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Reflectionless Sturm-Liouville Equations

We consider compactly supported perturbations of periodic Sturm-Liouville equations. In this context, one can use the Floquet solutions of the periodic background to define scattering coefficients. We prove that if the reflection coefficient is identically zero, then the operators corresponding to the periodic and perturbed equations, respectively, are unitarily equivalent. In some appendices, we also provide the proofs of several basic estimates, e.g. bounds and asymptotics for the relevant m-functions.


Introduction
Much can be understood about the properties of one dimensional Sturm-Liouville equations by analyzing the corresponding reflection and transmission coefficients. The purpose of this article is to present some results concerning the implications of trivial scattering, i.e., situations in which there is an absence of reflection.
Throughout our work, we will consider equations of the form −(pu ′ ) ′ + qu = λu (1.1) where λ ∈ C and the real-valued functions 1 p and q are locally integrable with p > 0 almost everywhere. As a first example, consider equation (1.1) in the case that the coefficients 1 − p and q are additionally assumed to have compact support; take supp(1 − p) ⊂ [0, 1] and supp(q) ⊂ [0, 1] for simplicity. For such equations, one defines the classical scattering coefficients by examining the Jost solution of (1.1) which satisfies u(x, k) = e ikx for x ≤ 0, a(k)e ikx + b(k)e −ikx for x ≥ 1.
( 1.2) for 0 < k with λ = k 2 . The coefficients a(k) and b(k) describe the reflected and transmitted parts of an incoming plane wave, a solution of the free equation, and in the physics literature one often defines the transmission coefficient by t(k) := 1 a(k) and the reflection coefficient by r(k) := b(k) a(k) . In our work, we find it more useful to deal with the coefficients a(k) and b(k) directly, and we label them the transmission and reflection coefficient, respectively. We will be specifically interested in cases where the equation (1.1) is reflectionless, i.e., in the event that b(k) = 0 for all k > 0. Let us briefly recall some of the known results. In the specific case mentioned above, in which both 1 − p and q are supported in [0, 1], one can consider the equation wherep andq are the 1-periodic extensions of p and q, respectively, to R. It is easy to see, e.g. Lemma 3.1 of [18], that if equation (1.1) is reflectionless, then the periodic operator corresponding to (1.3) has gapless spectrum equal to [0, ∞). If one assumes in addition that p ≡ 1, then a result due to Borg [3,4] for continuous q, later extended to integrable q by Hochstadt [12], may be used to prove that reflectionless, i.e., gapless spectrum for the periodic operator, implies q ≡ 0. Such results correspond to the well known fact that there are no compactly supported solitons. If p is sufficiently smooth, e.g. p and p ′ are absolutely continuous, then one may apply a unitary, Liouville-Green transformation, as is done in [10], and see that equation Using this equation, one can readily see that if q ≡ 0, 1 − p is compactly supported, p and p ′ are absolutely continuous, and equation (1.1) is reflectionless, then p ≡ 1. Moreover, one may establish that such a result remains true with no additional smoothness assumptions required on p. Specifically, Proposition 4.1 of [18] states that if q ≡ 0, 1 − p is compactly supported, and (1.1) is reflectionless then p ≡ 1. This result was proven by examining the asymptotics of the m-function corresponding to (1.1). In partcular, it did not use known results for the Schrödinger equation due to the fact that the classical Liouville-Green operator is ill-defined when p is not smooth.
In this article, we will generalize some of these results to reflectionless equations whose scattering coefficients are defined with respect to a periodic background. More specifically, we consider 1periodic, real-valued functions p 0 and q 0 for which both 1 p0 and q 0 are locally integrable with p 0 > 0 almost everywhere. It is well known, see e.g. [8,6,21], that the operator on L 2 (R) has spectrum which consists of a union of bands. For λ in a stability interval, there exist linearly independent solutions of which we label by φ ± (·, λ) and refer to as the Floquet solutions corresponding to (1.6), see Section 2 for a further discussion. Let f ≥ 0 and g be real-valued, integrable functions with compact support contained in [0, ∞). Define perturbations of the periodic coefficients introduced above by setting which satisfies again, for λ in a stability interval. Here D > 0 is chosen as inf{D Comparing with (1.2), we see that the Floquet solutions play the role of the plane waves when the periodic background is non-trivial. Analogously to equation (1.5), we will denote by H the operator on L 2 (R) corresponding to the coefficients p and q. We say that equation (1.7) (the operator H) is reflectionless with respect to (1.6) (the operator H 0 ) if there exists a non-empty stability interval in the spectrum of H 0 on which b p (λ) is identically zero. We note that due to the analyticity of b p , see Lemma 2.3 below, it is sufficient to assume that there exists a stability interval on which the zeros of b p have an accumulation point. We prove the following theorem. Our proof follows from a result of Bennewitz [2] which establishes the existence of a more general Liouville Transform. Moreover, since this unitary map is explicit, we may also prove Corollary 1.2. Let p 0 , q 0 , f , and g be given as above. Suppose that f ≡ 0 and H is reflectionless with respect to H 0 , then g ≡ 0.
The paper is organized as follows. In Section 2, we introduce scattering coefficients defined with respect to a periodic background. Next, we prove Theorem 1.1 and Corollary 1.2 in Section 3. In the appendices that follow, we provide a proof of the technical estimates necessary to apply the inverse results found in [2]. The first appendix, Appendix A, establishes some bounds on the growth of solutions to Sturm-Liouville equations. A convergence result for the corresponding m-functions is given in Appendix B. Appendix C contains the main results which describe the asymptotics of the m-function and thereby the Weyl solution.
Acknowledgements: The author wishes to thank the referee for the many useful comments made on a previous version of this paper. Moreover, the author is indebted to both Günter Stolz and Bruno Nachtergaele for many stimulating discussions.

