Soft Riemann-Hilbert problems and planar orthogonal polynomials

Riemann-Hilbert problems are jump problems for holomorphic functions along given interfaces. They arise in various contexts, e.g. in the asymptotic study of certain nonlinear partial differential equations and in the asymptotic analysis of orthogonal polynomials. Matrix-valued Riemann-Hilbert problems were considered by Deift et al. in the 1990s with a noncommutative adaptation of the steepest descent method. For orthogonal polynomials on the line or on the circle with respect to exponentially varying weights, this led to a strong asymptotic expansion in the given parameters. For orthogonal polynomials with respect to planar exponentially varying weights, the corresponding asymptotics was obtained by Hedenmalm and Wennman (2017), using a technically involved construction of an invariant foliation for the orthogonality. Planar orthogonal polynomials are characterized in terms of a matrix dbar-problem (Its, Takhtajan), which we call a soft Riemann-Hilbert problem. Here, we use this perspective to offer a simplified approach based not on foliations but instead on the ad hoc insertion of an algebraic ansatz for the Cauchy potential in the soft Riemann-Hilbert problem. This allows the problem to decompose into a hierarchy of scalar Riemann-Hilbert problems along the interface, which appears as the free boundary for an obstacle problem. Inspired by microlocal analysis, the method allows for control of the solution in such a way that for real-analytic weights, the asymptotics holds in the $L^2$ sense with error $O(e^{-\delta\sqrt{m}})$ in a fixed neighborhood of the closed exterior of the interface, for some constant $\delta>0$, where $m\to+\infty$. Here, $m$ is the degree of the polynomial, and in terms of pointwise asymptotics, the expansion dominates the error term in the exterior domain and across the interface a distance proportional to $m^{-\frac14}$.

1.The Riemann-Hilbert problem for orthogonal polynomials 1.1.Notation.We write C for the complex plane, D := {z ∈ C : |z| < 1} for the open unit disk, D e := {z ∈ C : |z| > 1} for the exterior disk, and T = ∂D for the unit circle.We let dA and ds stand for the normalized area and length measures: dA(z) = π −1 dxdy and ds(z) = (2π) −1 |dz|, for z = x + iy.Moreover, we use the complex derivative notation where, e.g., ∂ x stands for the partial derivative with respect to x.Also, we use ∆ z = ∂ z ∂z = 1 4 (∂ 2 x + ∂ 2 y ) for the Laplacian.When it is not needed for clarity, the subscript z may be omitted.We talk about the standard classes of smooth functions, C k , C ∞ , C ω , for k times differentiable, infinitely differentiable, and realanalytically smooth functions, respectively.Occasionally, we will meet the classes C k,α as well, where k is a nonnegative integer and 0 < α ≤ 1. Functions in C k,α have all their partial derivatives up to and including order k in the Hölder class with exponent α (locally).
We use A ≪ B to denote that A = O(B) in some limit procedure, and if both A = O(B) and B = O(A) then we write A ≍ B.
For 0 < τ < 1 + ε 0 , we consider the obstacle problem with Q being the obstacle from above, while we optimize over functions in Subh τ (C) which are dominated pointwise by the obstacle.More precisely, we consider the function Qτ (z) := sup q(z) : q ≤ Q on C, q ∈ Subh τ (C) .
