Harmonic extension technique for non-symmetric operators with completely monotone kernels

We identify a class of non-local integro-differential operators $K$ in $\mathbb{R}$ with Dirichlet-to-Neumann maps in the half-plane $\mathbb{R} \times (0, \infty)$ for appropriate elliptic operators $L$. More precisely, we prove a bijective correspondence between L\'evy operators $K$ with non-local kernels of the form $\nu(y - x)$, where $\nu(x)$ and $\nu(-x)$ are completely monotone functions on $(0, \infty)$, and elliptic operators $L = a(y) \partial_{xx} + 2 b(y) \partial_{x y} + \partial_{yy}$. This extends a number of previous results in the area, where symmetric operators have been studied: the classical identification of the Dirichlet-to-Neumann operator for the Laplace operator in $\mathbb{R} \times (0, \infty)$ with $-\sqrt{-\partial_{xx}}$, the square root of one-dimensional Laplace operator; the Caffarelli--Silvestre identification of the Dirichlet-to-Neumann operator for $\nabla \cdot (y^{1 - \alpha} \nabla)$ with $(-\partial_{xx})^{\alpha/2}$ for $\alpha \in (0, 2)$; and the identification of Dirichlet-to-Neumann maps for operators $a(y) \partial_{xx} + \partial_{yy}$ with complete Bernstein functions of $-\partial_{xx}$ due to Mucha and the author. Our results rely on recent extension of Krein's spectral theory of strings by Eckhardt and Kostenko.


Introduction
The purpose of this work is to characterise the class of non-local operators K that arise as Dirichlet-to-Neumann maps for certain second-order elliptic operators L in the half-plane R × (0, ∞) (or in a strip R × (0, R), with the Dirichlet boundary condition at y = R). We assume that L is translation-invariant with respect to the first variable x; that is, the coefficients of L only depend on the second variable y.
Thus, we consider elliptic equations of the form a(y)∂ xx u(x, y) + 2b(y)∂ xy u(x, y) + c(y)∂ yy u(x, y) + d(y)∂ x u(x, y) + e(y)∂ y u(x, y) = 0. (1.1) In general, the Dirichlet-to-Neumann operator associated to equation (1.1) is given by where u is a solution of (1.1) with boundary values u(x, 0) = f (x), σ is an appropriate scale function, and τ is an appropriate shearing profile; we refer to Sections 1.1 and 1.3 for further details. Building upon recent extension of Krein's spectral theory of strings due to Eckhardt and Kostenko [12], we prove that such Finally, we comment on the more general case, when the coefficients of the elliptic operator L are allowed to depend on both x and y. Although a lot is known about the corresponding Dirichlet-to-Neumann operators when the coefficients are sufficiently regular (see [17] and the references therein for a sample of such results), to the best knowledge of the author, in this generality, a complete description of the class of corresponding Dirichlet-to-Neumann operators is an open problem. This question for symmetric elliptic operators of the form Lu = ∇ x,y · (A(x, y)∇ x,y u) is closely related to the famous Calderón's question whether the conductivity A(x, y) can be reconstructed by measuring the resistance between different parts of the boundary; this is also known as the electrical impedance tomography or electrical resistivity tomography, and A(x, y) here can be either a scalar or a symmetric matrix. For a solution of Calderón's question in dimension two, we refer to [34]; see [3,40] for a general overview and further references.
No characterisation is known for the class of Dirichlet-to-Neumann operators corresponding to symmetric elliptic equations discussed in the above paragraph. A natural conjecture in the symmetric case is given in [19] in terms of a condition on signs of certain determinants, very similar to the concept of total positivity. In the same paper it is proved that this condition is indeed satisfied, given appropriate regularity of the coefficients. We also refer to [18] for a result closely related to the extension technique developed in [28].
Noteworthy, a similar question for symmetric elliptic operators on planar graphs with boundaries has been answered by Colin de Verdière: a complete characterisation of the corresponding discrete Dirichlet-to-Neumann maps is given in [10].
1.1. Reduction. Before we rigorously state our main result, Theorem 1.7, we explain how a general elliptic equation (1.1) can be transformed into a reduced elliptic equation of the form (1.4). In other words, we show that with no loss of generality we may assume that in the general equation (1.1) the linear term is missing (d(y) = e(y) = 0), and that the coefficient c(y) at ∂ yy is equal to 1. Within this class the correspondence between the associated Dirichlet-to-Neumann operators K defined by (1.2) and operators given by (1.3) is bijective.
For simplicity, in this section we ignore completely all regularity issues. These are discussed in detail in the next section for the class of reduced elliptic equations (1.4), and only briefly in Section 1.3 for other special cases of (1.1).
We begin with a general elliptic equation of the form (1.1). To be specific, we consider the equation L 0 u 0 = 0 for a function u 0 defined on R × (0, R 0 ), where the operator L 0 is given by L 0 = a 0 (y)∂ xx + 2b 0 (y)∂ xy + c 0 (y)∂ yy + d 0 (y)∂ x + e 0 (y)∂ y .
The reduction is divided into three steps.
Therefore, the rather non-standard definition (1.2) of the Dirichlet-to-Neumann operator K 0 associated to the equation L 0 u 0 = 0 agrees with the Dirichlet-to-Neumann operator K 3 that corresponds to the equation L 3 u 3 = 0 via the usual formula K 3 f (x) = ∂ y u 3 (x, 0); here u 0 (x, 0) = u 3 (x, 0) = f (x).
We stress again that in the above reduction we did not discuss the question of regularity of coefficients and solutions, and this is not merely a technicality. We come back briefly to this question in Section 1.3.

Assumptions and main result.
In the following part we first give a rigorous definition of the class of reduced elliptic equations (1.4) (Definition 1.1) and we carefully define the notion of a solution u of (1.4) (Definition 1.2). Then we prove that every boundary value f corresponds to a unique solution u (Proposition 1.3). This is used to define the Dirichlet-to-Neumann operator (Definition 1.4). Next, we give a rigorous meaning to the non-local operator K given by ( We say that L is of class L ⋆ if additionally a({0}) = 0.
We understand formula (1.5) purely formally: it does not define neither the domain, nor the action of L. In Definition 1.2 below, a rigorous meaning is given to the equation Lu = 0 for L ∈ L.
Note that we do not assume strict ellipticity of L: when a(dy) = (b(y)) 2 dy on some interval, then L becomes degenerate in the corresponding strip.
