papasymp

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Asymptotic Behaviour at Infinity of Solutions of

Second Kind Integral Equations on Unbounded

Regions ofR

n

Simon N. Chandler-WildeDepartment of Mathematics and Statistics

Brunel UniversityUxbridgeMiddlesexUB8 3PH

Andrew T. PeplowStructural Dynamics Group

Institute of Sound and Vibration ResearchUniversity of Southampton

SouthamptonS09 5NH

April 1994

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Abstract

We consider second kind integral equations of the form x(s) k(s, t)x(t) dt = y(s)

(abbreviated x K x = y), in which is some unbounded subset of Rn. LetXp denote the weighted space of functions x continuous on and satisfying x(s) =

O(|s|p

), s . We show that if the kernel k(s, t) decays like |st|q

as |st| for some sufficiently large q (and some other mild conditions on k are satisfied), thenK B(Xp) (the set of bounded linear operators on Xp), for 0 p q. If also(I K)1 B(X0) then (I K)

1 B(Xp) for 0 p < q, and (I K)1 B(Xq)

if further conditions on k hold. Thus, if k(s, t) = O(|s t|q), |s t| , andy(s) = O(|s|p), s , the asymptotic behaviour of the solution x may be estimatedas x(s) = O(|s|r), |s| , r := min(p,q). The case when k(s, t) = (s t), so thatthe equation is of Wiener-Hopf type, receives especial attention. Conditions, in termsof the symbol ofI K, for I K to be invertible or Fredholm on Xp are establishedfor certain cases ( a half-space or cone).

A boundary integral equation, which models three-dimensional acoustic propaga-

tion above flat ground, absorbing apart from an infinite rigid strip, illustrates thepractical application and sharpness of the above results. This integral equation mod-els, in particular, road traffic noise propagation along an infinite road surface sur-rounded by absorbing ground. We prove that the sound propagating along the rigidroad surface eventually decays with distance at the same rate as sound propagatingabove the absorbing ground.

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Asymptotic behaviour at infinity 1

1 Introduction

We consider integral equations of the form

x(s)

k(s, t)x(t) dt = y(s), s , (1.1)

where is some unbounded open subset ofRn, dt is n-dimensional Lebesgue measureand x, y X, the Banach space of bounded and continuous functions on . Weabbreviate (1.1) in operator form as

x K x = y (1.2)

where the operator K is defined as

K(s) =

k(s, t)(t) dt, s . (1.3)

Let ks(t) = k(s, t), s, t . We suppose throughout that 0 , ks L1() foreach s , and that k satisfies the following assumptions:

A. sups ks1 = sups |k(s, t)| dt < .

B. For all s , |k(s, t) k(s

, t)| dt 0 as s s with s .

These hypotheses imply that K B(X), the set of bounded linear operators on X,with norm K = sups ks1, and that if S X is bounded then KS is boundedand equicontinuous, but, since is unbounded, do not imply that K is compact.

For p 0 let wp(s) = (1 + |s|)p and let Xp denote the weighted space Xp :=

{x X : xp := wp x < } ( . denotes the supremum norm on X). Thenx Xp if and only ifx is continuous on and x(s) = O (|s|

p) as |s| , uniformlyin s.

We are concerned in this paper to develop sufficient conditions on the kernel k(in addition to A and B) to ensure that K B(Xp), the space of bounded linearoperators on Xp, for p > 0, and conditions which ensure that (I K)

1 B(Xp)or, at least, that I K is Fredholm as an operator on Xp. A main result is that if

k satisfies A and B and |k(s, t)| |(s t)|, s, t , where is locally integrableand (s) = O(|s|q) as |s| , for some sufficiently large q, then K B(Xp),0 p q. If also I K is Fredholm as an operator on X then it is Fredholm as anoperator on Xp for 0 p < q, and with the same index; and if (I K)

1 B(X)then (I K)1 B(Xp) for 0 p < q. With the help of further conditions on thekernel k we are able to sharpen these latter results to include the case p = q.

In terms of the integral equation (1.1), these results help us to bound the asymp-totic behaviour at infinity of the solution x: i f (I K)1 B(Xp), p > 0, and

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y(s) = O(|s|p), |s| , uniformly in s, then x(s) = O(|s|p), |s| , uniformlyin s. In the case of a pure convolution kernel, k(s, t) = (s t), our results show

that if (s) = O(|s|q) and y(s) = O(|s|p), |s| , then x(s) = O(|s|r), |s| ,where r = min(p,q).

Our results and methods of proof generalise and extend previous results for integralequations on the real line and on the half line in Chandler-Wilde [1,2].

A main step in the argument is to show that, with the assumptions we make on thekernel k, K K(p) is a compact operator on X for 0 p < q, where K(p) := wpK

1wp

.

In Section 2, preliminary to the main results, we present sets of sufficient conditionson k which ensure that K is a compact operator on X. These conditions are of someinterest in their own right.

