𝜕𝑚
𝜔𝐾∗
𝜔k𝐿2(Γ)→𝐿2(Γ)+ k𝜕𝑚
𝜔𝑆𝜔k𝐿2(Γ)→𝐿2(Γ). (4.79) For𝜔 > 𝜔
0, we have (second case in Equation (4.74))
𝜕𝑚
𝜔𝐴𝜔 =𝜕𝑚
𝜔𝐾∗
𝜔−i𝑚 𝜕𝑚−1
𝜔 𝑆𝜔−i𝜔𝜕𝑚
𝜔𝑆𝜔, and therefore,
k𝜕𝑚
𝜔𝐴𝜔k𝐿2(Γ)→𝐿2(Γ) ≤ k𝜕𝑚
𝜔𝐾∗
𝜔k𝐿2(Γ)→𝐿2(Γ) +𝑚k𝜕𝑚−1
𝜔 𝑆𝜔k𝐿2(Γ)→𝐿2(Γ) +𝜔k𝜕𝑚
𝜔𝑆𝜔k𝐿2(Γ)→𝐿2(Γ).
(4.80) Clearly, the desired bound (4.78) follows immediately from bounds on the norms of the operators in (4.79) and (4.80) and specifically, that the boundary integral operator𝜕𝑚
𝜔𝐾∗
𝜔 defined by (4.75) satisfies k𝜕𝑚
𝜔𝐾∗
𝜔k𝐿2(Γ)→𝐿2(Γ) ≤
𝐶1𝜔, if m = 1, 𝐶1𝜔+𝐶
2, otherwise,
(4.81) while the boundary integral operator𝜕𝑚
𝜔𝑆𝜔 defined by (4.76) satisfies k𝜕𝑚
𝜔𝑆𝜔k𝐿2(Γ)→𝐿2(Γ) ≤ 𝐶
2. (4.82)
To prove that (4.81) and (4.82) hold, we extend (using the same proof techniques—
namely upper-bounds on the integral operator kernel in conjunction with the Riesz- Thorin interpolation theorem) the results of [36] beyond the 𝑚 = 0 case proved there, and consequently we assume𝑚 ≥ 1 below. More specifically, for an integral operator
𝑇 𝜇(r) =
∫
Γ
𝜅(r,r0)𝜇(r0)d𝜎(r0),
the Riesz-Thorin interpolation theorem [64, Thm. 1.3.4] guarantees that k𝑇k𝐿2(Γ)→𝐿2(Γ) ≤ k𝑇k1/2
𝐿1(Γ)→𝐿1(Γ)k𝑇k1/2
𝐿∞(Γ)→𝐿∞(Γ), where
k𝑇k𝐿1(Γ)→𝐿1(Γ) =ess supr0∈Γ
∫
Γ
|𝜅(r,r0) | d𝜎(r), and
k𝑇k𝐿∞(Γ)→𝐿∞(Γ) =ess supr∈Γ
∫
Γ
|𝜅(r,r0) | d𝜎(r0).
Further, if|𝜅(r,r0) | ≤𝜅˜(r,r0)for allr,r0∈Γand ˜𝜅(r,r0) =𝜅˜(r0,r), then clearly k𝑇k𝐿2(Γ)→𝐿2(Γ) ≤ ess supr∈Γ
∫
Γ
˜
𝜅(r,r0)d𝜎(r0). We first show the result for𝑇 =𝜕𝑚
𝜔𝐾∗
𝜔. A computation shows 𝜅(r,r0) =𝜕𝑚
𝜔
𝜕 𝐺𝜔(r,r0)
𝜕n(r) = (r−r0) ·𝜈(r) 4𝜋|r−r0|3 ei𝜔|r−r
0|(i|r−r0|)𝑚(i𝜔|r−r0| +𝑚−1).
Defining ˜𝜅𝑚(r,r0) = 1
4𝜋|r−r0|2−𝑚 (𝜔diam(Γ) +𝑚−1), we see that|𝜅(r,r0) | ≤ 𝜅˜𝑚(r,r0), and therefore
k𝜕𝑚
𝜔𝐾∗
𝜔k𝐿2(Γ)→𝐿2(Γ) ≤ ess supr∈Γ
∫
Γ
˜
𝜅𝑚(r,r0)d𝜎(r0)
≤
𝐶1𝜔, if m = 1, 𝐶1𝜔+𝐶
2, otherwise, providedΓis a Lipschitz boundary. This shows Equation (4.81).
Next, we consider𝑇 =𝜕𝑚
𝜔𝑆𝜔. Clearly, 𝜅(r,r0) =𝜕𝑚
𝜔𝐺𝜔(r,r0) =(i|r−r0|)𝑚ei𝜔|r−r
0|
|r−r0|.
Defining ˜𝜅𝑚(r,r0) =diam(Γ)𝑚−1, we see that|𝜅(r,r0) | ≤ 𝜅˜𝑚(r,r0), and therefore k𝜕𝑚
𝜔𝐾∗
𝜔k𝐿2(Γ)→𝐿2(Γ) ≤ ess supr∈Γ
∫
Γ
˜
𝜅𝑚(r,r0)d𝜎(r0) ≤𝐶
3.
This shows Equation (4.82).
Lemma 8(Ramm [105]). Consider the parametrized linear bounded operator 𝐴𝜔 : 𝐿2(Γ) → 𝐿2(Γ) and parametrized function 𝑓(𝜔) ∈ 𝐿2(Γ), as well as the equation for 𝜇(𝜔) ∈ 𝐿2(Γ),
𝐴𝜔𝜇(𝜔)= 𝑓(𝜔), (4.83)
for each𝜔 ∈Δ, whereΔ⊂ Ris an open bounded set. Assume that
1. Equation(4.83)is uniquely solvable for every𝜔 ∈Δ0= {𝜔 : |𝜔−𝜔
0| ≤𝑟}, for some𝑟 >0and𝜔
0 ∈Δ,Δ0⊂ Δ,
2. 𝑓(𝜔) is continuous with respect to𝜔 ∈Δ0,sup𝜔∈Δ0k𝑓(𝜔) k𝐿2(Γ) ≤ 𝑐
0,
3. limℎ→0sup𝜔∈Δ0,𝑣∈𝑀 k (𝐴𝜔+ℎ−𝐴𝜔)𝑣k𝐿2(Γ) =0, where𝑀is an arbitrary bounded subset of𝐿2(Γ),
4. and that sup𝜔∈Δ0, 𝑓∈𝑁
𝐴−1
𝜔 𝑓
𝐿2(Γ) ≤ 𝑐
1, where 𝑁 is an arbitrary bounded subset of𝐿2(Γ)and where𝑐
1may depend on𝑁. Then,
ℎ→lim0
k𝜇(𝜔+ℎ) −𝜇(𝜔) k𝐿2(Γ) =0. Lemma 9. Let 𝑅 ∈ 𝐶𝑚(R+ \𝜔
0; 𝐿2(Γ)), and let 𝜇 be the solution of the integral equation
(𝐴𝜔𝜇) (r, 𝜔) =𝑅(r, 𝜔), and defined for negative𝜔by Hermitian symmetry. Then
𝜇∈𝐶𝑚(R;𝐿2(Γ)), and there exist constants{𝑐𝑚
𝑖 , 𝑖 =1, . . . , 𝑚} such that for all𝜔 ∈R+\𝜔
0, 𝐴𝜔 𝜕𝑚
𝜔𝜇 (r, 𝜔) =𝜕𝑚
𝜔𝑅(r, 𝜔) −
𝑚
Õ
𝑖=1
𝑐𝑚
𝑖 𝜕𝑖
𝜔𝐴𝜔
𝜕𝑚−𝑖
𝜔 𝜇
(r, 𝜔), (4.84)
where equality is in the sense of 𝐿2(Γ).