As usual, in Definition 1.1 we identify coefficients b which agree almost everywhere. Whenever we say that L is an operator of class L, we use a(dy), b(y) and R for the corresponding parameters described in Definition 1.1. We additionally denote the auxiliary parameters a(dy) = a(dy) − (b(y)) 2 dy and B(y) = y 0 b(t)dt for y ∈ [0, R).
In this general setting the notion of a solution of the equation Lu = 0 (or, in other words, a harmonic function for L) requires a careful formulation. Note that the value α = a({0}) has no effect on the following definition, and that the definition automatically requires harmonic functions to be sufficiently regular at infinity.

Definition 1.2.
For an operator L of class L, a Borel function u(x, y) on R × [0, R) is said to be harmonic with respect to L if: (a) for every y ∈ [0, R) the function u(·, y) is in L 2 (R), and it depends continuously (with respect to the L 2 (R) norm) on y ∈ [0, R); if R = ∞, then the L 2 (R) norm of u(·, y) is assumed to be a bounded function of y ∈ [0, ∞), while if R < ∞, then we additionally require that u(·, y) converges to zero in L 2 (R) as y → R − ; (b) the functionũ(x, y) = u(x + B(y), y) is weakly differentiable with respect to y on R × (0, R), with the weak derivative denoted by ∂ yũ (x, y), and (∂ yũ (x, y)) 2 is integrable over R × (y 1 , y 2 ) whenever 0 < y 1 < y 2 < R; (c) the equation Lu(x, y) = 0 is satisfied in the weak sense in R × (0, R).
The last item of the above definition requires clarification. If u is sufficiently regular, we can use the usual weak (or distributional) formulation of the equation Lu = 0, namely, we require that for every smooth, compactly supported function v(x, y) on R × (0, R) we have However, in the general case, ∂ y u may fail to exist: we only know that ∂ yũ is welldefined, whereũ(x, y) = u(x + B(y), y). Therefore, in the general case we understand condition (c) as for every smooth, compactly supported function v(x, y) on R × (0, R). If u is regular enough, it is straightforward to see that conditions (1.6) and (1.7) are equivalent.
We also clarify that the weak differentiability condition (b) for the L 2 (R)-valued function y →ũ(·, y) is understood in the usual way: there is a locally integrable Borel function ∂ yũ (x, y) on R × (0, R) such that for every smooth, compactly sup- Again we identify all functions u which are equal almost everywhere; however, we always require continuity of the L 2 (R)-valued function y → u(·, y).
The following preliminary result is needed for the definition of the Dirichlet-to-Neumann operator. Proposition 1.3. Suppose that L is an operator of class L. Then for every f ∈ L 2 (R) there is a unique function u harmonic with respect to L (in the sense of Definition 1.2) such that u(x, 0) = f (x) for almost all x ∈ R. Proposition 1.3 is proved in Section 3.

Definition 1.4.
For an operator L of class L ⋆ , the Dirichlet-to-Neumann operator K associated to the equation Lu = 0 is an unbounded operator on L 2 (R), defined by the formula where u is a harmonic function for L described in Proposition 1.3, with boundary values u(x, 0) = f (x). Here the limit in (1.9) is understood in the L 2 (R) sense, and f is in the domain D(K) of the operator K if and only if f ∈ L 2 (R) and the limit in (1.9) exists.
If L is an operator of class L with α = a({0}) > 0, then we use the definition and we say that f is in the domain D(K) if and only if f, f ′ , f ′′ ∈ L 2 (R) (with the second derivative understood in the weak sense) and the limit in (1.10) exists.
Our main result identifies Dirichlet-to-Neumann operators associated to elliptic equations Lu = 0 for L ∈ L with the following class of non-local operators.
Definition 1.5. We say that an operator K is of class K if and only if for every smooth, compactly supported function f (x) on R, where: (a) α 0, β ∈ R and γ 0; (b) ν(z) is a real-valued function on R \ {0} such that ν(z) and ν(−z) are completely monotone functions of z > 0, and ∞ −∞ min{1, z 2 }ν(z)dz < ∞. We say that K is of class K ⋆ if α = 0.
Whenever we consider an operator K of class K, we use the notation α, β, γ and ν(z) introduced above. Additionally, we always extend K to a closed unbounded operator on L 2 (R), as described below.
It is well-known that every operator K of class K is a Fourier multiplier with for ξ ∈ R; see, for example, [1,37]. By this we mean that if f is a smooth, compactly supported function on R, then the Fourier transform of Kf is given byK(ξ)f (ξ). Definition 1.6. Every operator K of class K is automatically extended to an unbounded operator on L 2 (R), with domain D(K) = {f ∈ L 2 (R) :K ·f ∈ L 2 (R)}, (1.13) and defined by Kf (ξ) =K(ξ)f (ξ). (1.14) We are now ready to state our main result.
(a) If L is an operator of class L, then the Dirichlet-to-Neumann operator K associated to the equation Lu = 0 is an operator of class K. (b) Every operator K of class K is the Dirichlet-to-Neumann operator associated to the equation Lu = 0 for a unique operator L of class L.
Theorem 1.7 is proved in Section 3. Here we observe that it is sufficient to prove Theorem 1.7 for classes L ⋆ and K ⋆ rather than L and K. Indeed, suppose that L is an operator of class L such that α = a({0}), and let L ⋆ be the corresponding operator of class L ⋆ , obtained by replacing a(dy) by a ⋆ (dy) = 1 (0,∞) (y)a(dy). The operators L and L ⋆ share the same class of harmonic functions. Thus, if K and K ⋆ are the corresponding Dirichlet-to-Neumann operators, then Kf = K ⋆ f + αf ′′ (see We note that very few explicit pairs of associated operators L and K are known; see Section 4 for examples and further discussion. We also remark that if f is in the domain of K and u is the harmonic extension of f , then the weak derivative ∂ y u is well-defined, and formula (1.9) in the definition of the Dirichlet-to-Neumann operator (Definition 1.4) can be equivalently written as Kf (x) = lim y→0 + ∂ y u(x, y), (1.15) with the limit in L 2 (R). This follows from Theorem 2.1(c) and Lemma 3.1 by an argument used in the proof of Theorem 4.3 in [28]; we omit the details.