Section 3 presents the main results of the paper. In Section 4 we consider furtherthe important case when k(s, t) = (s t), s, t , so that (1.1) is an equation ofWiener-Hopf type. Illustrating the results of Section 3 we give sufficient conditions,in terms of the behaviour of at infinity and the symbol of the operator I K, for(I K)1 B(Xp) in the case in which is the whole or half space, and for I K tobe Fredholm on Xp in the more general case when is a cone, extending the resultsof [3,4] to weighted function spaces.

In Section 5 we illustrate the general results of Sections 3 and 4 by a boundaryintegral equation in acoustics of Wiener-Hopf type which models acoustic scatteringby an infinite rigid strip set in an impedance plane. In particular this models soundpropagation from a motor vehicle along a road which is surrounded by sound absorb-ing ground. Using the results of Section 4 we are able to show that, at least if the

road is not too wide, the sound level eventually decays with distance at the same ratealong the rigid road surface as it does over the absorbing ground. This application inSection 5 also illustrates the sharpness of the results obtained in Section 3.

Throughout we shall use the following notation. Define, for A > 0 and s Rn,

BA(s) := {u Rn : |s u| < A}. (1.4)

Also, for A > 0, let

A := BA(0),

and let KA denote the finite section approximation to K, defined by

KA(s) :=A

k(s, t)(t) dt, s . (1.5)

2 Conditions for Compactness

Various conditions for the compactness of the integral operator K in the case n = 1and = R+ are given in Anselone and Sloan [5,6]. We generalise and modify theseresults to provide conditions for the compactness ofK for arbitrary Rn.

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Our first result (cf Anselone and Sloan [5]) is that K is compact if k satisfies Aand B and the following additional assumption:

C. |k(s, t)| dt 0 as |s| with s , uniformly in s.

Lemma 2.1 If k satisfies A, B and C then K is a compact operator on X.Proof Suppose that n is a bounded sequence in X and let n = Kn. Since Kmaps bounded sets onto bounded equicontinuous sets, n is bounded and equicon-tinuous on the whole of . Thus by the Arzela-Ascoli theorem applied to successiveregions 1, 2,..., and a diagonal argument, n has a subsequence m = nm whichconverges uniformly on A for every A > 0. Now, for any integers n and m,

m n supsA

|m(s) n(s)| + sups/A

|m(s) n(s)|.

For all > 0 the second term is less than /2 for A sufficiently large by AssumptionC. Also, for all A > 0, the first term is less than /2 for all sufficiently large n andm. Thus m is a Cauchy sequence and, since X is a Banach space, is convergent. Wehave shown that the image of every bounded sequence has a convergent subsequence,so that K is compact. 2

For the case = R+, Chandler-Wilde [2] shows that if the integral operator K iscompact then k satisfies A, B, and the following additional assumption:

D. K KA = sups\A |k(s, t)| dt 0 as A .

Since the subspace of compact operators is closed in B(X), K is compact ifk satisfiesA, B, and D and ifKA is compact for all A > 0. This is the case if the kernel of KAsatisfies C for all A > 0 ie if the following assumption (cf Atkinson [7]) is satisfied:

E. For all A > 0,A

|k(s, t)| dt 0 as |s| with s , uniformly in s.

Thus

A,B,D,E K compact. (2.1)

From our final lemma (cf Anselone and Sloan [6]) it follows that also

A,B,D,F K compact, (2.2)

where F is the following condition:

F. For all A > 0 there exists C > 0 such that, for all s \ C, k(s, .) is

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continuous in A, uniformly in s.

Lemma 2.2 If k satisfies A, B, and F then KA is compact on X for all A > 0.Proof For any subset G of let BC(G) denote the Banach space of bounded andcontinuous functions on G. Suppose that k satisfies F and write KA as

KA = EKA + EKA (2.3)

where KA : X BC(C) and KA : X BC( \ C) are defined by

KA(s) :=A

k(s, t)(t) dt, s C,

KA(s) :=A

k(s, t)(t) dt, s \ C,

and the extension operators

E : BC(C) L() and

E : BC( \ C) L()are defined by

E(s) :=

(s), s C,0, s \ C,

E(s) :=

(s), s \ C,0, s C.

Since k satisfies A and B, KA maps bounded sets in X onto bounded equicontin-uous sets in BC(C). Thus, by the Arzela-Ascoli theorem, KA is compact.

Choose a sequence of subdivisions of A =

ni=1

i,nA , such that the measurable sets

i,nA are disjoint and their diameters satisfy max1in{ diam i,nA } 0 as n .

Now select points ti,n i,nA and consider the sequence of operators K

(n)A : X

BC( \ C), defined by

K(n)A (s) :=

ni=1

k(s, ti,n)i,nA

(t) dt =A

kn(s, t)(t) dt

where kn(s, t) := k(s, ti,n), s \ C, t i,nA , i = 1,...,n. Then each K

(n)A is

bounded and compact since it has a finite dimensional range. Since k satisfies F, forall > 0 there exists an integer N() such that

|k(s, t) kn(s, t)|

A dt, s \ C, n N(),

so that KA K(n)A for n N(). Thus KA is the limit of a norm convergent

sequence of compact operators and so is compact.We have shown that KA and KA are compact and, clearly, E and E are bounded.