Proof. Because 𝐴𝜔 is an invertible linear operator that is continuous and bounded as a function of the parameter 𝜔, and 𝑅 is a continuous map into 𝐿2(Γ) for all nonnegative𝜔 ≠ 𝜔
0, the conditions of Lemma 8 are met (for the operator 𝐴𝜔 and right-hand side𝑅), and by that lemma, it follows that𝜇is also a continuous function of𝜔. Thus, Equation (4.84) is trivially true for𝑚 =0. The argument then proceeds by induction in 𝑝, where it is assumed that for each 𝑝 ≤ 𝑠 (and𝑠 < 𝑚, since the induction must terminate at 𝑠+1 = 𝑚 depending on the regularity of 𝑅), 𝜇 ∈ 𝐶𝑝 and that (4.84) holds. We must show that the function 𝜇 = 𝜇(r, 𝜔) is 𝑠+1 times differentiable in𝜔, or, more precisely, that 𝜇 ∈ 𝐶𝑠+1(R;𝐿2(Γ)). We must further show that there exists {𝑐𝑠+1
𝑖 , 𝑖 = 1, . . . , 𝑠+1} so that (4.84) holds for 𝑠+1. (The spatialr-dependence of 𝜇is suppressed below for clarity, but all equality is in the sense of 𝐿2(Γ).)
For an arbitrary nonnegative 𝜔∗ ≠ 𝜔
0, let 𝜔 = 𝜔∗ +Δ𝜔, Δ𝜔 ∈ R. We wish to show that 𝜕𝑠
𝜔𝜇is differentiable at𝜔∗, and do this by showing a certain quotient of increments has a finite limit. By assumption,𝜇satisfies
𝐴𝜔𝜕𝑠
𝜔𝜇(𝜔) =𝜕𝑠
𝜔𝑅(𝜔) −
𝑠
Õ
𝑝=1
𝑐𝑠
𝑝(𝜕
𝑝 𝜔𝐴𝜔)𝜕
𝑠−𝑝 𝜔 𝜇(𝜔),
and
𝐴𝜔
∗𝜕𝑠
𝜔𝜇(𝜔∗) =𝜕𝑠
𝜔𝑅(𝜔∗) −
𝑠
Õ
𝑝=1
𝑐𝑠
𝑝(𝜕
𝑝 𝜔𝐴𝜔
∗)𝜕
𝑠−𝑝
𝜔 𝜇(𝜔∗). Subtracting the second equation from the first, adding and subtracting 𝐴𝜔
∗𝜕𝑠
𝜔𝜇(𝜔) on the left, and moving some terms to the right-hand side yields
𝐴𝜔
∗𝜕𝑠
𝜔𝜇(𝜔) −𝐴𝜔
∗𝜕𝑠
𝜔𝜇(𝜔∗) =𝜕𝑠
𝜔𝑅(𝜔) −𝜕𝑠
𝜔𝑅(𝜔∗)
−
𝑠
Õ
𝑝=1
𝑐𝑠
𝑝
(𝜕
𝑝
𝜔𝐴𝜔)𝜕𝜔𝑠−𝑝𝜇(𝜔) − (𝜕
𝑝 𝜔𝐴𝜔
∗)𝜕𝑠−
𝑝
𝜔 𝜇(𝜔∗)
− (𝐴𝜔− 𝐴𝜔
∗)𝜕𝑠
𝜔𝜇(𝜔). Considering the expression in brackets in the preceding equation, adding zero results in the identity
(𝜕
𝑝
𝜔𝐴𝜔)𝜕𝑠−
𝑝
𝜔 𝜇(𝜔) − (𝜕
𝑝 𝜔𝐴𝜔
∗)𝜕𝜔𝑠−𝑝𝜇(𝜔∗) =
𝜕
𝑝
𝜔𝐴𝜔−𝜕
𝑝 𝜔𝐴𝜔
∗
𝜕𝜔𝑠−𝑝𝜇(𝜔) −𝜕𝜔𝑠−𝑝𝜇(𝜔∗)
− 𝜕
𝑝
𝜔𝐴𝜔−𝜕
𝑝 𝜔𝐴
𝑝 𝜔∗
𝜕
𝑠−𝑝
𝜔 𝜇(𝜔∗) + (𝜕
𝑝 𝜔𝐴𝜔
∗) 𝜕
𝑠−𝑝
𝜔 𝜇(𝜔) −𝜕
𝑠−𝑝 𝜔 (𝜔∗)
, which we use to rewrite that same equation as
𝐴𝜔
∗𝜕𝑠
𝜔𝜇(𝜔) − 𝐴𝜔
∗𝜕𝑠
𝜔𝜇(𝜔∗) =𝜕𝑠
𝜔𝑅(𝜔) −𝜕𝑠
𝜔𝑅(𝜔∗)