In [26], Fourier symbolsK of operators of class K are called Rogers functions, and a number of equivalent characterisations of this class of functions is given therein. For completeness, we list them in the following statement. Proposition 1.8 (Theorem 3.3 in [26]). Suppose that k(ξ) is a continuous function on R, satisfying k(−ξ) = k(ξ) for all ξ ∈ R. The following conditions are equivalent: (a) −k is the Fourier symbol of some operator K of class K, that is, k(ξ) = −K(ξ) for all ξ ∈ R, withK given by (1.12); (b) for all ξ ∈ R we have for someβ ∈ R and some non-negative measure µ on R \ {0} such that |s| ds (1.17) for some c > 0 and some Borel function ϑ on R with values in [0, π]; (d) k extends to a holomorphic function in the right complex half-plane {ξ ∈ C : Re ξ > 0} and Re(k(ξ)/ξ) 0 whenever Re ξ > 0 (that is, k(ξ)/ξ is a Nevanlinna-Pick function in the right complex half-plane).

Variants.
Our main result is stated for the reduced elliptic equation Lu = 0, with operator L of the form L = a(dy)∂ xx + 2b(y)∂ xy + ∂ yy (1.18) in R × (0, R), where R ∈ (0, ∞], a(dy) is a non-negative, locally finite measure, and b(y) is a real-valued function such that (b(y)) 2 is locally integrable and a(dy) − (b(y)) 2 dy 0. We choose this variant, because it leads to relatively few technical difficulties, and it is well-suited for a probabilistic interpretation. However, various reformulations of our result are possible, two of which are discussed below. More precisely, first we rephrase our main result for the operators of the form L =ã(dy)∂ xx + ∂ yy +d(y)∂ x , (1.19) and then we specialise our theorem to the class of operatorṡ Let us stress that, in principle, it is possible to reverse completely the reduction in Section 1.1 and state a result for general equations of the form (1.1). However, a complete description of the class of coefficients a, b, c, d, e, for which the corresponding Dirichlet-to-Neumann operator is well-defined, is somewhat problematic. Additionally, one loses the bijective correspondence between coefficients and Dirichlet-to-Neumann operators. For these reasons, we take a different perspective, and we focus on operators given by (1.19) and (1.20). Given enough regularity of a, b and u, it is now straightforward to show that Lu = 0 in R × (0, R) if and only ifLũ = 0 in R × (0, R). In the general case, however, some care is needed, as it was the case with Definition 1.2:d is the derivative of an arbitrary locally square-integrable function. Definition 1.9. Suppose that R ∈ (0, ∞),ã is a locally finite, non-negative measure on [0, R), andd is the distributional derivative of a locally square-integrable function on [0, R). We say that a functionũ(x, y) is harmonic with respect to the operatorL given by (1.19) if: (a) for every y ∈ [0, R) the functionũ(·, y) is in L 2 (R), and it depends continuously (with respect to the L 2 (R) norm) on y ∈ [0, R); if R = ∞, then the L 2 (R) norm ofũ(·, y) is assumed to be a bounded function of y ∈ [0, ∞), while if R < ∞, then we additionally require thatũ(·, y) converges to zero in L 2 (R) as y → R − ; (b) the functionũ(x, y) is weakly differentiable on R × (0, R) with respect to y, and (∂ yũ (·, y)) 2 is integrable over R × (y 1 , y 2 ) whenever 0 < y 1 < y 2 < R; (c) the equationLũ = 0 is satisfied in the weak sense in R × (0, R), that is, for every smooth, compactly supported function v on R × (0, R), We clarify that ifd = −b ′ , then the last integral in (1.22) should be understood as and in particular this is why weak differentiability ofũ with respect to y is needed. It is somewhat technical, but relatively straightforward to prove that u is harmonic with respect to L in the sense of Definition 1.2 if and only ifũ is harmonic with respect toL in the sense of Definition 1.9. In fact, the only difficulty lies in the proof that conditions (1.7) and (1.22) are equivalent. We omit the details.
Ifũ(x, y) = u(x + B(y), y) is a harmonic function forL with boundary values f (x) =ũ(x, 0), then, according to Definition 1.4, the corresponding Dirichlet-to-Neumann operator K is given by with α =ã({0}) and with the limit in L 2 (R). Note that given onlyL (that is, the coefficientsã andd), there is some ambiguity in the above definition: the function B is defined up to addition by a linear term only, and thus Kf is only defined up to addition by a first-order term Cf ′ for some C ∈ R.
As an immediate corollary of Theorem 1.7, we obtain the following result. Dirichlet-to-Neumann operator K associated to the equationLũ = 0, withL given by (1.19), is an operator of class K. (b) Every operator K of class K is the Dirichlet-to-Neumann operator associated to the equationLũ = 0 for a unique triplet of parameters R,ã and d = −b ′ satisfying the conditions listed in Definition 1.9.
Compared to the equation Lu = 0 in standard form, studied in Section 1.2, the Eckhardt-Kostenko formLũ = 0 is much more closely related to the ODE studied in [12]; see Section 2 for further discussion. Additionally, the definition of a solution of the elliptic equationLũ = 0 is somewhat simpler. On the other hand, the Eckhardt-Kostenko form presents a number of additional technical difficulties. First of all, one has to work with distributional derivatives of square-integrable functions, that is, with elements of the Sobolev space H −1 loc ([0, R)) of negative index; again see Section 2. Furthermore, the definition (1.23) of the Dirichlet-to-Neumann operator is less natural for the Eckhardt-Kostenko form. In fact, as described above, formula (1.23) is ambiguous: it depends on the function B, which is not uniquely determined by the coefficientd (this is the reason why we writed = −b ′ rather than simplyd in Theorem 1.10). Finally, the equation in standard form turns out to be more convenient than the Eckhardt-Kostenko variant in probabilistic applications, to be discussed in [27].
With the above arguments in mind, in this article we focus on the standard form (1.18) considered in Section 1.2, and we limit our discussion of the equation (1.19) in Eckhardt-Kostenko form to this section. Note, however, that finding an operator L which corresponds to a given operatorL, or vice versa, presents no difficulties; see formula (1.21).
1.3.2. Divergence-like form. We now move to the variant given by (1.20). In the symmetric case (corresponding to b(y) = 0), discussed in detail in [28], it is often convenient to work with the equation in the divergence form: ∇ x,y · (ȧ(y)∇ x,yu ) = 0, rather than the standard form: a(dy)∂ xx u + ∂ yy u = 0. Both equations are equivalent by an appropriate change of scale, which corresponds to a different choice of σ in Step 1 of the reduction. The equation in the divergence form, however, is less general: not every measure a(dy) corresponds to some coefficientȧ(y). We refer to [28] for a detailed discussion.