As a matter of definition, Qτ ≤ Q holds everywhere.It is a consequence of the general methods of obstacle theory that Qτ is C 1,1 -smooth, which means that its second order partial derivatives are all locally bounded in the sense of distribution theory.Moreover, Qτ is in Subh τ (C) as well.The contact set is compact, and off the contact set, Qτ is harmonic.The contact set S τ will also be referred to as the spectral droplet, and it grows with the parameter τ : Moreover, in the sense of distribution theory, we have the equality where 1 E denotes the indicator function of E ⊂ C, which equals 1 on E and vanishes elsewhere.In particular, ∆Q ≥ 0 holds on the spectral droplet S τ .The growth of S τ with the parameter τ is called weighted Laplacian growth, as the increase in mass in 1 Sτ ∆Q is infinitesimally given by harmonic measure for the point at infinity along the interface Γ τ := ∂S τ .For background material, we refer the reader to, e.g., [17], [18], [19], and the book [15].We shall use the potential Q to form the family of exponentially varying weights e −2mQ , where m is a positive real parameter, and we are interested in asymptotic properties as m → +∞.We form the associated Hilbert space L 2 mQ of of equivalence classes of Borel measurable functions with For n = 0, 1, 2, . .., the closed finite-dimensional subspace Pol mQ,n of L 2 mQ consisting of polynomials of degree ≤ n is of interest.If all polynomials of degree ≤ n are represented in Pol mQ,n , that is, if (1.2.4) which is a consequence of (1.2.1) for n < (1 + ε 0 )m − 1, the reproducing kernel of Pol mQ,n can be used to model a system of n + 1 particles that repel each other by Coulomb interaction, while at the same time being confined by the potential Q.To be more precise, the determinantal point process induced by the correlation kernel of Pol mQ,n (a variant of the reproducing kernel) models 2D Coulomb gas with a special value of the temperature parameter in the Gibbs model.For details, see, e.g., [17].The reproducing kernel of Pol mQ,n , call it K m,n , may be written in the form K m,n (z, w) = n j=0 e j (z)e j (w), where e 0 , . . ., e n constitutes an orthonormal basis of the space Pol mQ,n .A particular orthonormal basis is the one that consists of the orthogonal polynomials, normalized so that each one has norm equal to 1. Let P k be the monic orthogonal polynomial of degree k, which means that the leading coefficient equals 1: P k (z) = z k + O(z k−1 ), as |z| → +∞, while P k is orthogonal to the polynomials of lower degree, p, P k mQ = 0, p ∈ Pol mQ,k .
Then the connection between the orthogonal polynomials and the polynomial reproducing kernel becomes K m,n (z, w) = n j=0 P j −2 mQ P j (z)P j (w).
In recent work with Wennman [20], an asymptotic expansion formula for the orthogonal polynomial P k was found, in the instance when k grows in proportion with m as m → +∞.This led to the universal appearance of error function transition for the Coulomb gas along the smooth interface Γ τ = ∂S τ for τ = k/m (see [20], [21]).For background material, we refer to, e.g., [1], [2], and [17].For related work on the real line in the determinantal case, when orthogonal polynomials are relevant and enjoy a three-term recursion, we refer to [12], [13], [10], and the book [9].The proof of the expansion formula in [20] is based on the idea of a foliation of a neighborhood of the interface Γ τ by smooth curves where we have (approximately) the required orthogonality along each curve in the foliation.While that approach has its merits it may be difficult to expand it beyond the given setting, especially given that the algorithm which provides the foliation is rather unwieldy.Here, we supply an alternative approach, which avoids foliations and instead finds appropriate ad-hoc algebraic formulae for the solution of a matrix ∂-problem.In so doing, we obtain a simple algebraic framework for the terms of the asymptotic expansion, expressed in terms of a Neumann series.This permits us to control the growth of the terms in the asymptotic expansion using a scale of Banach spaces, analogous to the Nishida-Nirenberg approach to the Cauchy-Kovalevskaya theorem.We would like to couple the soft Riemann-Hilbert problem (1.3.3) with the following asymptotics at infinity, for a given nonnegative integer n: In connection with the Riemann-Hilbert analysis of orthogonal polynomials on the real line, two rows are traditionally used instead of one, and then the asymptotic requirement (1.3.5) may be written in a more suggestive form (omitted here).The condition (1.3.5)involves asymptotics of both P and Ψ : (1.3.6) as |z| → +∞.
Remark 1.3.1.A word should be said about the terminology.After all, Riemann-Hilbert problems are jump problems across smooth interfaces, and not ∂-problems like (1.3.3).What speaks in favor of using this terminology is the fact that a holomorphic jump problem may be viewed as a ∂-problem in the sense of distribution theory.Indeed, if we apply ∂ to a holomorphic function with a jump along an interface, the result is, in the sense of distribution theory, a measure supported along the interface which amounts to the jump.More importantly, in fact, our method of analysis actually decomposes the problem at hand (given by (1.3.3) and (1.3.5))into a hierarchy of actual Riemann-Hilbert problems, see [20], [21], [22].