Thus, from (2.3), we see that KA is compact as an operator from X onto L(). ButKA maps X onto X and any sequence in X convergent in L() is convergent in X.Thus KA is compact also as an operator from X onto X. 2

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3 Asymptotic Behaviour at Infinity

Let

Q() := {q [0, ) : Gq := sups

(1 + |s t|)q dt < } (3.1)

and let q = q() := infQ(). Then 0 q n : for example, q = n if = Rn,q = 1 if = {t = (t1, t2,...,tn) : t1 R, |t2|,..., |tn| < 1}, and q

= 0 if has finitemeasure.

The following assumption is stronger than Assumption A and imposes a boundon the rate of decay of the kernel as |s t| :

A. |k(s, t)| |(s t)|, s , t , where is locally integrable on Rn and,for some q > q, (s) = O (|s|q) as |s| , uniformly in s.

If k satisfies A then, for some M,C > 0 and all s, t ,

|k(s, t)| | (s t) | M

(1 + |s t|)q, |s t| > C, (3.2)

and, since q > q and is locally integrable,

:= sups

|(s t)| dt < . (3.3)

Note that equation (1.1) is equivalent to

xp(s)

k(p)(s, t)xp(t) dt = yp(s), s , (3.4)

where xp := wp x, yp := wp y and k(p)(s, t) := wp(s)wp(t)k(s, t). Defining the integral

operator K(p) by (1.3) with k replaced by k(p) we may abbreviate (3.4) as

xp K(p)xp = yp. (3.5)

From the equivalence of equations (1.1) and (3.4) and the observation that, for X and p 0,

K(p) = wpK(/wp) , K = (1/wp)K(p)(wp), (3.6)

it follows straightforwardly that

K B(Xp) K(p) B(X), (3.7)

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(I K)1 B(Xp) (I K(p))1 B(X). (3.8)

Also notice thatI K injective on X I K injective on Xp (3.9)

I K(p) injective on X. (3.10)

Before we proceed with the first theorem the following technical lemmas are re-quired.

Lemma 3.1 For q > q\B

|s|

1

2

(s)(1 + |s t|)q dt 0 as |s|

with s , uniformly in s.Proof Let = (q q)/2 > 0. Then (see (3.1) and the definition of q) Gq+ isfinite. Also, for all s, t ,

(1 + |s t|)q = (1 + |s t|)(q+) (1 + |s t|)

and hence\B

|s|12

(s)(1 + |s t|)q dt (1 + |s|1/2)

\B

|s|12

(s)(1 + |s t|)(q

+) dt

Gq+ (1 + |s|1/2)

0

as |s| , uniformly in s. 2

Lemma 3.2 For , 0, + > q, define

f(s) :=

(1 + |t|) (1 + |s t|) dt, s .

Then

F := sups

|f(s)| < ,

and, moreover, if , > 0, |f(s)| 0 as |s| with s , uniformly in s.Proof Since + > q,

G+ = sups

(1 + |s t|)(+) dt < .

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Clearly f(s) G+ if either = 0 or = 0.Suppose now that > 0, > 0. Define the functions

H1s (t) := min

1,

1 + |t|

1 + |s t|

,

H2s (t) := min

1,

1 + |s t|

1 + |t|

.

Then

(1 + |t|) (1 + |s t|) =

(1 + |t|)(+) H1s (t), |s t| |t|,

(1 + |s t|)(+) H2s (t), |s t| |t|,

and hence

f(s) I1(s) + I2(s)

where

I1(s) :=

(1 + |t|)(+) H1s (t) dt, s ,

I2(s) :=

(1 + |s t|)(+) H2s (t) dt, s .

Now H1s (t), H2s (t) 1, s, t , so

Ij(s) G+, j = 1, 2,

so that F 2G+. Also

I1(s) =\B

|s|12

(0)(1 + |t|)(+) H1s (t) dt +

B

|s|12

(0)(1 + |t|)(+) H1s (t) dt

\B

|s|12

(0)(1 + |t|)(+) dt +

1 + |s| 12

1 + |s| |s|1

2

G+

0

as |s| with s , uniformly in s. Similarly

I2(s) =\B

|s|12

(s)(1 + |s t|)(+) H2s (t) dt +

B

|s|12

(s)(1 + |s t|)(+) H2s (t) dt

\B

|s|12

(s)(1 + |s t|)(+) dt +

1 + |s| 12

1 + |s| |s|1

2

G+

0

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as |s| with s , uniformly in s, by Lemma 3.1. 2

Our first main result is that A and B are sufficient conditions to ensure thatK B(Xp) for 0 p q.