−
𝑠
Õ
𝑝=1
𝑐𝑠
𝑝
𝜕
𝑝
𝜔𝐴𝜔−𝜕
𝑝 𝜔𝐴𝜔
∗
𝜕
𝑠−𝑝
𝜔 𝜇(𝜔) −𝜕
𝑠−𝑝
𝜔 𝜇(𝜔∗)
− 𝜕
𝑝
𝜔𝐴𝜔−𝜕
𝑝 𝜔𝐴
𝑝 𝜔∗
𝜕𝜔𝑠−𝑝𝜇(𝜔∗) + (𝜕
𝑝 𝜔𝐴𝜔
∗) 𝜕𝜔𝑠−𝑝𝜇(𝜔) −𝜕𝜔𝑠−𝑝𝜇(𝜔∗)
− (𝐴𝜔− 𝐴𝜔
∗)𝜕𝑠
𝜔𝜇(𝜔). Multiplying this equation by 𝐴−1
𝜔∗, we next show that the quotient of differences
Δ𝜔1 𝜕𝑠
𝜔𝜇(𝜔) −𝜕𝑠
𝜔𝜇(𝜔∗)
inherent in this equation has a finite limit for each 𝜔∗. Using the triangle inequality, we have
Δlim𝜔→0
1 Δ𝜔
𝜕𝑠
𝜔𝜇(𝜔) −𝜕𝑠
𝜔𝜇(𝜔∗)
− 𝐴−1
𝜔∗𝜕𝑠+1
𝜔 𝑅(𝜔∗)
−𝐴−1
𝜔∗ 𝑠+1
Õ
𝑝=1
𝑐𝑠+1
𝑝 (𝜕
𝑝
𝜔𝐴𝜔)𝜕𝜔𝑠+1−𝑝𝜇(𝜔∗) 𝐿2(Γ)
≤ lim
Δ𝜔→0
𝐴−1
𝜔∗
𝐿2(Γ)→𝐿2(Γ)
1 Δ𝜔
𝜕𝑠
𝜔𝑅(𝜔) −𝜕𝑠
𝜔𝑅(𝜔∗)
−𝜕𝑠+1
𝜔 𝑅(𝜔∗) 𝐿2(Γ)
+ lim
Δ𝜔→0
𝐴−1
𝜔∗
𝐿2(Γ)→𝐿2(Γ)
𝑠
Õ
𝑝=1
𝑐𝑠
𝑝
𝜕
𝑝
𝜔𝐴𝜔−𝜕
𝑝 𝜔𝐴𝜔
∗
Δ𝜔
𝜕
𝑠−𝑝
𝜔 𝜇(𝜔) −𝜕
𝑠−𝑝
𝜔 𝜇(𝜔∗)
− 𝜕
𝑝
𝜔𝐴𝜔−𝜕
𝑝 𝜔𝐴𝜔
∗
Δ𝜔
𝜕
𝑠−𝑝
𝜔 𝜇(𝜔∗) + 𝜕
𝑝 𝜔𝐴𝜔
∗
𝜕
𝑠−𝑝
𝜔 𝜇(𝜔) −𝜕
𝑠−𝑝 𝜔 𝜇(𝜔∗) Δ𝜔
+𝐴𝜔− 𝐴𝜔
∗
Δ𝜔
𝜕𝑠
𝜔𝜇(𝜔) +
𝑠+1
Õ
𝑝=1
𝑐𝑠+1
𝑝 (𝜕
𝑝 𝜔𝐴𝜔
∗)𝜕
𝑠+1−𝑝
𝜔 𝜇
𝐿2(Γ)
Since by hypothesis 𝑅 ∈ 𝐶𝑚 and 𝑠+1 ≤ 𝑚, the first limit on the right-hand side of this inequality vanishes. In the second limit expression, the first term in the sum also vanishes since firstly by Lemma 7,
𝜕
𝑝
𝜔𝐴𝜔−𝜕𝜔𝑝𝐴𝜔∗
Δ𝜔 → 𝜕𝑝+1
𝜔 𝐴𝜔
∗, while at the same time𝜕
𝑠−𝑝
𝜔 𝜇is continuous by the inductive hypothesis in order that the quantity k𝜕
𝑠−𝑝
𝜔 𝜇(𝜔) −𝜕
𝑠−𝑝
𝜔 𝜇(𝜔∗) k𝐿2(Γ) →0 asΔ𝜔 → 0. We thus have
Δlim𝜔→0
1 Δ𝜔
𝐴𝜔
∗𝜕𝑠
𝜔𝜇(𝜔) −𝐴𝜔
∗𝜕𝑠
𝜔𝜇(𝜔∗)
−𝜕𝑠+1
𝜔 𝑅(𝜔∗)
−
𝑠+1
Õ
𝑝=1
𝑐𝑠+1
𝑝 (𝜕
𝑝
𝜔𝐴𝜔)𝜕𝜔𝑠+1−𝑝𝜇(𝜔∗) 𝐿2(Γ)
≤ lim
Δ𝜔→0
𝐶
𝑠
Õ
𝑝=1
𝑐𝑠
𝑝
h (𝜕
𝑝 𝜔𝐴𝜔
∗)𝜕𝜔𝑠+1−𝑝𝜇(𝜔∗) − (𝜕𝜔𝑝+1𝐴𝜔
∗)𝜕𝜔𝑠−𝑝𝜇(𝜔∗)i +𝐴𝜔− 𝐴𝜔
∗
Δ𝜔
𝜕𝑠
𝜔𝜇(𝜔) +
𝑠+1
Õ
𝑝=1
𝑐𝑠+1
𝑝 (𝜕
𝑝 𝜔𝐴𝜔
∗)𝜕
𝑠+1−𝑝 𝜔 𝜇(𝜔∗)
𝐿2(Γ)
.
Using the identity 𝐴𝜔−𝐴𝜔
∗
Δ𝜔
𝜕𝑠
𝜔𝜇(𝜔) = 𝐴𝜔−𝐴𝜔
∗
Δ𝜔
𝜕𝑠
𝜔𝜇(𝜔) −𝜕𝑠
𝜔𝜇(𝜔∗)
+ 𝐴𝜔 −𝐴𝜔
∗
Δ𝜔
𝜕𝑠
𝜔𝜇(𝜔∗) and the triangle inequality we further have
Δlim𝜔→0
1 Δ𝜔
𝐴𝜔
∗𝜕𝑠
𝜔𝜇(𝜔) − 𝐴𝜔
∗𝜕𝑠
𝜔𝜇(𝜔∗)
−𝜕𝑠+1
𝜔 𝑅(𝜔∗)
−
𝑠+1
Õ
𝑝=1
𝑐𝑠+1
𝑝 (𝜕
𝑝 𝜔𝐴𝜔)𝜕
𝑠+1−𝑝 𝜔 𝜇(𝜔∗)
𝐿2(Γ)
≤ lim
Δ𝜔→0
𝐶
𝑠
Õ
𝑝=1
𝑐𝑠
𝑝
h (𝜕
𝑝 𝜔𝐴𝜔
∗)𝜕
𝑠+1−𝑝
𝜔 𝜇(𝜔∗) − (𝜕
𝑝+1 𝜔 𝐴𝜔
∗)𝜕
𝑠−𝑝
𝜔 𝜇(𝜔∗)i +𝐴𝜔− 𝐴𝜔
∗
Δ𝜔
𝜕𝑠
𝜔𝜇(𝜔∗) +
𝑠+1
Õ
𝑝=1
𝑐𝑠+1
𝑝 (𝜕
𝑝 𝜔𝐴𝜔
∗)𝜕𝜔𝑠+1−𝑝𝜇(𝜔∗) 𝐿2(Γ)
+ lim
Δ𝜔→0
𝐶
𝐴𝜔− 𝐴𝜔
∗
Δ𝜔
𝜕𝑠
𝜔𝜇(𝜔) −𝜕𝑠
𝜔𝜇(𝜔∗) 𝐿2(Γ)
.