Below we implement a similar strategy in the non-symmetric case, and again we need to impose additional restrictions on the coefficients a(dy) and b(y); in other words, this approach leads to the representation as Dirichlet-to-Neumann operators for a class of operators K strictly smaller than K.
These conditions assert that the integrals in (1.25) make sense. Ifȧ andḃ are sufficiently regular (for example, locally bounded on (0,Ṙ)), then every smooth, compactly supported function is a suitable test function. We will shortly see that also under our more general assumptions onȧ andḃ, the class of suitable test functions for (1.25) is sufficiently rich.
Given the parametersṘ,ȧ andḃ ofL, we first construct the parameters R,ã andd of the corresponding equationLũ = 0 in the Eckhardt-Kostenko form. This involves an appropriate change of scale. Only later we switch to the standard form Lu = 0, with appropriately chosen coefficients a and b.
Note that the formula a(dy) = a(y)dy defines a locally finite measure on [0, R) with a positive almost everywhere density function a(y). It follows that ifu is a harmonic function forL in the sense of Definition 1.11, then u is a harmonic function for L in the sense of Definition 1.2.
Suppose now thatu is a harmonic function forL with boundary values f (x) = u(x, 0). According to Definition 1.4, the Dirichlet-to-Neumann operator K associated tou is given by (1.29) with the limits in L 2 (R); the second inequality is a consequence of (1.15). As an immediate corollary of Theorem 1.7, we obtain the following result. Dirichlet-to-Neumann operator K associated to the equationLu = 0, withL given by (1.20), is an operator of class K.
(b) Every operator K of class K is the Dirichlet-to-Neumann operator associated to the equationLu = 0 for at most one triplet of parametersṘ,ȧ andḃ satisfying the conditions listed in Definition 1.11.
Note that the counterpart of Theorem 1.7(b) is incomplete: not all operators of class K can be realised as described above. This is the main reason for us to focus on the equation Lu = 0 in standard form studied in Section 1.2. On the other hand, some examples take a particularly simple form when written as in (1.24), and this form is also more suitable for some constructions; further discussion can be found in Section 4.
It is easy to find an operator L (orL) which corresponds to a given operatoṙ L, using (1.26) and (1.28) (or (1.26) and (1.27)). The converse is slightly more complicated. Let L be an operator in the standard form (1.18), with coefficients a and b, and suppose that a(dy) has a positive almost everywhere density function, denoted by a(y).
If ϕ is an absolutely continuous function on an interval, then ϕ is differentiable almost everywhere, and the weak (or distributional) derivative of ϕ corresponds to a function equal almost everywhere to the point-wise derivative. If ϕ is a function of bounded variation, then the distributional derivative of ϕ corresponds to a measure. Here we take special care about the endpoints of the domain of ϕ: if ϕ is defined on [0, R) and ϕ(0) = ϕ(0 + ), then we understand that ϕ ′ contains an atom at 0 of mass ϕ(0 + ) − ϕ(0), as if ϕ was extended to a constant function ϕ(t) = ϕ(0) for t < 0. In particular, the value of ϕ at a single point 0 does influence the distributional derivative of ϕ.
A locally integrable function u(x, y) is said to be weakly differentiable with respect to x if there is a locally integrable function v(x, y) such that for every smooth, compactly supported (test) function w. As remarked above, a function u of one variable is weakly differentiable if and only if it is (locally) absolutely continuous. In higher dimensions, we will use the following characterisation of weak differentiability, known as absolute continuity on lines (ACL): u(x, y) is weakly differentiable with respect to x if and only if there is a functionũ(x, y) which is equal to u(x, y) almost everywhere, which is absolutely continuous with respect to x for every y, and such that the point-wise derivative ∂ xũ (x, y) (which necessarily exists almost everywhere) is a locally integrable function. In this case ∂ xũ (x, y) is the weak derivative of u(x, y). We use the same notation ∂ x u for both the usual (point-wise) and the weak derivative. Whenever this convention may lead to ambiguities, we will explicitly state which derivative we have in mind.

Auxiliary ODE
As it will become apparent in the next section, Fourier transform reduces our problem to the study of a second-order linear ordinary differential equation Here ϕ is a function on [0, R) with R ∈ (0, ∞], ξ is a 'spectral' parameter, and the coefficients a(dt) and b(t) are as in the definition of class L (Definition 1.1): a(dt) is a non-negative measure on [0, R) (we allow for an atom at 0), the coefficient b(t) is locally square-integrable on [0, R), and a(dt) − (b(t)) 2 dt is assumed to be a non-negative measure on [0, R). For our later needs it is enough to assume that ξ ∈ R; however, we stress that in the proof of Theorem 2.1 arbitrary complex ξ need to be considered. We understand (2.1) in the sense of distributions; more precisely, we assume that ϕ is an absolutely continuous function such that the first distributional derivative of ϕ corresponds to a left-continuous function (which we denote ϕ ′ (t)), the second distributional derivative of ϕ is a complex-valued measure (that we denote by ϕ ′′ (dt)), and we have equality of measures given by (2.1).
As already mentioned, for our purposes we only need to study the properties of solutions of (2.1) when ξ is a real number. It is in fact sufficient to consider ξ > 0: if ϕ is a solution of (2.1) for some ξ > 0, then ϕ satisfies (2.1) with ξ replaced by −ξ. Furthermore, for ξ = 0, equation (2.1) requires ϕ to be an affine function. For this reason, we restrict our attention to ξ > 0 in the following statement. We refer to [12] and to Appendix A for results that cover general complex ξ.
The following statement summarizes some of the main results of [12], which play a crucial role in our development. The function k(ξ)/ξ introduced in item (b) is often called the principal Weyl-Titchmarsh function for the equation (2.1).

Theorem 2.1.