Scattering at a Periodic Background
In this section, we recall some of the basic facts concerning periodic Sturm-Liouville equations. For a more detailed discussion of periodic problems, we refer the reader to [8]. We also note that many of the results stated below are proven, for Schrödinger equations, in [7].

The Floquet Solutions
Let p 0 and q 0 be 1-periodic, real valued functions for which both 1 p0 and q 0 are locally integrable and p 0 > 0 almost everywhere. Consider the self-adjoint operator on L 2 (R) given by The domain of this operator is the set of all f ∈ L 2 (R) for which both f and p 0 f ′ are absolutely continuous and H 0 f ∈ L 2 (R). Since p 0 is not assumed to be smooth, in this generality we allow p 0 to be a step function for example, we note that the smoothness of p 0 f ′ is not necessarily inherited by f ′ . For any z ∈ C, let u N (·, z) and u D (·, z) denote the solutions of Take g 0 (z) to be the transfer matrix of (2.2) from x = 0 to x = 1, i.e., the matrix for which for any solution u of (2.2). Set ρ ± (z) to be the eigenvalues of g 0 (z), i.e., the roots of (2.5) Lemma 2.1. Let p 0 , q 0 , and S (c,d) be taken as above. i) As functions of z, the eigenvalues ρ ± of g 0 may be chosen analytic in S (c,d) with, at most, algebraic singularities at the points z = c and z = d.
ii) For the choices of ρ ± taken in i) above, one may define eigenvectors v ± of g 0 , corresponding to ρ ± , that are analytic in S (c,d) with, at most, algebraic singularities at c and d.
Using Lemma 2.1, one defines the Floquet solutions to be the solutions of (2.2) which satisfy the initial conditions for any z ∈ S (c,d) . We state the properties of these solutions as a separate lemma.
The proofs of both Lemma 2.1 and Lemma 2.2 are provided in Section 2.1 of [7] in the context of Schrödinger equations. Substituting the definitions given above, one may easily translate these proofs to the Sturm-Liouville equations we consider here.

Periodic Scattering Coefficients
Fix p 0 , q 0 , and S (c,d) as defined in the previous subsection. As in the introduction, let f ≥ 0 and g be real-valued, integrable functions whose supports are contained in [0, D]. Define perturbations of p 0 and q 0 by the equations 1 p := 1 p0 + f and q := q 0 + g, respectively. For any z ∈ S (c,d) , let u + be the solution of (2.8) As indicated by Lemma 2.2 ii), the Floquet solutions are linearly independent for z ∈ S (c,d) , and therefore, a p (z) and b p (z) are uniquely defined. In this setting, b p and u + take on the role of a modified reflection coefficient and Jost solution, respectively, relative to the periodic background.
The scattering coefficients a p (·) and b p (·) are analytic for z ∈ S (c,d) with, at most, algebraic singularities at the points z = c and z = d.
Proof. Let g 1 (z) denote the transfer matrix corresponding to (2.7) from x = 0 to x = D, in analogy with (2.4). Clearly then, It is well known that these solutions and their derivatives are, at fixed x, analytic in z, and therefore, the entries of g 1 (z) are entire as functions of z. Using Lemma 2.2 i), we conclude that the left hand side of (2.9) is analytic in S (c,d) with, at most, algebraic singularities at the points z = c and z = d. From (2.8), one sees that In what follows, we will denote by H the self-adjoint operator in L 2 (R) corresponding to equation (2.7) in analogy to (2.1). We will say that equation (2.7) (the operator H) is reflectionless with respect to equation