Theorem 3.3 If k satisfies A and B and 0 p q then k(p) satisfies A and B andK B(Xp), K

(p) B(X).Proof For s, t Rn,

wp(s)

wp(t)=

1 +

|s| |t|

1 + |t|

p

2p

1 +

|s t|

1 + |t|

p. (3.11)

From the above inequality and equation (3.2),

|k(p)(s, t)|

2pM

(1 + |s t|)q + (1 + |t|)p (1 + |s t|)pq

, |s t| > C,

(1 + C)p|(s t)|, |s t| C.

(3.12)

Hence we have, for s , where f and F are defined in Lemma 3.2,

|k(p)(s, t)| dt (1 + C)p + 2p {f0,q(s) + fp,qp(s)}

(1 + C)p + 2p {F0,q + Fp,qp} .

Thus k(p) satisfies Assumption A. To show that k(p) satisfies Assumption B, note thatwp(s)wp(t) ks(t) wp(s

)

wp(t)ks(t)

|wp(s)ks(t) wp(s)ks(t)|since |wp(t)| 1. Hence

|k(p)(s, t) k(p)(s, t)| dt |wp(s) wp(s

)| ks1 + |wp(s)| ks ks1 .

But wp is continuous and k satisfies Assumptions A and B, so k(p) satisfies Assump-

tion B. The rest of the lemma follows from the equivalence in (3.7).2

The next theorem shows that, under the same assumptions, K K(p) is in factcompact for 0 p < q.

Theorem 3.4 If Assumptions A and B are satisfied by k and 0 p < q then

|k(s, t) k(p)(s, t)| dt 0

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as |s| with s , uniformly in s, so that K K(p) is a compact operator on X.Proof Define

F(s) :=

|k(s, t) k(p)(s, t)| dt.

We have immediately from Assumption A that, for s ,

F(s)

wp(s)wp(t) 1 |(s t)| dt (3.13)

F1(s) + F2(s)

where

F1(s) :=\B

|s|12

(s)

wp(s)wp(t) 1 |(s t)| dt, F2(s) :=

B

|s|12

(s)

wp(s)wp(t) 1 |(s t)| dt.

From (3.2) and (3.11), for |s| sufficiently large we have

F1(s) (2p + 1)M

\B

|s|12

(s)(1 + |s t|)q dt + 2pM|fp,qp(s)|

0

as |s| with s , uniformly in s, by Lemmas 3.1 and 3.2. Let

cp(s) := suptB

|s|12

(s)

1 wp(s)wp(t) =

1 + |s|

1 + |s| |s|1

2

p

1 . (3.14)

Then

F2(s) cp(s)

0

as |s| , uniformly in s. We have just shown that F(s) 0 as |s| , ie thatk k(p) satisfies Assumption C. Hence K K(p) is compact from Theorem 3.3 and

Lemma 2.1.2

Theorem 3.5 If k satisfies A and B, 0 p < q, and (I K)1 B(X), then(I K(p))1 B(X) and (I K)1 B(Xp).Proof Suppose that Assumptions A and B are satisfied by k and that (I K)1 B(X). Then, for 0 p < q, K(p) B(X) and K(p) K is a compact operator onX, by Theorems 3.3 and 3.4 respectively. Moreover, from (3.10), I K(p) is injectiveon X. Thus (I K(p))1 B(X) since I K(p) = (I K) + (K K(p)) is the

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sum of an invertible operator and a compact operator and so satisfies the Fredholmalternative. That (I K)1 B(Xp) then follows from (3.8). 2

As a corollary to the above theorem we have

Corollary 3.6 Suppose that the conditions of the previous theorem are satisfied andthat y Xp for some 0 p < q. Then equation (1.1) has a unique solution x Xpand

|x(s)| Cpyp(1 + |s|)

p, s , (3.15)

where Cp denotes the norm of (I K)1 B(Xp).

The previous results do not extend as they stand to the case p = q, since A andB are not sufficient conditions on k to ensure that K K(q) is compact: see [2]. Wenow examine the case p = q further. Define

ks(t) = k(s, t) :=wq(s t)

wq(t)k(s, t), (3.16)

and the operator K, with kernel k, by (1.3) with K(k) replaced by K(k).

Lemma 3.7 If k satisfies A and B then k satisfies Assumptions A, B and D.Proof Let A := \ BA(0). Recalling the inequality (3.2), for A 0 and s we

have A

|k(s, t)| dt ABC(s)

wq(s t)

wq(t)|(s t)| dt + M

A\BC(s)

dt

wq(t)

1 + C

1 + A

q + M

A

dt

wq(t).

Thus k satisfies Assumptions A and D. Further, since k satisfies Assumption B sodoes k (cf proof of Theorem 3.3). 2

We now show that K K(q) + K is a compact operator.

Theorem 3.8 If k satisfies Assumptions A and B then

|k(s, t) k(q)(s, t) + k(s, t)| dt 0 (3.17)

as |s| with s , uniformly in s, so that K K(q) + K is a compact operatoron X.