(a) Suppose that the coefficients a(dt) and b(t), defined on [0, R), satisfy the conditions of Definition 1.1. For every ξ > 0 there is a unique solution ϕ ξ of (2.1) on [0, R) which satisfies ϕ ξ (0) = 1 and such that ϕ ξ is bounded when R = ∞ (in this case every other solution diverges to infinity at ∞), and ϕ ξ (R − ) = 0 if R < ∞ (in this case every other solution is bounded away from zero in some left neighbourhood of R). (b) If ϕ ξ is the solution defined above and k(ξ) = −ϕ ′ ξ (0), then k extends to a Rogers function; that is, k has a holomorphic extension to the the right complex half-plane, and this extension satisfies Re(k(ξ)/ξ) 0 whenever Re ξ > 0. (c) If ϕ ξ is the solution defined above, then |ϕ ξ | 2 is positive, non-increasing and convex on [0, R), and |ϕ ′ ξ | is non-increasing on [0, In particular, |φ ′ ξ | 2 is integrable on [0, R). In [12], Eckhardt and Kostenko study the equation (2.1) in a different form, for the functionφ ξ rather than ϕ ξ . For this reason, we include below a brief discussion of equivalence of these two forms. The direct part of Theorem 2.1 (that is, items (a) through (c)) is proved in Sections 3-5 of [12]. For a less general class of coefficients a(dt) and b(t), this goes back to [24,29]. The inverse part of Theorem 2.1 (item (d)) is the main contribution of [12]; its proof involves deep ideas due to de Branges. For reader's convenience, in Appendix A we include an alternative, less abstract proof of parts (a) through (c) of Theorem 2.1, written in the language of (2.1) rather than that of [12].
Proof of equivalence of Theorem 2.1 and the results of [12]. We transform equation (2.1) in a way that corresponds to shearing in Section 1.1: as usual, we denote B(t) = t 0 b(s)ds, and whenever ϕ is a functon on [0, R), we writẽ ϕ(t) = e iξB(t) ϕ(t).
On a formal level, ϕ is a solution of (2.1) if and only ifφ satisfies By assumption,ã is a non-negative measure on [0, R). However, b is only assumed to be locally squareintegrable, and therefore the distributional derivative b ′ need not correspond to a function or a measure: it is an element of the Sobolev space H −1 loc ([0, R)) on [0, R) with negative index −1, that is, the dual of the Sobolev space H 1 c ([0, R)) of compactly supported and weakly differentiable functions f on [0, R) such that f and f ′ are in L 2 ([0, R)).
Under the above assumptions (that is,ã a non-negative measure andd an element of the Sobolev space H −1 loc ([0, R))), the equation satisfied byφ: is precisely the equation studied systematically by Eckhardt and Kostenko in [12], see equation (1.2) therein. With the notation used there, z, ν and ω in [12] correspond to iξ,ã and −d used here, respectively.
Equivalence of (2.1) and (2.2) can be rigorously proved by writing both equations in an integral form. Indeed, suppose thatφ solves (2.2). In [12], this is understood as for every test function g in H 1 c ([0, R)) and some constant C; see Definition 3.1 in [12]. Recall thatã(dt) = a(dt) − (b(t)) 2 dt, andd = b ′ , that is, by definition, which is clearly equivalent to (2.1). By essentially reversing the steps of the above argument, we find that if ϕ satisfies (2.1), thenφ is a solution of (2.2) (we omit the details), and it follows that (2.1) and (2.2) are indeed equivalent. Part (a) of the theorem is now essentially Lemma 4.2 in [12], part (b) follows from Lemma 5.1 in [12], and part (c) is essentially given in the proof of Lemma 5.1 in [12] (see the last display in p. 954 therein). As mentioned above, alternative proofs are given in Appendix A. Finally, part (d) is stated as Theorem 6.1 in [12].

Harmonic extensions
In this section we describe the class of functions harmonic with respect to operators L of class L in terms of Fourier transform and solutions ϕ ξ (t) of the ODE (2.1), described in Theorem 2.1.
We assume, as in Definition 1.1, that a(dy) is a non-negative measure on [0, R), b(y) is a locally square-integrable real-valued function on [0, R), and a(dy)−(b(y)) 2 dy is non-negative. We commonly use the auxiliary measureã(dy) = a(dy) − (b(y)) 2 dy and function B(y) = y 0 b(t)dt. We study functions u(x, y) on R × [0, R) which are harmonic with respect to the elliptic operator L in the sense of Definition 1.2. We denote byû(ξ, y) the Fourier transform of u(x, y) in variable x, whenever well-defined. We equally often work with the functionũ(x, y) = u(x + B(y), y). Observe thatû(ξ, y) = e iξB(y)û (ξ, y).
We begin with the proof of Proposition 1.3, which asserts the existence and uniqueness of harmonic extensions. The argument is divided into two steps, which correspond to uniqueness and existence, respectively. Lemma 3.1. Suppose that L is an operator of class L. For ξ ∈ R let ϕ ξ be the solution of (2.1) discussed in Section 2. If u is harmonic with respect to L, then for all y ∈ [0, R) we have, for almost all ξ ∈ R, u(ξ, y) =û(ξ, 0)ϕ ξ (y).
Here is the philosophy of the proof: if u is sufficiently regular, then, by (1.6) and Plancherel's theorem, we have (0, R)). By a density argument, this implies that (after a modification on a set of zero Lebesgue measure) for almost all ξ ∈ R the function u(ξ, ·) is a solution of the ODE (2.1), and henceû(ξ, y) =û(ξ, 0)ϕ ξ (y), as desired. Our goal is to make the above idea rigorous in the general case, where only minimal smoothness of u is assumed. By Definition 1.2 (or, more precisely, by (1.7)) and Plancherel's theorem, for every The the ACL characterisation of weak differentiability implies that, after modifyinĝ u(ξ, y) and ∂ yû (ξ, y) on a set of zero Lebesgue measure, we may assume that for every ξ ∈ R the functionû(ξ, ·) is absolutely continuous on [0, R), and the pointwise derivative of this uni-variate function agrees almost everywhere on [0, R) with the weak derivative ∂ yû (ξ, ·) of the bi-variate function. We temporarily work with this modification, and a similar modification ofû(ξ, y) = e iξB(y)û (ξ, y). For every ξ ∈ R, the functionφ(y) =û(ξ, y) is absolutely continuous on [0, R). It follows that also ϕ(y) =û(ξ, y) = e −iξB(y)û (ξ, y) = e −iξB(y)φ (y) is absolutely continuous on [0, R), andφ ′ (y) = e iξB(y) (ϕ ′ (y) + iξb(y)ϕ(y)) for almost all y ∈ [0, R) (we stress that u(x, y) need not be weakly differentiable with respect to y; nevertheless, it turns out thatû(ξ, y) is necessarily weakly differentiable with respect to y).