Proofs of the Main Results
In this section, we will prove Theorem 1.1 and Corollary 1.2. Our proofs are based on a recent inverse result of Bennewitz [2] in which he constructs a more general Liouville transform; specifically, one that is applicable even in the case of non-smooth p. We will begin by describing his transformations and then verify that we may apply his inverse results.

Liouville Transforms
Bennewitz's results apply to half-line operators. To state his main theorem, let p and q be realvalued functions on [0, ∞) with both 1 p and q in L 1 loc (0, ∞). Consider the Sturm-Liouville equation subject to the boundary condition for some α ∈ [0, π). Bennewitz's results are applicable under rather general assumptions. For the results we wish to present, we will assume more than is necessary as we indicate briefly below. i) We assume that we are working on a half-line; the results also apply in the case that p and q are as above, yet defined ii) We assume that p > 0 almost everywhere. Technically, one need only assume 1 p ∈ L 1 loc . iii) We will assume that equation (3.1) is limit-point at +∞, see Chapter 9 of [6] for details.
One may easily check that such a map is unitary and that F −1 , the inverse of F , is also a unitary Liouville transform. Now, for i = 1, 2, let p i and q i be functions which satisfy the conditions stated above and let α i ∈ [0, π). Denote by H i the self-adjoint operator on L 2 (0, ∞) generated by equation (3.1) and (3.2) with coefficients p i and q i and boundary conditions α i ∈ [0, π). We use the following result from [2]: Theorem 3.2. Suppose the operators H 1 and H 2 have the same spectral measure. Then there is a unitary Liouville transform F mapping H 2 to H 1 ; specifically, F H 2 = H 1 F .

Bennewitz explicitly constructs this unitary Liouville transformation in terms of the mappings
) and subsequently, for almost every We note that although the spectral measures of H 1 and H 2 depend on the boundary conditions α 1 and α 2 , respectively, the functions s and t, and therefore the Liouville transform F , do not.

Proofs
We may now provide the proofs of Theorem 1.1 and Corollary 1.2. The proof uses well known results concerning the m-function corresponding to the Sturm-Liouville equations we are considering. We refer the interested reader to Chapter 9 of [6] for a complete discussion. Proof of Theorem 1.1 If H is reflectionless with respect to H 0 , then there exists a stability interval (c, d) in the spectrum of H 0 for which b p (λ) = 0 for all λ ∈ (c, d). As b p is analytic on S (c,d) , we conclude that b p (z) = 0 for all z ∈ S (c,d) . Using (2.8), we see that for λ ∈ (c, d) and η > 0 the modified Jost solution satisfies for all x ≥ D. By Lemma 2.2 iii), the Floquet solution φ + (·, λ + iη) is square integrable at +∞. Thus, the modified Jost solution u + coincides, up to a complex multiple, with the Weyl solution.
Appealing again to (2.8), it is clear that for λ + iη ∈ S (c,d) with η > 0, the m-function for the perturbed equation (1.7) satisfies where m 0 is the corresponding m-function for the periodic equation ( Proof of Corollary 1.2: Suppose f ≡ 0, i.e., p ≡ p 0 and H is reflectionless with respect to H 0 . In this case, it is clear from (3.4) that the mappings t 1 ≡ t 2 , for this application we take t 1 to be defined in terms of p 0 and t 2 in terms of p, and therefore t(x) = x. Using (3.5), we see that s ≡ 1, and therefore, the unitary Liouville transform F is the identity. From this, the equation