The second limit on the right-hand side clearly vanishes since𝜇is𝑠-times continu- ously differentiable at𝜔∗, while, conversely, the limit of the expression
𝐴𝜔−𝐴𝜔∗
Δ𝜔 →
𝜕𝜔𝐴𝜔
∗. Finally, therefore, the limit obeys the inequality
Δlim𝜔→0
1 Δ𝜔
𝐴𝜔
∗𝜕𝑠
𝜔𝜇(𝜔) −𝐴𝜔
∗𝜕𝑠
𝜔𝜇(𝜔∗)
−𝜕𝑠+1
𝜔 𝑅(𝜔∗)
−
𝑠+1
Õ
𝑝=1
𝑐𝑠+1
𝑝 (𝜕
𝑝
𝜔𝐴𝜔)𝜕𝜔𝑠+1−𝑝𝜇(𝜔∗) 𝐿2(Γ)
≤𝐶
𝑠
Õ
𝑝=1
𝑐𝑠
𝑝
h (𝜕
𝑝 𝜔𝐴𝜔
∗)𝜕
𝑠+1−𝑝
𝜔 𝜇(𝜔∗) − (𝜕
𝑝+1 𝜔 𝐴𝜔
∗)𝜕
𝑠−𝑝 𝜔 𝜇(𝜔∗)
i
+(𝜕𝜔𝐴𝜔
∗)𝜕𝑠
𝜔𝜇(𝜔∗) +
𝑠+1
Õ
𝑝=1
𝑐𝑠+1
𝑝 (𝜕
𝑝 𝜔𝐴𝜔
∗)𝜕𝑠+1−
𝑝 𝜔 𝜇(𝜔∗)
𝐿2(Γ)
.
Making the selections 𝑐𝑠+1
1 = −1+𝑐𝑠
1, 𝑐𝑠+1
𝑝 = −𝑐𝑠
𝑝+𝑐𝑠
𝑝−1(𝑝 =2, . . . , 𝑠), 𝑐𝑠+1
𝑠+1 = 𝑐𝑠
𝑠
ensures that the right-hand side, a fixed quantity independent of Δ𝜔, vanishes.
This shows that𝜕𝑠
𝜔𝜇is differentiable and establishes that Equation (4.84) holds for arbitrary𝜔 =𝜔∗. Continuity of𝜕𝑠+1
𝜔 𝜇follows, as in the case for𝜇, by application of the result of Lemma 8 since the right-hand-side of (4.84) is continuous by the inductive hypothesis, and, as previously mentioned, the operator 𝐴𝜔 satisfies the
required conditions for that lemma.
Lemma 10. Assume the obstacle satisfies a𝑞-growth condition. Let 𝑅 ∈𝐶𝑚(R+ \ 𝜔0;𝐿2(Γ)), and let𝜇be the solution of the integral equation
(𝐴𝜔𝜇) (r, 𝜔) =𝑅(r, 𝜔),
for each𝜔 ≥ 0and defined for negative𝜔by Hermitian symmetry. Then there exist
coefficients𝑏𝑚
𝑖 𝑗 > 0and𝑐𝑚
𝑖 > 0such that for𝜔 ≠±𝜔
0, we have
𝜕𝑚
𝜔𝜇(·, 𝜔) 𝐿2(Γ) ≤
𝑚−1
Õ
𝑖=0
©
«
(𝑖+1) (𝑞+1)−1
Õ
𝑗=0
𝑏𝑚
𝑖 𝑗|𝜔|𝑗 𝜕𝑚−𝑖
𝜔 𝑅(·,|𝜔|) 𝐿2(Γ)ª
®
¬ +
𝑚(𝑞+1)
Õ
𝑖=0
𝑐𝑚
𝑖 |𝜔|𝑖k𝜇(·, 𝜔) k𝐿2(Γ),
(4.85)
and coefficients𝑑𝑚
𝑖 𝑗 > 0and𝑒𝑚
𝑖 >0so that for𝜔 ≠ ±𝜔
0, we have
𝜕𝑚
𝜔𝜇(·, 𝜔) 2
𝐿2(Γ) ≤
𝑚−1
Õ
𝑖=0
©
«
(𝑖+1) (𝑞+1)−1
Õ
𝑗=0
𝑑𝑚
𝑖 𝑗𝜔2𝑗 𝜕𝑚−𝑖
𝜔 𝑅(·,|𝜔|) 2
𝐿2(Γ)
ª
®
¬ +
𝑚(𝑞+1)
Õ
𝑖=0
𝑒𝑚
𝑖 𝜔2𝑖k𝜇(·, 𝜔) k2
𝐿2(Γ).
(4.86)
Proof. By Lemma 9, 𝜇 ∈ 𝐶𝑚(R; 𝐿2(Γ)) so for 𝜔 ≠ ±𝜔
0 all quantities in the inequalities are well-defined, and there exist 𝑎𝑠+1
𝑘 such that for 𝜔 ∈ R+ \ 𝜔
0 the equation
𝜕𝑠+1
𝜔 𝜇
(r, 𝜔) = 𝐴−1
𝜔 𝜕𝑠+1
𝜔 𝑅(r, 𝜔) −
𝑠+1
Õ
𝑘=1
𝑎𝑠+1
𝑘 (𝜕𝑘
𝜔𝐴𝜔) (𝜕𝑠+1−𝑘
𝜔 𝜇) (r, 𝜔)
!