Suppose that R = ∞. By Theorem 2.1, in this case any solution of (3.3) is either a multiple of ϕ ξ or it diverges to infinity at R − . Since the L 2 (R) norm ofû(·, y) is bounded uniformly with respect to y in [0, ∞) except a set of zero Lebesgue measure (recall that we have modifiedû on a set of zero Lebesgue measure!), |û(ξ, y)| cannot diverge to infinity as y → ∞ for all ξ in a set of positive Lebesgue measure (otherwise, by Fatou's lemma, the L 2 (R) norm ofû(·, y) would diverge to infinity as y → ∞). It follows that for almost all ξ ∈ R there is a number c ξ ∈ C such that for all y ∈ (0, R) we haveû (ξ, y) = c ξ ϕ ξ (y).
The same equality necessarily holds almost everywhere for the original version ofû, before modification on a set of zero Lebesgue measure. Since y →û(·, y) is a continuous map from [0, R) to L 2 (R), and ϕ ξ (0) = 1 for all ξ ∈ R, we have c ξ =û(0, ξ) for almost all ξ ∈ R, and the assertion of the lemma follows.
When R < ∞, the proof is very similar. In this case we know that the L 2 (R) norm ofû(·, y) converges to zero as y → R − except for a set of y of zero Lebesgue measure, and by Theorem 2.1, any solution ϕ of (3.3) is either a multiple of ϕ ξ or |ϕ| has a positive lower limit at R − . Fatou's lemma again implies that for almost every ξ ∈ R the functionû(ξ, ·) is a multiple of ϕ ξ , and the remaining part of the argument is the same as in the case R = ∞. Lemma 3.2. Suppose that L is an operator of class L. For ξ ∈ R let ϕ ξ be the solution of (2.1) discussed in Section 2. If f ∈ L 2 (R), then the formulâ u(ξ, y) =f (ξ)ϕ ξ (y) defines a function u on R × [0, R) harmonic with respect to L.
Proof. We need to verify the conditions listed in Definition 1.2. By Theorem 2.1, for every ξ ∈ R the function ϕ ξ is continuous and bounded by 1. In particular, for every y ∈ [0, R),û(·, y) is in L 2 (R) with norm bounded by the L 2 (R) norm off , and so it is the Fourier transform of some function u(·, y) with L 2 (R) norm no greater than the L 2 (R) norm of f . Since ϕ ξ is continuous on [0, R) for every ξ ∈ R, by the dominated convergence theorem, y →û(·, y) is a continuous map from [0, R) into L 2 (R); thus y → u(·, y) has the same property. A similar argument implies that if R < ∞, then u(·, y) converges in L 2 (R) to zero as y → R − . This proves that condition (a) of Definition 1.2 is satisfied.
The above two lemmas prove Proposition 1.3.

Lemma 3.3.
Suppose that L is an operator of class L ⋆ . For ξ ∈ R let ϕ ξ be the solution of (2.1) discussed in Section 2, and let k(ξ) = −ϕ ′ ξ (0) be the associated Rogers function. If u is a harmonic function for L (in the sense of Definition 1.2) with boundary values f ∈ L 2 (R), then the L 2 (R) limit in the definition of the Dirichlet-to-Neumann operator Kf = ∂ y u(·, 0) = lim y→0 + u(·, y) − u(·, 0) y (3.4) exists if and only if k(ξ)f (ξ) is square integrable, and in this case Proof. By Lemma 3.1, for every y 0 and ξ ∈ R we haveû(ξ, y) =f (ξ)ϕ ξ (y) (after choosing the right representative ofû(·, y)); and conversely, by Lemma 3.2, for every f ∈ L 2 (R) there is a corresponding function u harmonic with respect to L. By Theorem 2.
The above lemma proves the first statement of Theorem 1.7 for operators L of class L ⋆ . As explained after the statement of Theorem 1.7, extension to the class L is immediate. The other part of Theorem 1.7 is a consequence of item (d) of Theorem 2.1.

Examples
In this section we discuss a number of non-local operators and corresponding extension problems. More precisely, we prescribe the coefficients a and b of the reduced elliptic equation Lu = 0, and evaluate, often omitting the technical details, the corresponding solution ϕ ξ (t) of the ODE (2.1). This allows us to identify the corresponding Fourier symbol −k(ξ) = ϕ ′ ξ (0), and eventually leads to the explicit form of the Dirichlet-to-Neumann operator K. Whenever possible, we discuss all three variants: the standard form L, the Eckhardt-Kostenko formL and the divergencelike formL, discussed in Section 1.3. For the convenience of the reader, we recall that Lu = a(dy)∂ xx u + 2b(y)∂ xy u + ∂ yy u, Lũ =ã(dy)∂ xxũ + ∂ yyũ +d(y)∂ xũ , Lu = ∇ x,y · (ȧ(y)∇ x,yu ) + 2ḃ(y)∂ xyu .
The corresponding operatorL in Eckhardt-Kostenko form is simplyLũ = p 2 ∂ xxũ + ∂ yyũ , with coefficientsã(dy) = p 2 dy andd(y) = 0 that do not depend on q. The firstorder term qf ′ in the expression for Kf (x) comes from the somewhat artificial definition (1.23) of the Dirichlet-to-Neumann operator: the function B is defined bỹ d up to a linear term only, and we choose B(y) = −qy in order that B ′ (y) = −q = b(y).

4.3.
Degenerate equations corresponding to one-sided operators without first-order term. As explained in the introduction, there is a one-to-one correspondence between measures a 0 (dy) on [0, R) and complete Bernstein functions ψ. Namely, the Dirichlet-to-Neumann operator associated to the equation Lu = 0 with coefficients a(dy) = a 0 (dy) and b(y) = 0 is K = −ψ(−∂ xx ). By this we mean that the corresponding symbol is equal to −k(ξ) = −ψ(ξ 2 ). We refer to [28] for a detailed discussion.
It is known that ∂ xx can be replaced by a more general non-positive definite operator D x acting in variable x: every operator of the form −ψ(−D x ) arises as the Dirichlet-to-Neumann map for the equation a 0 (dy)D x u + ∂ yy u = 0. In particular, we can set D = −∂ x . We refer to [14] for a related discussion.