Appendix
The goal of the appendices that follow is to prove the key technical estimate, Lemma 2.4 of [2], which enables Bennewitz to prove Theorem 3.2. Such estimates were originally proven by Bennewitz in [1], see specifically Theorem 6.1 and Corollary 6.2, and they are applicable under rather general assumptions; for example, the coefficients of the basic Sturm-Liouville equation may be taken to be measures. Our approach is more pedestrian, in particular, we assume the coefficients are in L 1 loc (R), but we hope our streamlined presentation is more easily accessible.
The heart of the matter is contained in Appendix C where we prove Theorem C.5, the analogue of Lemma 2.4 of [2]. The proof of Theorem C.5 uses a convergence result for m-functions which are defined with respect to a sequence of Sturm-Liouville equations whose coefficients converge in L 1 loc (R). In Appendix B, we prove Lemma B.1 and Theorem B.2 which demonstrate the desired convergence of the m-functions. These results are new. Moreover, as is discussed in the appendix of [13] for equations with coefficients in L 2 loc (R), they constitute a generalization of the applicability of Kotani Theory, see [14,15,16], to the Sturm-Liouville equations considered here. Lastly, all the results presented in these appendices rely heavily on certain basic solution estimates. The proofs of these results, which are simple generalizations of estimates well known in the context of Schrodinger equations, i.e., when p ≡ 1, are presented in Appendix A.

A A Priori Solution Estimates
In this first appendix, we provide several standard solution estimates which we will use frequently in the appendices that follow. Although results of this type are well-known, see e.g. [6,17,21], we include the proofs here for the convenience of the reader.
The basic Sturm-Liouville equation we consider is where it is assumed throughout that 1 p ∈ L 1 loc (R), p > 0 almost everywhere, and q ∈ L 1 loc (R) may be complex valued. In most of our applications, we have q = q 0 − λ for some real valued, locally integrable function q 0 and λ ∈ C a constant. For any f ∈ L 1 loc (R) and I ⊂ R, a bounded interval, we will denote by Lemma A.1. Let u be a solution of (A.1). For any x, y ∈ R, we have that where the interval I := [min(x, y), max(x, y)].
Lemma A.2. For i = 1, 2, let p i and q i be functions with 1 pi ∈ L 1 loc (R), p i > 0 almost everywhere, and q i ∈ L 1 loc (R). Suppose u i are solutions of −(p i u ′ i ) ′ + q i u i = 0 which satisfy u 1 (y) = u 2 (y) and p 1 u ′ 1 (y) = p 2 u ′ 2 (y) for some y ∈ R. Then, for any x ∈ R there exists a constant C > 0 for which and one may take Here I := [min(x, y), max(x, y)].
Proof. Without loss of generality, we consider the case of y ≤ x. For i = 1, 2, define the vector for any t ∈ R. Using this notation, the solutions u 1 and u 2 clearly satisfy for any y ≤ s ≤ x. With the usual vector norm · , one may estimate that By Lemma A.1, we may conclude that for ψ ∈ {u 1 , p 1 u ′ 1 }, sup t∈I |ψ(t)| ≤ exp 1 2 An application of Gronwall's lemma, see e.g. [20], to inequality (A.9) yields (A.5) as desired.
Since the function x → In this case, we conclude that for any |t − x 0 | ≤ δ, by Lemma A.1. Let 0 < a < 1 be given; we will choose such an a below. We claim that there exists an x 0 ∈ [x, x + 1] for which |u(x 0 )| ≥ aC 1 .

(A.26)
If we establish the existence of an a and an x 0 for which (A.26) holds, then (A.23) is true as one may calculate that and therefore We can now estimate that where we have used (A.27), the boundary condition Re[u(x)pu ′ (x)] = 0, To bound I 2 , we use (A.29) as follows where for a real valued function f , f − (t) := max(−f (t), 0). Collecting our bounds from (A.32) to (A.37), we have proven that which is an obvious contradiction for a sufficiently small. We have established the claim (A.26), and thus the lemma is proven.
Remark A.6. One added difficulty in the proof of Lemma A.5 above, is the possibility that the solution u may vanish at x. If one knows the complex valued solution satisfies |u(x)| 2 = 1, in contrast to Re[u(x)pu ′ (x)] = 0, then arguments as in (A.28) readily provide lower bounds of the type found in (A.23).

