We consider the solvability of linear integral equations on the real line, in operator form (λ − K)φ = ψ, where λ ∈ C and K is an integral operator. We impose conditions on the kernel, k ,o fK which ensure that K is bounded as an operator on Lp(R), 1 ≤ p ≤∞ ,a nd on BC(R). We establish conditions on families of operators, {Kk : k ∈ W } ,w hich ensure that ifλ � =0a ndλφ = Kkφ has only the trivial
... [Show full abstract] solution in BC(R), for all k ∈ W ,t hen for 1 ≤ p ≤∞ ,( λ − K)φ = ψ has exactly one solution φ ∈ Lp(R) for every k ∈ W and ψ ∈ Lp(R). The results of considerable generality apply in particular to kernels of the form k(s, t )= κ(s − t)z(t )a ndk(s, t )=˜ κ(s − t)˜ z(s, t), where κ, ˜ κ ∈ L1(R), z ∈ L∞(R), ˜ z ∈ BC(R2 )a nd ˜κ(s )= O(s−b )a s|s |→∞ ,f or some b> 1. As a significant application we consider the problem of acoustic scattering by a sound-soft, unbounded one-dimensional rough surface which we reformulate as a second kind boundary integral equation. Combining the general results of earlier sections with a uniqueness result for the boundary value problem, we establish that the integral equation is well-posed as an equation on Lp(R), 1 ≤ p ≤∞ , and on weighted spaces of continuous functions.