The appearance of the complex conjugation of the matrix Y on the right-hand of (1.3.3)results from the need to characterize the monic orthogonal polynomial P of degree n with respect to the inner product of the Hilbert space L 2 mQ given by (1.2.3).The complex conjugation does not make the ∂-problem any easier, as already the scalar equation ∂u = W ū is notoriously difficult, see e.g.[24].The 2 × 2 matrix analogue of the combined problem (1.3.3)-(1.3.5) was considered by Its and Takhtajan [23], and shown to be a way to characterize the orthogonal polynomials.We briefly recapture the basic argument.Given that P is entire, the first condition in (1.3.6) says that P is a monic polynomial of degree n.Any solution Ψ to (1.3.4) with the decay Ψ (z) = O(|z| −1 ) at infinity must be of the form of the Cauchy potential where dA is normalized area measure, provided that the integral converges.The growth condition (1.2.1) gives that so the condition on n to guarantee the convergence of the integral ( Next, we apply the finite geometric series expansion , For this formula to be correct we need all the involved integrals to converge.In view of (1.3.7), this is the case provided that n < (1 + ε 0 )m − 1.We also need to estimate the integral appearing on the right-hand side of (1.3.9).If n ≤ (1 + ε 0 )m − 2 we can estimate for some positive constant C, so that So, we assume that n ≤ (1 + ε 0 )m − 2 holds.Identifying terms in (1.3.9), it now follows that the second condition in (1.3.6)amounts to having so that P is orthogonal to all polynomials of degree ≤ n − 1.It is now uniquely determined by the first condition in (1.3.6), i.e., that it is a monic polynomial of degree n.In conclusion, the solution Y to (1.3.1) with asymptotics (1.3.5)exists and is uniquely determined: it is given in terms of P and Ψ , where Ψ is given by (1.3.7), and P = P n is the monic orthogonal polynomial of degree n with respect to the inner product of L 2 mQ .
1.4.Further assumptions on the confining potential Q.If we assume 0 < τ < 1 + ε 0 , and that Q is real-analytically smooth in a neighborhood of the droplet boundary Γ τ = ∂S τ , while ∆Q > 0 holds on Γ τ , it follows from work of Sakai that Γ τ consists of real-analytically smooth arcs with certain types of possible singularities and isolated points.
To simplify the presentation, we shall focus on the parameter value τ = 1.At τ = 1, we now make the following assumptions: (i) Γ 1 is a real-analytically smooth Jordan curve, and hence S 1 is the simply connected region enclosed by Γ 1 , (ii) Q is real-analytically smooth with ∆Q > 0 in a neighborhood of Γ 1 .
Then Q1 equals Q on S 1 and is harmonic in the complement C \ S τ , with a C 1,1 -smooth transition across the interface Γ 1 .Moreover, if we let Q1 denote the restriction Q1 | C\S1 , this harmonic function extends harmonically across Γ 1 , so that Q1 defines a harmonic function in C \ K, for some compact subset K of the interior of the spectral droplet S 1 .We then form the function R = R 1 := Q − Q1 , which is well-defined and real-valued in C \ K.The function R vanishes along with its normal derivative along the interface Γ 1 while ∆R = ∆Q > 0 holds on Γ 1 .The real-analytic smoothness gives that R is real-analytically smooth near Γ 1 and we know that R > 0 on C \ S 1 .Putting this information together, we see that R(z) ≍ [dist C (z, Γ 1 )] 2 near Γ 1 , and that there exists a real-analytically smooth function R in a neighborhood of Γ 1 which is a square root of R (so that R2 = R), with R > 0 on the exterior side of Γ 1 while R < 0 holds on the interior side.This function has a natural C 2 -smooth extension to C \ K, also denoted by R, which keeps the property R2 = R, provided the compact subset K ⊂ int S 1 is big enough.The extended function is positive exteriorly to Γ 1 and negatively interiorly, while it vanishes precisely along Γ 1 .
1.5.Approach to the solution of the soft Riemann-Hilbert problem.The approach in [20] is based on two things: (i) the collapsed orthogonality equations, and (ii) the the notion of the orthogonal foliation flow.The collapsed orthogonality equations supply algorithmically the successive terms of the expansion formula for the orthogonal polynomial P , but are unable to show that the expansion formula is valid in the first place.The orthogonal foliation flow resolves that difficulty by supplying an invariant foliation of a neighborhood of Γ 1 by curves on which approximate orthogonality to lower order polynomials holds, and integration over the flow parameter gives approximate orthogonality with respect to e −2mQ dA near Γ 1 .Appropriate glueing procedures, supported in part by Hörmander's ∂-estimates, then show that the approximate orthogonal polynomial is close to the actual orthogonal polynomial P .The argument is quite involved and moreover does not connect P with the ∂-potential Ψ in (1.3.1).Here, we propose a novel approach which calculates directly candidates for P and Ψ from a hierarchy of Riemann-Hilbert problems along Γ 1 , based on an ad-hoc hypothesis on the shape of P and Ψ .Once the guess is accurate enough, the algorithm kicks in and produces the successive terms of the asymptotic expansion.The algorithm also allows for control of the growth of the terms in the expansions as a Neumann series, which gives asymptotic control of the error terms, and shows that the expansions are valid in a mesoscopic band of width proportional to m − 1 4 around Γ 1 , which is substantially wider than the microscopic basic length scale m − 1 2 .The width established in [20] was only Am − 1 2 log 1 2 m for an arbitrarily large positive constant A.