B The m-function
The goal of this section is to prove Theorem B.2 below which concerns the compact uniform convergence of m-functions corresponding to a sequence of Sturm-Liouville equations whose coefficients converge in L 1 loc (R). For a more indepth discussion of m-function theory, we refer the reader to Chapter 9 of [6], and for convenience, we adopt the notation used therein.
Let {p n } ∞ n=0 and {q n } ∞ n=0 be sequences of real-valued functions which satisfy p n > 0, 1 pn ∈ L 1 loc (R), and q n ∈ L 1 loc (R) for all n ≥ 0. Let K ⊂ C + , the complex upper half plane, be compact and consider solutions of −(p n u ′ ) ′ + q n u = λu (B.1) for λ ∈ K. For n ≥ 0 and x ∈ R, let φ n,x and θ n,x be the solutions of (B.1) which satisfy the boundary conditions Given any y > x denote by D y n,x,λ := m ∈ C + : the disc of radius r y n,x,λ and center c y n,x,λ in the upper half plane. In the limit as y → ∞, the boundary of these discs is given by the solutions, m n (x, λ), of the equation where C may be choosen uniform in the parameters: n ≥ 0, x ∈ I, and λ ∈ K.
Proof. We will prove that the union of discs D is bounded by deriving uniform estimates on the center and radius of each D b+2 n,x,λ . Recall from [6] that and where [φ n,x , θ n,x ](y) = φ n,x (y)p n θ ′ n,x (y)−p n φ ′ n,x (y)θ n,x (y). WithĨ = [a, b+2], the L 1 loc -convergence of the coefficients of (B.1) implies that: max sup with a constant C > 0 which is uniform in the parameters: x ∈ I, n ≥ 0, and λ ∈ K. From this bound, it is clear that , (B.14) where we note that the constant C ′ , appearing in (B.14), incorporates a repeated application of Lemma A.1 to bound the generalized Wronskian [φ n,x , θ n,x ](b + 2). This proves the boundedness of D.
Theorem B.2. Let {p n } ∞ n=0 and {q n } ∞ n=0 be sequences of real-valued functions which satisfy p n > 0, 1 pn ∈ L 1 loc (R), and q n ∈ L 1 loc (R) for all n ≥ 0. Fix I = [a, b] ⊂ R and K ⊂ C + compact. If 1 pn → 1 p0 and q n → q 0 in L 1 loc (R) and the equation is limit point at +∞, then uniformly for (x, λ) ∈ I × K.
Proof. To prove this theorem, we will first establish pointwise convergence. Fix x 0 ∈ I and λ 0 ∈ K. As (B.16) is limit point at +∞, the disc D y 0,x0,λ0 shrinks to a unique point in the limit y → ∞. Thus, by taking y large, we can make the radius r y 0,x0,λ0 arbitrarily small. For y ′ large but fixed, Lemma A.2 implies that r y ′ n,x0,λ0 → r y ′ 0,x0,λ0 as the functions θ n,x0 all satisfy the same normalization. From this we conclude that for all sufficiently large n, the radii r y ′ n,x0,λ0 can be made arbitrarily small as well. Moreover, since m 0 (x 0 , λ 0 ) is in the interior of D y ′ 0,x0,λ0 for all y ′ > x 0 , we see from Appealing again to Lemma A.2, we conclude that for n sufficiently large, as well. In other words, for large enough n, m 0 (x 0 , λ 0 ) ∈ D y ′ n,x0,λ0 , and therefore, |m n (x 0 , λ 0 ) − m 0 (x 0 , λ 0 )| ≤ 2 · r y ′ n,x0,λ0 . As we have already argued, these radii become arbitrarily small, and therefore we have proven pointwise convergence. In fact, as the functions m n (x 0 , ·) are analytic in C + , the uniform bounds proven in Lemma B.1 combined with Montel's Theorem, see e.g. [19], imply that the pointwise convergence is uniform for λ ∈ K.
The full result now follows from the Ricatti equation, i.e., the fact that For any a ≤ t ≤ b, integration of (B.20) yields the following estimate: As each m n (s, λ) ∈ D b+2 n,s,λ , Lemma B.1 implies that there exists M > 0 for which |m n (s, λ)| ≤ M uniformly in the parameters n ≥ 0, s ∈ I, and λ ∈ K. Inserting this upper bound into (B.21) implies that Applying Gronwall's Lemma, see e.g. [20], we find that for any t ∈ I, The pointwise result we proved above demonstrates that C(I, λ, n) → 0 uniformly for λ ∈ K. We have proven the theorem.