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Proof Since k satisfies A, from (3.13) and (3.14),

B

|s|12

(s) |k(s, t) k(q)

(s, t)| dt cq(s) 0 (3.18)

as |s| . Thus, and since k satisfies Assumption D, it remains only to show that

I1(s) :=\B

|s|12

(s)|k(s, t)| dt 0, |s| ,

I2(s) :=\B

|s|12

(s)|k(q)(s, t) k(s, t)| dt 0, |s| .

By (3.2) and Lemma 3.1, for |s| sufficiently large,

I1(s) M\B

|s|12

(s)(1 + |s t|)q dt 0, |s| . (3.19)

Also

I2(s) =\B

|s|12

(s)|k(s, t)|

wq(s t)

wq(t)

1 wq(s)wq(s t) dt

M\B

|s|12

(s)

1 wq(s)wq(s t) dtwq(t)

J1(s) + J2(s) (3.20)

where

J1(s) := MB

|s|12

(0)

1 wq(s)wq(s t) dtwq(t)

and

J2(s) := M\(B

|s|12

(s)B|s|

12

(0))

1 wq(s)wq(s t) dtwq(t) .

Now

J1(s) M cq(s)

dt

wq(t) 0 (3.21)

as |s| , uniformly in s. Further, from (3.11),

wq(s)

wq(t) 2q

1 +

wq(s t)

wq(t)

,

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so

1 wq(s)wq(s t) 1wq(t) (2q

+ 1)wq(t)+ 2

q

wq(s t)

so that

J2(s) (2q + 1)

\B

|s|12

(0)

dt

wq(t)+ 2q

\B

|s|12

(s)

dt

wq(s t)

0 (3.22)

as |s| with s , uniformly in s, by Lemma 3.1. 2

Define KA, a finite section version of the operator K, by

KA(s) =A

k(s, t)(t) dt, s . (3.23)

We have the following extension of Theorem 3.5 to the case p = q.

Theorem 3.9 Suppose that k satisfies A and B, that (I K)1 B(X) and thatKA is compact for all A > 0. Then (I K)

1 B(Xq) and (I K(q))1 B(X).

Proof Since KA is compact for all A > 0 and, by Lemma 3.7, k satisfies A, B and D,K is compact. Thus, and by Theorem 3.8, K K(q) is compact. The result followsas in the proof of Theorem 3.5. 2

From Lemmas 2.1 and 2.2, KA is compact for all A > 0 ifk satisfies E or F. ApplyingLemmas 2.1 and 2.2 we obtain the following additional criterion for compactness ofKA, utilised in Sections 4 and 5.

Lemma 3.10 Suppose that k satisfies A and B and that, for every A > 0,wq(s t)k(s, t) = k

(s, t) + o(1) as |s| with s , uniformly in s and tfor t A, and that k

is continuous and bounded on and satisfies F. ThenKA is compact for all A > 0 so that K is compact.Proof Define k1(s, t) = k

(s, t)/wq(t), k2(s, t) = k(s, t) k1(s, t), and let K1,A,

K2,A, denote the integral operators defined by (1.3) with k replaced by k1 and k2,respectively. Then it is easy to see that k1 satisfies A, B, and F, so that, by Lemma2.2, K1,A is compact for all A > 0. Hence and by Lemma 3.7, k2 satisfies A and B.Moreover, k2(s, t) = (wq(s t)k(s, t) k

(s, t))/wq(t) 0 as s with s ,uniformly in s and t for t A, so k2 also satisfies E. Thus K2,A is compact and soKA = K1,A + K2,A is compact for all A > 0. 2.

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Combining Theorem 3.9 and Lemma 3.10 we have the following extension of Corol-lary 3.6.

Corollary 3.11 Suppose that the conditions of the previous lemma are satisfied, that(I K)1 B(X), and that y Xp for some 0 p q. Then equation (1.1) has aunique solution x Xp, and this solution satisfies the inequality (3.15).

The above theorems, 3.5 and 3.9, give conditions for the invertibility of I Kin the weighted space Xp. In cases where we do not know that (I K)

1 B(X)these results do not apply, but we may still be able to obtain information about theFredholm properties of I K. For p 0 let (Xp) B(Xp) denote the set of allFredholm operators on Xp (see egJorgens [8] for definitions). We have the followingresult:

Theorem 3.12 Suppose that k satisfies A and B and that I K (Xp) for somep in the range 0 p < q. Then I K (Xp) for all 0 p < q, and the index ofI K is the same in each of these spaces.Proof Note that the inverse operations of multiplication by wp and multiplication by1/wp are isometric isomorphisms from Xp to X and from X to Xp, respectively. Thuseach of these operations is a Fredholm operator of index zero. It therefore followsfrom (3.6) and a standard result on the composition of Fredholm operators (see eg[8,Theorem 5.6]) that

I K (Xp) I K

(p)

(X) (3.24)

and that if I K (Xp) then the indices of I K (Xp) and I K(p) (X)

are the same.Now suppose that I K (Xp) for some p with 0 p < q. Then, by the above

remarks, I K(p) (X) with the same index, and since, by Theorem 3.4, K K(p)

is compact, it follows (see [8,Theorem 5.12]) that I K (X) with the same index.Reversing this argument we can show that, if I K (X) then, for any p with0 p < q, I K (Xp) with the same index. The result follows. 2.