holds in𝐿2(Γ). To prove the bound (4.85), assume the result holds for𝑚 ≤ 𝑠, and consider 𝑚 = 𝑠+ 1 (the base case 𝑚 = 0 is trivially satisfied as the first sum in the inequality is dropped). Using Definition 2 of the𝑞-growth condition as well as operator norms from Lemma 7 (Equation (4.78)), there exist positive𝐶
1, 𝐶
2, 𝛼𝑘
0, and𝛼𝑘
1such that for all𝜔 ∈R+\𝜔
0, the inequalities 𝐴−1
𝜔
𝐿2(Γ)→𝐿2(Γ) ≤ 𝐶
1+𝐶
2𝜔𝑞 and
𝜕𝑘
𝜔𝐴𝜔
𝐿2(Γ)→𝐿2(Γ) ≤ 𝛼𝑘
0+𝛼𝑘
1𝜔 hold, and therefore we obtain for𝜔 ≠±𝜔
0,
𝜕𝑠+1
𝜔 𝜇(·, 𝜔)
𝐿2(Γ) ≤ (𝐶
1+𝐶
2|𝜔|𝑞) 𝜕𝑠+1
𝜔 𝑅(·, 𝜔) 𝐿2(Γ)
+
𝑠+1
Õ
𝑘=1
|𝑎𝑠+1
𝑘 | (𝛼𝑘
0+𝛼𝑘
1|𝜔|) 𝜕𝑠+1−𝑘
𝜔 𝜇(·, 𝜔) 𝐿2(Γ)
! .
(4.87)
Substituting in the result of the inductive hypothesis (that is, that Equation (4.85)
holds for𝑚 ≤ 𝑠),
𝜕𝑠+1
𝜔 𝜇(·, 𝜔)
𝐿2(Γ) ≤ (𝐶
1+𝐶
2|𝜔|𝑞) 𝜕𝑠+1
𝜔 𝑅(·, 𝜔) 𝐿2(Γ)
+
𝑠+1
Õ
𝑘=1
|𝑎𝑠+1
𝑘 | 𝐶1𝛼𝑘
0+𝐶
2𝛼𝑘
0|𝜔|𝑞+𝐶
1𝛼𝑘
1|𝜔| +𝐶
2𝛼𝑘
1|𝜔|𝑞+1
·
·
𝑠−𝑘
Õ
𝑖=0
(𝑖+1) (𝑞+1)−1
Õ
𝑗=0
𝑏𝑠+1−𝑘
𝑖 𝑗 |𝜔|𝑗
𝜕𝑠+1−𝑘−𝑖
𝜔 𝑅(·, 𝜔) 𝐿2(Γ)
+
(𝑠+1−𝑘) (𝑞+1)
Õ
𝑖=0
𝑐𝑠+1−𝑘
𝑖 |𝜔|𝑖k𝜇(·, 𝜔) k𝐿2(Γ)
# ,
and then, expanding the products results in the inequality
𝜕𝑠+1
𝜔 𝜇(·, 𝜔)
𝐿2(Γ) ≤ (𝐶
1+𝐶
2|𝜔|𝑞) 𝜕𝑠+1
𝜔 𝑅(·, 𝜔) 𝐿2(Γ)
+
𝑠+1
Õ
𝑘=1 𝑠−𝑘
Õ
𝑖=0
(𝑖+1) (𝑞+1)−1
Õ
𝑗=0
|𝑎𝑠+1
𝑘 |𝑏𝑠+1−𝑘
𝑖 𝑗
𝐶1𝛼𝑘
0|𝜔|𝑗+𝐶
2𝛼𝑘
0|𝜔|𝑞+𝑗 +𝐶
1𝛼𝑘
1|𝜔|𝑗+1+𝐶
2𝛼𝑘
1|𝜔|𝑞+𝑗+1
𝜕𝑠+1−𝑘−𝑖
𝜔 𝑅(·, 𝜔) 𝐿2(Γ)
+
𝑠+1
Õ
𝑘=1
(𝑠+1−𝑘) (𝑞+1)
Õ
𝑖=0
|𝑎𝑠+1
𝑘 |𝑐𝑠+1−𝑘
𝑖
𝐶1𝛼𝑘
0|𝜔|𝑖+𝐶
2𝛼𝑘
0|𝜔|𝑞+𝑖 +𝐶
1𝛼𝑘
1|𝜔|𝑖+1+𝐶
2𝛼𝑘
1|𝜔|𝑞+𝑖+1
k𝜇(·, 𝜔) k𝐿2(Γ).
Considering this final inequality, it can be seen that the maximal power of|𝜔|in the first sum-term expression is at the indices 𝑘 =1,𝑖 =𝑠−1 and 𝑗 =𝑠(𝑞+1) −1, for which the term present in the above inequality is|𝜔|𝑞+(𝑠(𝑞+1)−1)+1k𝜕𝜔𝑅(·, 𝜔) k𝐿2(Γ) =
|𝜔|(𝑠+1) (𝑞+1)−1k𝜕𝜔𝑅(·, 𝜔) k𝐿2(Γ), and which is found in (4.85) for𝑚 = 𝑠+1. Sim- ilarly, the maximal power of |𝜔| in the second sum-term expression is at the in- dices 𝑘 = 1 and 𝑖 = 𝑠(𝑞 +1), for which the term in the above inequality above equals |𝜔|(𝑠+1) (𝑞+1) k𝜇(·, 𝜔) k𝐿2(Γ), which is also present in (4.85) for 𝑚 = 𝑠+ 1.
Since inspection of this final inequality shows that there is no term of the form
|𝜔|𝑗
𝜕𝑠+1−𝑘−𝑖
𝜔 𝑅(·, 𝜔)
𝐿2(Γ) or |𝜔|𝑖k𝜇(·, 𝜔) k𝐿2(Γ) that is not also present in Equa- tion (4.85) for𝑚 =𝑠+1, inequality (4.85) is established.
The inequality (4.86) follows immediately from (4.85) using the formula Í𝑚
𝑖=1 𝑓𝑖 2 ≤ 𝑚Í𝑚
𝑖=1k𝑓𝑖k2.
Lemma 11. For𝜔 ≥ 0let
𝑅(r, 𝜔) = 𝛾−𝜕n−i𝜂𝛾+
𝐻𝑡(r, 𝜔), r∈Γ,
where 𝐻𝑡 is defined by Equation (4.40) and 𝜂 is as defined in Definition 1, and assume 𝑅 ∈ 𝐶𝑚(R+ \𝜔
0; 𝐿2(Γ)) and𝜓∗ ∈ 𝐻𝑛+1(𝐼;𝐿2(Γ)) for some𝑛 > 0. Then for0≤ 𝑖 ≤ 𝑚and0≤ 𝑗 ≤ 𝑛
∫ ∞ 0
𝜔2𝑗 𝜕𝑚−𝑖
𝜔 𝑅(·, 𝜔) 2
𝐿2(Γ) d𝜔 ≤ 𝐶k𝜓∗k2
𝐻𝑗+1(𝐼;𝐿2(Γ)), where𝐶 is independent of𝜓.