In a similar way, we can find the corresponding operatorL in the divergence-like form (1.20), as long as A 0 (y) = a 0 ([0, y)) is strictly positive for y > 0. Let Then B(y) = y 0 a(s)ds, so thatσ(y) = B −1 (y) (see Section 1.3 for the notation), and consequentlẏ It is not difficult to verify that the Dirichlet-to-Neumann operator K associated to the equation Lu = 0 (with L as above) is indeed the operator −ψ(∂ x ). As usual, let B(y) = y 0 b(s)ds, and let ϕ 0,ξ be the solution of the ODE (2.1) with coefficients a 0 (dy) and b 0 (y) = 0, for an arbitrary complex parameter ξ. Then, for ξ > 0, the formulaφ ξ (y) = ϕ 0, √ iξ (y) defines a solution of the ODEφ ′′ = (iξ)a 0 (dy)φ, and thus (by equivalence of (2.1) and (2.2)) ϕ ξ (y) = e −iξB(y)φ ξ (y) = e −iξB(y) ϕ 0, √ iξ (y) is a solution of (2.1) with coefficients a(dy) and b(y) defined above. It is more complicated to show that this is the solution discussed in Section 2, that is, that ϕ ξ is bounded if R = ∞, and ϕ ξ has a zero left limit at R when R < ∞; we omit the details. Since B ′ (0) = b(0) = 0, we find that the symbol −k(ξ) of the corresponding Dirichlet-to-Neuman operator is given by and consequently Kf (x) = −ψ(∂ x )f (x).
We remark that the condition a 0 ([0, y)) > 0 for every y > 0 (required in order to properly define the operatorL in divergence-like form) is equivalent to ψ being unbounded on (0, ∞). This follows, for example, from formula (2.14) in [23]; we omit the details.

Complementary equations and operators.
Following Section 5.7 in [28], where symmetric operators are studied, we say that the operators K and K ♯ of class K are complementary, if their composition KK ♯ is equal to −∂ xx , the onedimensional Laplace operator. In terms of the corresponding symbols −k and −k ♯ , we require that k(ξ)k ♯ (ξ) = ξ 2 for all ξ ∈ R. We note that if k is a Rogers function, then the formula k ♯ (ξ) = ξ 2 /k(ξ) also defines a Rogers function (see Proposition 1.8); therefore, every operator K of class K has a unique complementary operator K ♯ of class K.
In this part it is convenient to work with the equation in a divergence-like forṁ Lu = 0, whereL is given by (1.20). Below we argue that if K is the corresponding Dirichlet-to-Neumann operator and K ♯ is an operator complementary to K, then K ♯ is the Dirichlet-to-Neumann operator associated to the complementary equatioṅ L ♯u♯ = 0, with coefficientṡ a ♯ (y) = 1/ȧ(y),ḃ ♯ (y) = −ḃ (y) (ȧ(y)) 2 , The proof of this claim consists of two steps.
First, we observe that ifu is a harmonic function forL, theṅ u ♯ (x, y) =ȧ(y)∂ yu (x, y) +ḃ(y)∂ xu (x, y) is a harmonic function forL ♯ . If the coefficients are smooth, this is almost immediately verified using the expression (1.20) forL, because the operatorȧ(y)∂ y +ḃ(y)∂ x commutes with ∂ xx . A rigorous proof in the general case is more involved, and we omit the details.
In the second step, we evaluate the Dirichlet-to-Neumann operator K ♯ associated to the equationL ♯u = 0. We already know that K ♯ is an operator of class K. Let f be a smooth, compactly supported function, letu be the harmonic function forL with boundary values f , and letu ♯ be defined as above. Then, by (1.29), (with all limits understood in the sense of L 2 (R)). Therefore,u ♯ is a harmonic extension of Kf forL ♯ . Again using (1.29), we find that ∂ y −ḃ (y) (ȧ(y)) 2 ∂ x u ♯ (x, y) (with the limit again understood in the sense of L 2 (R)). Using the definitions ofu ♯ andL, we conclude that , as desired (once again with all limits in L 2 (R)). As in the first step, we omit the technical details related to regularity of u and u ♯ .
Note that the functionsσ andσ ♯ (describing appropriate change of variables) and coefficients a and a ♯ only depend on the 'symmetric' coefficientȧ, and not on the 'non-symmetric' coefficientḃ. Therefore, just as it was the case for symmetric operators (see Section 5.7 in [28]), we have a([0,σ(y))) = σ(y) 0 a(s)ds = y 0 a(σ(r))(σ) ′ (r)dr = y 0ȧ (r)dr =σ ♯ (y), dr =σ(y), so that y → a([0, y)) and y → a ♯ ([0, y)) is a pair of inverse functions. With the terminology of Krein's spectral theory of strings, this means that a(dy) and a ♯ (dy) are a pair of dual strings.
The above argument only covers a limited class of operators K, namely those operators which are Dirichlet-to-Neumann maps for equations in the divergencelike formLu = 0. However, the corresponding result in the standard form (1.5) (involving dual Krein's strings) is fully general. A detailed proof is based on the theory of dual Krein's strings and it falls beyond the scope of the present article.
It is immediate to see that the coefficients a and b of the equation Lu = 0 in standard form (1.5) correspond to the coefficients a(y) = p 2 y 2/µ−2 ,d(y) = q(1 − µ) µ y 1/µ−2 of the equationLũ = 0 in Eckhardt-Kostenko form (1.19). This leads to a certain simplification of the above expressions for A and B (see below). Similarly, one easily finds that the coefficients of the equationLu = 0 in the divergence-like form are given bẏ The results of this section can be summarised as follows, with a slightly changed notation: we replace C ± by |C ± |. Dirichlet-to-Neumann operators related to the following elliptic equations: (p 2 + q 2 )y 2/µ−2 ∂ xx u − 2qy 1/µ−1 ∂ xy u + ∂ yy u = 0, p 2 y 2/µ−2 ∂ xxũ + ∂ yyũ +qy 1/µ−2 ∂ xũ = 0, ∂ xxu + y µ−1 (ṗ∂ y −q∂ y ) y 1−µ (ṗ∂ y +q∂ y )u = 0, where p,p,ṗ 0 and q,q,q ∈ R, and |q| ṗ, are Fourier multipliers with symbol −k(ξ) = −(A + iB sign ξ)|ξ| µ , where A 0, B ∈ R and | arg(A + iB)| min{µ, 2 − µ}, and can be represented as where C + , C − 0. More precisely, the elliptic equations are all equivalent if note that when µ = 1, thenq is always 0, see Section 4.2 for further discussion. The corresponding coefficients A and B are given by In Section 2 we discussed the properties of solutions of the second-order ordinary differential equation (2.1)). Here ϕ is assumed to be a continuous function on [0, R) such that the second distributional derivative ϕ ′′ corresponds to a measure. In this case necessarily ϕ is absolutely continuous, and the distributional derivative ϕ ′ corresponds to a function of bounded variation, equal almost everywhere to the pointwise derivative of ϕ. Throughout this section, we denote by ϕ ′ (t) the left-continuous version of the point-wise derivative of ϕ. Note that with this convention, if ϕ is a solution of (A.1), then ϕ ′ (0 + ) − ϕ ′ (0) = ξ 2 a({0})ϕ(0).