Remark 3.13 The above result depends on the compactness of K K(p). If the

conditions of Lemma 3.10 are satisfied then K K(q)

is also compact and Theorem3.12 holds with the range 0 p < q extended to 0 p q.

4 Wiener-Hopf Integral Equations

We consider the important special case when k(s, t) = (s t), s, t , so that (1.1)is an equation of Wiener-Hopf type (see eg[3,4,8-11]). The conditions of the previous

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section simplify somewhat in this special case as the following lemma illustrates.

Lemma 4.1 Ifk(s, t) = (st), s, t , is locally integrable, and, for some q > q,(s) = O(|s|q) as |s| , uniformly in s, then k satisfies A and B.Proof Clearly k satisfies A. To see that k satisfies B note that, for all A > 0,

BA+1(s)|k(s, t) k(s, t)| dt

BA+1(s)

|(s t) (s t)| dt

=BA+1(0)

|(t) (t (s s))| dt

0

as s s, since is locally integrable. Also, for some M, C > 0 and all s, t , satisfies (3.2). Thus, for |s s| 1 and A C,

\BA+1(s)

|k(s, t) k(s, t)|dt M\BA+1(s)

1

(1 + |s t|)q+

1

(1 + |s t|)q

dt

2Msups

\BA(s)

dt

(1 + |s t|)q

0

as A by Lemma 3.1. 2.

Note further that ifk satisfies the conditions of Lemma 4.1 and also

|s|q(s) = (s) + o(1) as |s| , uniformly in s, (4.1)

with bounded and uniformly continuous on Rn,

then the conditions of Lemma 3.10 are satisfied.Throughout the remainder of the section we suppose that k(s, t) = (s t),

s, t , with locally integrable.Combining the above lemma and remark with Theorems 3.5, 3.9, and Lemma 3.10

we obtain

Theorem 4.2 Suppose that, for some q > q, (s) = O(|s|q) as |s| , uniformly

in s. Then K B(Xp), 0 p q. If also (IK)

1

B(X) then (IK)

1

B(Xp),0 p < q. If, moreover, (4.1) holds, then (I K)1 B(Xq).

In the specific cases = Rn and = Rn+ := {(s1,...,sn) Rn : sn > 0}, necessary

and sufficient conditions for (I K)1 B(X) are known in terms of the Fouriersymbol of the operator I K, defined by

() = 1 Rn

ei.s(s) ds, Rn

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(.s the scalar product of and s). If L1(Rn) and either (a) = Rn; or (b)

= Rn+ and n 2; then (I K)1 B(X) if and only if

() = 0, Rn (4.2)

(see [8,3]). (Note that if = Rn or Rn+ then q = n so that L1(R

n) if the condi-tions of Lemma 4.1 are satisfied.) Thus, as a corollary of the previous theorem, wehave the following extension of the results of Wiener [8, page 340] and of Goldensteinand Gohberg [3] to the weighted spaces Xp.

Theorem 4.3 Suppose that = Rn or Rn+ (with n 2 in the second case). Supposefurther that, for some q > n, (s) = O(|s|q) as |s| , uniformly in s, and that(4.2) holds. Then (I K)1 B(Xp), 0 p < q. If, moreover, (4.1) holds, then also(I K)1 B(Xq).

Note that the case = R+ is excluded from the above result. Sufficient conditionsfor the invertibility of I K on Xp in the case = R+ are given in [2].

For a larger class of regions the non-vanishing of the symbol , while not knownto guarantee the invertibility ofI K on X, still ensures that I K is Fredholm. Forexample, this is the case if is a connected open conic set, provided the boundaryof , except at the point 0, is a smooth surface (the case if is a circular cone, etc.).

Combining these observations with Theorem 3.12 and Remark 3.13 we have thefollowing extension of the results of Simonenko [4] to the weighted space Xp.

Theorem 4.4 Suppose that Rn

(n 2) is a connected open conic set, andthat the boundary of , except at the point 0, is a smooth (C1) surface. Supposealso that, for some q > n, (s) = O(|s|q) as |s| , uniformly in s, and that (4.2)holds. Then I K (Xp), 0 p < q, and has index zero in each of these spaces.If, moreover, (4.1) holds, then also I K (Xq) with index zero.Proof Simonenko [4] established that if (4.2) holds then I K (Lp()) for1 p < , and this result is established also for p = in [12]. Since K is acontinuous mapping from L() to the closed subspace X it is easy to see thatI K (L()) I K (X). Thus I K (X). Further, Simonenko[4] shows that I K has index zero as an operator on Lp(), 1 p < , and thehomotopy argument he uses applies equally to I K as an operator on X. Thus

I K (X) with index zero. The result now follows from Theorem 3.12 andRemark 3.13. 2.