Proof. We have
∫ ∞ 0
𝜔2𝑗 𝜕𝑚−𝑖
𝜔 𝑅(·, 𝜔) 2
𝐿2(Γ) d𝜔=
∫
Γ
∫ ∞ 0
b𝑆𝑖 𝑗 𝑚𝐻𝑡(r, 𝜔)
2
d𝜔d𝜎(r), (4.88) where the operator ˆ𝑆𝑖 𝑗 𝑚 is defined as
b𝑆𝑖 𝑗 𝑚 =𝜔𝑗𝜕𝑚−𝑖
𝜔 𝛾−𝜕n−i𝜂𝛾+ . Now,𝜂 is defined piecewise as𝜂 =𝜔 for𝜔 > 𝜔
0 and𝜂 =1 for 0 ≤ 𝜔 < 𝜔
0. The argument requires certain estimates to be made in time-domain, and, for that reason, it becomes useful to consider the splitting of (4.88)
∫ ∞ 0
𝜔2𝑗 𝜕𝑚−𝑖
𝜔 𝑅(·, 𝜔) 2
𝐿2(Γ) d𝜔
=
∫
Γ
∫ 𝜔
0
0
+
∫ ∞ 𝜔0
𝑆b𝑖 𝑗 𝑚𝐻𝑡(r, 𝜔)
2
d𝜔d𝜎(r).
Then, with a view to matchingb𝑆𝑖 𝑗 𝑚 on each integration region, define the operators with Fourier symbols
𝑆b1
𝑖 𝑗 𝑚 =𝜔𝑗𝜕𝑚−𝑖
𝜔 (𝛾−𝜕n−i𝜔𝛾+), and 𝑆b2
𝑖 𝑗 𝑚 =𝜔𝑗𝜕𝑚−𝑖
𝜔 (𝛾−𝜕n−i𝛾+) and which are in time-domain
𝑆1
𝑖 𝑗 𝑚 =(i
𝜕
𝜕 𝑡
)𝑗(𝑖𝑡)𝑚−𝑖(𝛾−𝜕n+ 𝜕
𝜕 𝑡
𝛾+), and 𝑆2
𝑖 𝑗 𝑚 =(i
𝜕
𝜕 𝑡
)𝑗(𝑖𝑡)𝑚−𝑖(𝛾−𝜕n−i𝛾+). These definitions are used in conjunction with Plancherel’s theorem to observe that
∫ ∞ 0
𝜔2𝑗 𝜕𝑚−𝑖
𝜔 𝑅(·, 𝜔) 2
𝐿2(Γ) d𝜔
≤
∫
Γ
∫ ∞
−∞
b𝑆1
𝑖 𝑗 𝑚𝐻𝑡(r, 𝜔)
2
d𝜔d𝜎(r) +
∫
Γ
∫ ∞
−∞
b𝑆2
𝑖 𝑗 𝑚𝐻𝑡(r, 𝜔)
2
d𝜔d𝜎(r)
=
∫
Γ
∫ ∞
−∞
𝑆1
𝑖 𝑗 𝑚ℎ(r, 𝑡)
2
d𝑡d𝜎(r) +
∫
Γ
∫ ∞
−∞
𝑆2
𝑖 𝑗 𝑚ℎ(r, 𝑡)
2
d𝑡d𝜎(r)
=
∫
Γ
∫ 𝑇∗
−𝜏
𝑆1
𝑖 𝑗 𝑚𝑢∗(r, 𝑡)
2
d𝑡d𝜎(r) +
∫
Γ
∫ 𝑇∗
−𝜏
𝑆2
𝑖 𝑗 𝑚𝑢∗(r, 𝑡)
2
d𝑡d𝜎(r),
where the last equality follows by using the result of Lemma 5, that onΓthe function ℎis temporally supported in [−𝜏, 𝑇∗] and is equal to𝑢∗ onΓ× [−𝜏, 𝑇∗]. Defining for clarity the functions ˜𝑢
1= (𝛾−𝜕n+ 𝜕
𝜕 𝑡𝛾+)𝑢∗and ˜𝑢
2 =(𝛾−𝜕n−i𝛾+)𝑢∗, we have by the Leibniz product rule
∫ ∞ 0
𝜔2𝑗 𝜕𝑚−𝑖
𝜔 𝑅(·, 𝜔) 2
𝐿2(Γ) d𝜔
≤
∫
Γ
∫ 𝑇∗
−𝜏
(i
𝜕
𝜕 𝑡
)𝑗(𝑖𝑡)𝑚−𝑖𝑢˜
1(r, 𝑡)
2
d𝑡d𝜎(r) +
∫
Γ
∫ 𝑇∗
−𝜏
(i
𝜕
𝜕 𝑡
)𝑗(𝑖𝑡)𝑚−𝑖𝑢˜
2(r, 𝑡)
2
d𝑡d𝜎(r)
=
∫
Γ
∫ 𝑇∗
−𝜏
𝑗
Õ
ℓ=0
𝑎ℓ 𝜕ℓ
𝜕 𝑡ℓ
(𝑖𝑡)𝑚−𝑖 𝜕𝑗−ℓ
𝜕 𝑡𝑗−ℓ𝑢˜
1(r, 𝑡)
2
d𝑡d𝜎(r)
+
∫
Γ
∫ 𝑇∗
−𝜏
𝑗
Õ
ℓ=0
𝑎ℓ 𝜕ℓ
𝜕 𝑡ℓ
(i𝑡)𝑚−𝑖 𝜕𝑗−ℓ
𝜕 𝑡𝑗−ℓ𝑢˜
2(r, 𝑡)
2
d𝑡d𝜎(r). Considering the derivative 𝜕
ℓ
𝜕 𝑡ℓ(i𝑡)𝑚−𝑖in these expressions, we further have
∫ ∞ 0
𝜔2𝑗 𝜕𝑚−𝑖
𝜔 𝑅(·, 𝜔) 2
𝐿2(Γ) d𝜔
=
∫
Γ
∫ 𝑇∗
−𝜏
𝑗
Õ
ℓ=0
˜ 𝑎ℓ
𝑖𝑚−𝑖𝑡𝑚−𝑖−ℓ
𝜕𝑗−ℓ
𝜕 𝑡𝑗−ℓ𝑢˜
1(r, 𝑡)
2
d𝑡d𝜎(r)
+
∫
Γ
∫ 𝑇∗
−𝜏
𝑗
Õ
ℓ=0
˜ 𝑎ℓ
𝑖𝑚−𝑖𝑡𝑚−𝑖−ℓ
𝜕𝑗−ℓ
𝜕 𝑡𝑗−ℓ𝑢˜
2(r, 𝑡)
2
d𝑡d𝜎(r), where ˜𝑎ℓ = (𝑚−𝑖−ℓ)(𝑚−𝑖)!