Unlike in Section 2, here we omit the arguments of functions and measures whenever this causes no confusion. For example, we write equations as in (A.1) rather than as in (2.1).
We also need to prove various properties of this solution; most notably, that the mapping ξ → −ϕ ′ (0) extends to a Rogers function of ξ.
We divide the argument into a number of lemmas. The first one is a completely standard application of Picard's iteration. For the convenience of the reader, we provide full details.
Proof. Clearly, ϕ is a solution of (A.1) with initial conditions ϕ(0) = α, ϕ ′ (0) = β if and only if for t ∈ [0, R) we have Existence of the solution of (A.2) on [0, R) follows by Banach's fixed point theorem. In order to define an appropriate Banach space, we choose C |ξ| and we introduce an auxiliary function M, defined by . We now consider the Banach space X of absolutely continuous functions ϕ such that the second distributional derivative ϕ ′′ corresponds to a measure, and the norm in X, defined by is finite. Here, as usual, ϕ ′ corresponds to the left-continuous version of the derivative of ϕ. Observe that if ϕ ∈ X, then |ϕ(t)| |ϕ(0)| + First of all, I is a well-defined operator on X: if ϕ ∈ X, then |ϕ|/M and |ϕ ′ |/M are bounded by ϕ X , and hence and consequently Iϕ X |α| + |β| + 1 2 ϕ X . In particular, indeed Iϕ belongs to X. In a similar way, if ϕ 1 , ϕ 2 ∈ X, then and therefore It follows that I is a contraction on X, and thus, by Banach's fixed point theorem, I has a unique fixed point ϕ in X. By definition, ϕ(0) = Iϕ(0) = α and ϕ ′ (0) = (Iϕ) ′ (0) = β, and since ϕ = Iϕ, we conclude that ϕ is a solution of (A.2) with the desired initial conditions.
In addition, ϕ is the limit in X of the iterates ϕ n = I n ϕ 0 of I applied to ϕ 0 (t) = 0.
Observe that Therefore, ϕ X is uniformly bounded with respect to ξ such that |ξ| C, and the convergence of ϕ n to ϕ in X is uniform in this region. It follows that for every r ∈ [0, R), ϕ n (t) and ϕ ′ n (t) are uniformly bounded with respect to t ∈ [0, r) and ξ such that |ξ| C, and in this region ϕ n (t) and ϕ ′ n (t) converge uniformly to ϕ(t) and ϕ ′ (t). By Morera's theorem and induction, for every t ∈ [0, r), ϕ n (t) and ϕ ′ n (t) are holomorphic functions of ξ in the region |ξ| < C, and by Morera's theorem and the dominated convergence theorem, ϕ(t) and ϕ ′ (t) have a similar property. Since C > 0 and r ∈ [0, R) are arbitrary, we conclude that ϕ(t) and ϕ ′ (t) are entire functions of ξ for every t ∈ [0, R).
By setting α = 1 and β = 0, we obtain existence of ϕ N . Similarly, α = 0 and β = 1 lead to existence of ϕ D . Clearly, these functions are linearly independent, and their linear combinations are solutions to (A.1). Furthermore, for every t ∈ [0, R), ϕ D (t), ϕ ′ D (t), ϕ N (t) and ϕ ′ N (t) are entire functions of ξ. Banach's fixed point theorem asserts that ϕ D and ϕ N are unique in X. To prove uniqueness of ϕ D and ϕ N in the general class of admissible functions ϕ, one observes that if ϕ is a solution of (A.1), then ϕ ′ is a function with bounded variation, so that |ϕ ′ |/M is bounded on every interval [0, r), where r ∈ [0, R). Repeating the above proof with X replaced by the Banach space X r defined in a similar way, but with R replaced by r, one obtains uniqueness of solutions on every interval [0, r), with r ∈ [0, R). Of course this implies that ϕ D and ϕ N are unique solutions on [0, R), and every solution is a linear combination of ϕ D and ϕ N . The next lemma is a key technical result. Recall that we writeã(dt) = a(dt) − (b(t)) 2 dt and B(t) = t 0 b(s)ds. Lemma A.2. Suppose that Re ξ > 0 and ϕ is a solution of (A.1). Then e −2B Im ξ Re(ξϕϕ ′ ) is a non-decreasing function on [0, R).
Note that although the left-hand side is always a measure, the distributional derivative of ξφφ ′ (rather than the real part of this function) need not correspond to a measure.
Proof. The first assertion follows directly from (A.3) and (A.4). Furthermore, by a direct calculation, has the same sign as (|ϕ| 2 ) ′ .
Lemma A.4. If ξ > 0 and ϕ D , ϕ N are defined as in Lemma A.1, then |ϕ D | 2 is convex and increasing, while |ϕ N | 2 is convex and non-decreasing. If R = ∞, then ϕ D is unbounded, and ϕ N is either unbounded or constant.
We now come to the main results of this section, which we split into the following two lemmas.
Lemma A.5. If R = ∞ and ξ > 0, then there is a unique bounded solution ϕ of (A.1) such that ϕ(0) = 1, and every other solution diverges to infinity at ∞. If R < ∞ and ξ > 0, then there is a unique solution ϕ of (A.1) such that ϕ(0) = 1 and ϕ(R − ) = 0, and every other solution is bounded away from zero in some left neighbourhood of

R.
Proof. Let ϕ D and ϕ N be the solutions described in Lemma A.1. For r ∈ (0, R) we define β r = − ϕ N (r) ϕ D (r) and ϕ r = ϕ N + β r ϕ D , so that ϕ r is a solution of (A.1) satisfying ϕ r (0) = 1 and ϕ r (r) = 0. Note that ϕ D (r) = 0, so that β r and ϕ r are well-defined. Our goal is to prove that ϕ = lim r→R − ϕ r is the desired solution of (2.1).
It remains to prove the other part of item (d) of the lemma. Observe that if ξ > 0, then |φ| 2 = |ϕ| 2 is convex by Lemma A.3, and hence Thus, as in formula (A.7), we have |ξ| 2 [t,R)ã Part (a) of Theorem 2.1 is now a consequence of Lemma A.5, while parts (b) and (c) follow from Lemma A.6.