5 An Application in Acoustics

Consider the following boundary value problem for the Helmholtz equation in thehalf space R3+ := {s = (s1, s2, s3) R

3 : s3 > 0}:

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u + u = F inR3

+,u

n+ iu = 0 on R2 = R3+, (5.1)

u satisfies the Sommerfeld radiation condition.

In (5.1) the functions L(R2) and F are supposed given, with F L2(R

3+)

compactly supported and e 0.Let R2 = R3+ denote the strip {(s1, s2) : 0 < s1 < d, s2 R}. Define by

(s) =

0, s ,c, s R

2 \ ,(5.2)

where c C with ec > 0.The boundary value problem (5.1) models outdoor sound propagation from the

source region (the support of F) over a flat ground plane. In this context |u| is theamplitude of the pressure fluctuation due to the sound wave and the relative surfaceadmittance of the ground plane: its value (s) at a particular point s R2 dependson the frequency of the sound source and on local properties of the ground at thatpoint [13]. Where = 0 the ground is perfectly rigid while where e > 0 theground is energy absorbing. Thus the choice (5.2) models a rigid infinite strip () inan otherwise homogeneous energy absorbing plane. In particular, the boundary valueproblem (5.1) with given by (5.2) is a good model of sound propagation above a

long straight road (the rigid strip ) surrounded by absorbing ground (for examplegrassland) [14].

An interesting practical question is at what rate the sound generated by a motorvehicle decays with distance along the road. Using the results of the previous sectionwe shall show that, at least if the width of the road is not too large, for an observeron the road surface, while the decay in the sound pressure with distance may initiallybe the same as above a completely rigid ground (O(|s|1)) the decay with distancemust ultimately be that for a completely absorbing ground (O(|s|2)). To the bestof our knowledge this result has not been previously been established.

The Greens function Gc(s, t) which satisfies (5.1) with F(s) = (s t) and(s) = c, s R

2, is given by [15]

Gc(s, t) = ei|st|

4|s t|

ei|st|

4|s t|

+ic2

ei|st|0

e|st|u du

u2 2i(1 + c)u (+ c)2

+ c2

H(1)0 (r

1 2c )e

ic(s3+t3), (5.3)

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for s = (s1, s2, s3), t = (t1, t2, t3) R3+, s = t. Here t

= (t1, t2, t3) denotes the

image of t in the ground plane, r = (s1 t1)2 + (s2 t2)2, = (s3 + t3)/|s t|,and is given by

= H[mc]H[e{c(s3 + t3) +

1 2c r |s t|}],

H[u] =

1, u > 0,0, u 0.

In (5.3), e(

1 2c ) 0 and the branch cut for the square root in the integrandshould be chosen so that the square root depends continuously on u and takes thevalue i(+ c) at u = 0.

From (5.3) it is easy to see, using Watsons lemma and the asymptotic behaviourof the Hankel function H

(1)0 for large argument in the case ec > 0, that, for any

constant C > 0, as |s t| with s3 + t3 C,

Gc(s, t)

exp(i|st|)

2|st|, c = 0,

= O(|s t|2), ec > 0,(5.4)

uniformly in s and t. A full asymptotic expansion for Gc in the limit |s t| is given in [16]. The asymptotic result (5.4) illustrates the faster decay rate overabsorbent ground (ec > 0) than over rigid ground (c = 0).

Applying Greens theorem to u and Gc in R3+, the boundary value problem (5.1),

with given by (5.2), can be reformulated as the following boundary integral equationfor x, the restriction of u to [7]:

x(s) = y(s) + ic

gc(s t) x(t) dt, s , (5.5)

with, for s = (s1, s2) R2,

y(s) :=R3+

Gc ((s1, s2, 0), t) F(t) dt, (5.6)

gc(s) := Gc ((s1, s2, 0), 0)

=

ei|s|

2|s| +

ic

2 e

i|s| 0

e|s|u duu2 2iu 2c

+ c2

H(1)0 (|s|

1 2c ). (5.7)

Equation (5.5), a convolution equation on the strip , is identical to equation (1.1)if we define

k(s, t) := (s t) := icgc(s t), s, t . (5.8)

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From (5.4), since ec > 0, it follows that y(s) = O(|s|2), as |s| , uniformly

in s. Further it can be seen from (5.3), or more conveniently from an inverse Hankel

transform representation of Gc(s, t) [17] that Gc(., t) is continuous in R+

3 except att. Thus y Xp for 0 p 2. Also, from (5.7) it can be seen that gc is continuousin R2 except for an integrable singularity at 0. Thus k and satisfy the conditionsof Lemma 4.1 with q = 2, so that k satisfies A and B.