!𝑎ℓ for 𝑚−𝑖−ℓ ≥ 0 and ˜𝑎ℓ = 0 for𝑚−𝑖−ℓ < 0. Since the 𝑡-integration region is limited to the bounded region [−𝜏, 𝑇∗] the factors𝑡𝑚−𝑖−ℓ can be bounded above by a constant, and thus
∫ ∞ 0
𝜔2𝑗 𝜕𝑚−𝑖
𝜔 𝑅(·, 𝜔) 2
𝐿2(Γ) d𝜔
≤ 𝐶
1 𝑗
Õ
ℓ=0
∫
Γ
∫ 𝑇∗
−𝜏
𝜕ℓ
𝜕 𝑡ℓ𝑢˜
1(r, 𝑡)
2
+
𝜕ℓ
𝜕 𝑡ℓ𝑢˜
2(r, 𝑡)
2!
d𝑡d𝜎(r)
≤ 𝐶
1 𝑗
Õ
ℓ=0
∫
Γ
∫ ∞
−∞
𝜕ℓ
𝜕 𝑡ℓ𝑢˜
1(r, 𝑡)
2
+
𝜕ℓ
𝜕 𝑡ℓ𝑢˜
2(r, 𝑡)
2!
d𝑡d𝜎(r),
where the last inequality estimates above the𝐿2norm on the finite region[−𝜏, 𝑇∗]by the full𝐿2(R)norm. Recalling the definitions of ˜𝑢
1and ˜𝑢
2as ˜𝑢
1= (𝛾−𝜕n+ 𝜕
𝜕 𝑡𝛾+)𝑢∗
and ˜𝑢
2= (𝛾−𝜕n−i𝛾+)𝑢∗, we thus continue in the frequency domain and estimate,
∫ ∞ 0
𝜔2𝑗 𝜕𝑚−𝑖
𝜔 𝑅(·, 𝜔) 2
𝐿2(Γ) d𝜔
≤ 𝐶
1 𝑗
Õ
ℓ=0
∫
Γ
∫ ∞
−∞
𝜔ℓ(𝛾−𝜕n−i𝜔𝛾+)𝑈∗𝑡(r, 𝜔)
2 d𝜔d𝜎(r) +
∫
Γ
∫ ∞
−∞
𝜔ℓ(𝛾−𝜕n−i𝛾+)𝑈∗𝑡(r, 𝜔)
2 d𝜔d𝜎(r)
≤ 𝐶
1
∫
Γ
∫ ∞
−∞
(1+𝜔2)𝑗/2(𝛾−𝜕n −i𝜔𝛾+)𝑈∗𝑡(r, 𝜔)
2
d𝜔d𝜎(r) +𝐶
1
∫
Γ
∫ ∞
−∞
(1+𝜔2)𝑗/2(𝛾−𝜕n−i𝛾+)𝑈∗𝑡(r, 𝜔)
2
d𝜔d𝜎(r).
We thus have shown the estimate
∫ ∞ 0
𝜔2𝑗 𝜕𝑚−𝑖
𝜔 𝑅(·, 𝜔) 2
𝐿2(Γ) d𝜔 ≤ 𝐶
1
∫ ∞
−∞
(1+𝜔2)𝑗 (𝜕−
n −i𝜔)𝑈∗𝑡(·, 𝜔) 2
𝐿2(Γ) d𝜔 +𝐶
1
∫ ∞
−∞
(1+𝜔2)𝑗
(𝛾−𝜕n −i𝛾+)𝑈∗𝑡(·, 𝜔) 2
𝐿2(Γ) d𝜔.
(4.89) By Lemma 6, the frequency-wise operator bounds
(𝛾−𝜕n−i𝜔𝛾+)𝑈∗𝑡(·, 𝜔)
𝐿2(Γ) ≤ 𝐷(1+𝜔2)1/2
𝜓𝑡∗(·, 𝜔)
𝐿2(Γ) (4.90) and
(𝛾−𝜕n−i𝛾+)𝑈∗𝑡(·, 𝜔)
𝐿2(Γ) ≤ 𝐸(1+𝜔2)1/2
𝜓∗𝑡(·, 𝜔)
𝐿2(Γ), (4.91) hold for some𝐷 , 𝐸 > 0 independent of𝜔and𝜓. Using Equations (4.90) and (4.91) in Equation (4.89), we conclude
∫ ∞ 0
𝜔2𝑗
𝜕𝜔𝑞−𝑖𝑅(·, 𝜔)
2
𝐿2(Γ) d𝜔
≤
∫ ∞
−∞
𝐶2(1+𝜔2)𝑗+1
𝜓∗𝑡(·, 𝜔) 2
𝐿2(Γ) d𝜔 +
∫ ∞
−∞
𝐶3(1+𝜔2)𝑗+1
𝜓𝑡∗(·, 𝜔) 2
𝐿2(Γ) d𝜔.
≤𝐶 𝜓𝑡∗
𝐻𝑗+1(R;𝐿2(Γ)) =𝐶k𝜓∗k𝐻𝑗+1(𝐼;𝐿2(Γ)).
With Lemmas 7 through 11 having been established, it is now possible to conclude with the proof of Theorem 5.
Proof of Theorem5. Define𝑤(𝑡), a nonnegative bounded𝐶∞
𝑐 (R)function,𝑤(𝑡) ≤ 1, satisfying (for given window width parameters𝜏and𝑇𝑤),
𝑤(𝑡) =
0 if𝑡 <−𝜏 1 if 0 ≤ 𝑡 ≤𝑇𝑤 0 if𝑡 > 𝑇𝑤+𝜏 .
(4.92)
Define also𝑤𝑇(𝑡) =𝑤(𝑡−𝑇).
In order to study the decay of k𝜓+(·, 𝑡) k𝐿2(Γ) on𝑡 > 𝑇, we consider first the bound k𝜓𝑘k2
𝐿2( [𝑇 ,𝑇+𝑇𝑤);𝐿2(Γ)) =
∫ 𝑇+𝑇𝑤
𝑇
k𝜓+(r, 𝑡) k2
𝐿2(Γ)d𝑡
≤
∫ ∞
−∞
k𝑤𝑇(𝑡)𝜓+(·, 𝑡) k2
𝐿2(Γ)d𝑡 = k𝑤𝑇𝜓+k2
𝐿2(R;𝐿2(Γ)), (4.93) and show that𝜓𝑘 in this norm decays as𝑇 → ∞. The result on [𝑇 ,∞) then follows by summing an infinite number of the norms of the desired quantity over such bounded-intervals.