C m-Asymptotics and Solution Estimates
We may now reproduce certain estimates found in [1]. We will consider the equation where p and q are real valued functions with p > 0 almost everywhere, both 1 p and q are locally integrable, and λ ∈ C. We impose a further condition on the coefficients, namely that (C.1) is limit point at +∞. The crux of Bennewitz's argument is a clever rescaling of the coefficients in (C.1) as the energy parameter λ varies along a ray in the complex plane. From this, he derives an asymptotic formula for the m-function and a related result for the Weyl solution.

C.1 The Scaling
Fix x ∈ R and denote by P x : [x, ∞) → [0, As both P x and W x are continuous and strictly increasing, we may define their inverses P −1 x and W −1 x , respectively, each of which is also continuous and strictly increasing. Consider the functioñ Observe thatf x is continuous and strictly decreasing with lim t→0 +f x (t) = ∞ andf x (t) → 0 as t → Moreover, for any pair of numbers (r, t) with r > 0 and t > x, In particular, for such a pair (r, t), f x (r) = P x (t).
Proof. Let x ∈ R be fixed. Asf x is invertible, for each r > 0 there exists a unique t r > 0 for which r =f x (t r ). Using the above observations, as r → ∞, t r → 0 proving the first claim in (C.4). Direct substitution shows that from which the later portion of (C.4) is clear. Next, suppose that r > 0 and t > x are choosen to satisfy the equation r P x (t) W x (t) = 1. In this case, .

C.2 m-function asymptotics
Fix x ∈ R and let φ x and θ x be the solutions of (C.1) which satisfy the boundary conditions The following theorem is proven in [6]: Let φ x and θ x be the solutions of (C.1), corresponding to λ ∈ C + , which satisfy the boundary conditions given by (C.8). The linear combination ψ x = φ x + mθ x has the property that m is the m-function, m(x, λ), of (C.1) if and only if lim is the modified Wronskian corresponding to (C.1).
This theorem is a direct consequence of the equation which is used to define the discs D y x,λ introduced previously in (B.3); we refer to [6,9] for the details.
We will now consider the properties of solutions of equation (C.1) as the energy parameter is varied along a ray in the complex upper half plane. Fix µ ∈ C + , take r > 0, and suppose u(t, rµ) is a solution of (C.1) corresponding to λ = rµ. For fixed x ∈ R set s := 1 rfx(r) where f x is as defined above. Denote by and u r (t) := u(st + x, rµ). A short calculation shows that and we see that if u solves (C.1), then u r solves Proof. Let ψ r (t) := φ r (t)+ m r (0, µ)θ r (t), where m r (0, µ) is as defined in (C.15). One easily verifies that [ψ r , ψ r ](t) = f x (r)[ψ x , ψ x ](ts + x), (C. 16) where, as before, ψ x = φ x + m(x, rµ)θ x . Using Theorem C.2, if m(x, rµ) is the m-function corresponding to (C.1) with λ = rµ, then (C.16) implies that lim t→∞ [ψ r , ψ r ](t) = 0, and therefore the lemma follows from another application of Theorem C.2.
Theorem C.4. Suppose p and q are real valued functions for which p > 0, 1 p ∈ L 1 loc (R), q ∈ L 1 loc (R), and equation (C.1) is limit point at ∞. Assume in addition that x is a Lebesgue point of where the square root above is the principle branch and the convergence is uniform for µ in compact subsets of C + .
Proof. Fix an arbitrary c > 0. One may calculate that By Lemma C.1, both s and f x (r) go to zero as r goes to infinity, and so the same is true for the above integral. A similar result holds for 1 pr . To see this, sets := W −1 x (s). Clearly then, and from Lemma C.1 we may conclude that f x (r) = P x (s). In this case, one may calculate that Observe that as r → ∞,s → x. Thus, since x is a Lebegue point of 1 p , the product of the ratios Wx(s) Px(s) and Px(cs+x) cs goes to one as r → ∞. These calculations show that as r → ∞, q r → 0 and 1 pr → 1 in L 1 loc (0, ∞). Using Theorem B.2, we may conclude that which is the well-known m-function for the free equation. Applying Lemma C.1 again, (C. 19) implies not only that f x (r) = P x (s), but also r P x (s) W x (s) = 1. From these equations it is easy to see that
Formally, equation (C.28) follows readily by combining (C.17) and (C.27) above. Justifying the use of dominated convergence, however, requires a bit of work. To prove this theorem, we use a lemma, due to Hardy, concerning maximal functions.