We have shown that k satisfies the conditions of Theorem 3.3 so that, where K isthe integral operator (1.3) with k given by (5.8), we have

Theorem 5.1 For 0 p 2, K B(Xp) and K(p) B(X).

To obtain similar mapping properties for the inverse operator (I K)1 andestablish the asymptotic behaviour of the solution x of equation (5.5) we need firstthat (I K)1 B(X). Now it is easily seen that

K = sups

|k(s, t)| dt

2|c|

|gc(t)| dt. (5.9)

Thus K < 1 provided d (the width of ) is sufficiently small. Thus we obtain

Theorem 5.2 Provided d is sufficiently small so that the right hand side of (5.9)is < 1, (I K)1 B(X) so that equation (5.5) has a unique bounded continuous

solution x.

As y Xp for 0 p 2 and k satisfies A and B with q = 2 we can combine

Theorems 3.6 and 5.2 to immediately obtain that also (I K)1 B(Xp) and that

x(s) = O(|s|p), s ,

for 0 p < 2. To sharpen this result, note that, from Rawlins [16],

gc(s) aei|s||s|2, a := i/(22c ), (5.10)

as s , uniformly in s R2. Thus (4.1) is satisfied with (s) := aei|s|. We cantherefore apply Theorem 4.2 to obtain

Theorem 5.3 If the condition of the previous theorem is satisfied then (I K)1 B(Xp), 0 p 2, so that the solution of equation (5.5) satisfies x(s) = O(|s|

2) as|s| , uniformly in s.

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This example we have given, of practical interest, also serves to illustrate thesharpness of the results we have obtained in Section 3. From (5.10) it follows that if

X is compactly supported then

K(s) a

eis.t(t)dt ei|s||s|2, (5.11)

as |s| , uniformly in s = s/|s|, where s.t denotes the scalar product of s and t.In general the integral on the right hand side of (5.11) will not vanish for any s so that K Xp for p > 2 even though Xp, for all p > 0. Thus K B(Xp) for

p > 2.

6 References[1] S.N.CHANDLER-WILDE On the behavior at infinity of solutions of integral equa-tions on the real line, J. Integral Equations Appl. 4 (1992) pp. 153-177.

[2] S.N.CHANDLER-WILDE On asymptotic behavior at infinity and the finite sec-tion method for integral equations on the half-line, to appear in J. Integral EquationsAppl.

[3] L.S. GOLDENSTEIN & I.C. GOHBERG On a multidimensional integral equa-tion on a half-space whose kernel is a function of the difference of the arguments, and

on a discrete analogue of this equation, Sov. Math. Dokl. 1 (1960), pp. 173-176.

[4] I.B. SIMONENKO Operators of convolution type in cones, Math. USSR-Sbornik3 (1967), pp. 279-293.

[5] P.M. ANSELONE & I.H. SLOAN Integral equations on the half-line, J. Inte-gral Equations 9 (1985), pp. 3-23.

[6] P.M. ANSELONE & I.H. SLOAN Numerical solutions of integral equations onthe half-line. I. The compact case, Numer. Math. 51 (1987), pp. 599-614.

[7] K.E. ATKINSON The numerical solution of integral equations on the half-line,SIAM J. Numer. Anal. 6 (1969), pp. 375-397.

[8] K. JORGENS Linear Integral Operators. Pitman, London 1982.

[9] E. MEISTER & F.-O. SPECK Some multidimensional Wiener-Hopf equationswith applications, Trends Appl. Pure Math. Mech. 2 (1977), pp. 217-262.

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[10] F.-O. SPECK General Wiener-Hopf Factorization Methods. Pitman, Boston

1985.

[11] A. BOTTCHER & B. SILBERMANN Analysis of Toeplitz Operators.Akademie-Verlag, Berlin, 1989. Springer-Verlag, New York, 1990.

[12] A. BOTTCHER, N. KRUPNIK, & B. SILBERMANN A general look at localprinciples with special emphasis on the norm computation aspect, Int. Equs. Op.Th. 11 (1988), pp. 455-479.

[13] K. ATTENBOROUGH Acoustical impedance models for outdoor ground surfaces,J. Sound Vib. 99 (1985), pp. 521-544.

[14] D.C. HOTHERSALL & S.N. CHANDLER-WILDE Prediction of the attenua-tion of road traffic noise with distance, J. Sound Vib. 115 (1987), pp. 459-472.

[15] S.-I. THOMASSON Reflection of waves from a point source by an impedanceboundary, J. Acoust. Soc. Amer. 59 (1976), pp. 780-785.

[16] A.D. RAWLINS The field of a spherical wave reflected from a plane absorbentsurface expressed in terms of an infinite series of Legendre polynomials, J. SoundVib. 89 (1983), pp. 359-363.

[17] D. HABAULT & P.J.T. FILIPPI Ground effect analysis: surface wave and layerpotential representations, J. Sound Vib. 79 (1981), pp. 529-550.

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