The function (𝑤𝑇𝜓+) (r, 𝑡)can be written using the convolution theorem as (𝑤𝑇𝜓+) (r, 𝑡) =
∫ ∞
−∞
ˆ
𝑤𝑇 ∗𝜓+𝑡
(r, 𝜔)e−𝑖𝜔𝑡d𝜔. (4.94) But, in view of the definition of𝑤 in Equation (4.92) and the subsequent definition of𝑤𝑇, we have ˆ𝑤𝑇 =ei𝜔𝑇𝑤ˆ(𝜔), and, thus, integration by parts yields
ˆ
𝑤𝑇 ∗𝜓𝑡+
(r, 𝜔) =
∫ ∞
−∞
e𝑖 𝜏𝑇𝑤ˆ(𝜏)𝜓+𝑡(r, 𝜔−𝜏)𝑑 𝜏
=−
∫ ∞
−∞
1
𝑖𝑇e𝑖 𝜏𝑇 𝑤ˆ0(𝜏)𝜓+𝑡(r, 𝜔−𝜏) −𝑤ˆ(𝜏)𝜕𝜔𝜓𝑡+(r, 𝜔−𝜏) 𝑑 𝜏, (4.95) where the boundary terms at ±∞ in the integration by parts calculation vanish because |𝜓𝑡+| → 0 as |𝜔| → ∞. This can be repeated given sufficient smoothness of𝜓𝑡+(r,·). Indeed, since𝜓+𝑡 ∈𝐶𝑛by Lemma 9,
ˆ
𝑤𝑇 ∗𝜓𝑡+
(r, 𝜔) =
− 1 𝑖𝑇
𝑛∫ ∞
−∞
e𝑖 𝜏𝑇
𝑛
Õ
𝑚=0
𝑎𝑖(𝜕𝑛−𝑚𝑤ˆ(𝜏)) 𝜕𝑚
𝜔𝜓+𝑡(r, 𝜔−𝜏)
! 𝑑 𝜏, (4.96)
where all boundary terms vanish since |𝜕𝑚
𝜔𝜓𝑡+| → 0 as |𝜔| → ∞ for all 𝑚 < 𝑛. The vanishing of such boundary terms results from Lemma 10, which ensures that for 𝑚 < 𝑛, the limit of
𝜕𝑚
𝜔𝜓𝑡+(·, 𝜔)
𝐿2(Γ) → 0 as |𝜔| → ∞ if the limit
|𝜔|𝑚(𝑞+1)
𝜓𝑡+(·, 𝜔)
𝐿2(Γ)) → 0—a fact satisfied for𝜓+ ∈ 𝐻𝑚(𝑞+1)(R;𝐿2(Γ))which is itself ensured by Lemma 1 together with the hypothesis of the present theorem that𝛾+𝑏𝑘 ∈ 𝐻(𝑛+1) (𝑞+1)(R;𝐿2(Γ))and𝛾+𝜕n𝑏𝑘 ∈ 𝐻𝑛(𝑞+1)+𝑞(R;𝐿2(Γ)).
Therefore, using Plancherel’s theorem, k𝑤𝑇𝜓+k2
𝐿2(R;𝐿2(Γ)) =
(𝑤𝑇𝜓+)
2
𝐿2(R;𝐿2(Γ)) =
𝑤ˆ𝑇 ∗𝜓𝑡+ 2
𝐿2(R;𝐿2(Γ))
=
∫ ∞
−∞
∫
Γ
𝑤ˆ𝑇 ∗𝜓𝑡+
(r, 𝜔)
2 d𝜎(r)d𝜔
=
∫
Γ
𝑤ˆ𝑇 ∗𝜓+𝑡(r,·) 2
𝐿2 d𝜎(r).
≤
∫
Γ
(𝑛+1)𝑇−2𝑛
𝑛
Õ
𝑚=0
𝑎𝑚
(ei𝑇·𝜕𝑛−𝑚𝑤ˆ) ∗ (𝜕𝑚
𝜔𝜓𝑡+) (r,·)
2
𝐿2 d𝜎(r), where we used the fact that kÍ𝑛
𝑖=1 𝑓𝑖k2 ≤ 𝑛Í𝑛
𝑖=1k𝑓𝑖k2. Note the presence of ei𝑇· in the final convolution expression above, arising from the ei𝜏𝑇 in Equation (4.96);
this term has unit absolute value and so is irrelevant to the 𝐿1estimates that follow.
Indeed, because of the fact that ei𝜏𝑇𝜕𝑛−𝑚
𝜔 𝑤ˆ(𝜏) is an element of 𝐿1(R) (and with a norm value independent of𝑇), application of Young’s inequality yields a bound on the temporal 𝐿2 norm of the convolution in terms of ei𝑇·𝜕𝑛−𝑚𝑤ˆ ∈ 𝐿1(R) and
𝜕𝑚𝜓𝑡+(r,·) ∈ 𝐿2(R), k𝑤𝑇𝜓+k2
𝐿2(R;𝐿2(Γ)) ≤
∫
Γ
(𝑛+1)𝑇−2𝑛
𝑛
Õ
𝑚=0
|𝑎𝑖|2k𝜕𝑛−𝑚𝑤ˆk2
𝐿1
𝜕𝑚
𝜔𝜓+𝑡(r,·) 2
𝐿2 d𝜎(r)
≤ 𝐶(𝑛, 𝜏, 𝑇𝑤)𝑇−2𝑛
𝑛
Õ
𝑚=0
∫
Γ
𝜕𝑚
𝜔𝜓+𝑡(r,·) 2
𝐿2 d𝜎(r)
=𝐶(𝑛, 𝜏, 𝑇𝑤)𝑇−2𝑛
𝑛
Õ
𝑚=0
𝜕𝑚
𝜔𝜓𝑡+ 2
𝐿2(R;𝐿2(Γ)).
(4.97) Indeed𝑤 ∈𝐶∞
𝑐 , so it is assured that ˆ𝑤and its derivatives are in𝐿1, since the Fourier transform is a continuous operator mapping the Schwartz class of test functions into itself. The constant 𝐶(𝑛, 𝜏, 𝑇𝑤) depends on𝑇𝑤 only through the 𝐿1 norm of
ˆ
𝑤 and its derivatives, and is independent of 𝑇. Note that (in a slight abuse of notation) the
𝜕𝑚
𝜔𝜓𝑡+
𝐿2(R;𝐿2(Γ))terms on the right-hand-sides of Equation (4.97) are