1.6.The ad-hoc shape of P, Ψ .We let Q = Q 1 denote the bounded holomorphic function in C\S 1 with ReQ 1 = Q on Γ 1 and ImQ 1 (∞) = 0. Given the real-analytic smoothness of Q near Γ 1 and the smoothness of Γ 1 , Q 1 extends analytically across is the punctured exterior disk, and φ = φ 1 : C \ S 1 → D e is the conformal mapping which is onto and preserves the point at infinity, with φ ′ 1 (∞) > 0, then (1.6.1) In the sequel, we frequently suppress the subscript 1 and prefer to write φ in place of φ 1 .We solve the soft Riemann-Hilbert problem (1.3.3)approximately in an iterative fashion.The algorithm begins with assuming that P is of the form where c m,n is a constant, which is fixed by the requirement that In view of the requirement (1.3.6), the value of the constant is then We keep the ratio τ = n/m fixed, and since we focus on the parameter value τ = 1, we take n = m, so that m is a positive integer which tends to infinity: (1.6.3) where we use the shorthand notation c m := c m,m .For j = 0, 1, 2, . .., the functions F j are all supposed to be independent of the parameter m and as functions bounded and holomorphic in a domain C \ K, on which Q and φ are well-defined (holomorphic and conformal, respectively), for some compact subset K ⊂ int S 1 .In the representation (1.6.3),we do not really care what happens in the compact K, which may seem odd at first glance.However, this is actually a typical feature of the asymptotics of unweighted Bergman polynomials studied by Carleman [5] (see also [6]), following the lead of Szegő [31] (see also [32]).The analogous instance of a fixed weight was only recently analyzed in terms of asymptotic expansion by Hedenmalm and Wennman [22].We should mention that in (1.6.3) the series for F need not converge.Instead, the precise statement would be to say that that for any integer k = 1, 2, 3, . .., (1.6.4) with a uniform error bound in C \ K.In fact, we shall obtain rather strong control of the growth of the error term which in turn tells us how big is the optimal k in the expansion (1.6.4), which gives an expansion with k proportional to m and rapidly decaying error O(e −ǫ √ m ) for some small positive ǫ.The complete analysis requires an analysis of the function Ψ as well.We look for Ψ of the form where V is a regularized version of R: we require that V = R near Γ 1 but allow for V to be different further away.Here, we use the notation for the standard Gaussian error function.Like the function F , the functions A and B are understood in terms of asymptotic expansion, (1.6.6) Like the F j , the functions A j are all defined and bounded holomorphic in C\K, with asymptotics O(|z| −1 ) at infinity, but the B j are only smooth and also have a more fleeting existence: they need only be well-defined in a fixed neighborhood of the loop Γ 1 .The cut-off function χ 1 of (1.6.5)solves this issue.It has 0 ≤ χ 1 ≤ 1 globally, while χ 1 = 1 on a fixed small neighborhood of Γ 1 and χ 1 = 0 off a slightly bigger neighborhood of Γ 1 .We make sure the latter neighborhood is contained strictly inside the region where B is smoothly defined.This way we get a globally welldefined smooth function Bχ 1 by declaring that it vanishes wherever B is undefined.The final step is to show that the thus defined approximate solution is close to the true solution Y = (P , Ψ ) where P is the monic orthogonal polynomial of degree n and Ψ is given by (1.3.7).
As a side remark, if we think of the expressions as quantum waves, then the first is the principal wave and the second serves as a "local glue".Both types of quantum waves are needed for the algorithm discussed below.1.7.Glimpses of the algorithm.Assuming that Ψ is given by the formula (1.6.5),where A is holomorphic, its ∂-derivative equals , where we agree to the notation B 1 := Bχ 1 .On the other hand, if P is given by (1.6.3),we may use the fact (1.6.1): (1.7.2) The equation (1.3.4) requires the expressions (1.7.1) and (1.7.2) to coincide, which reduces to the equation We restrict to Γ 1 and use that R vanishes on the curve: (1.7.4) Taken together with the asymptotics (1.3.6), which amounts to having the asymptotics as |z| → +∞ (1.7.5) the equation (1.7.4) amounts to a Riemann-Hilbert problem in an alternative coordinate chart (where the unit circle T replaces Γ 1 ).We explain the details of this step in Section 3 below.For the moment we observe that the equation involves an unknown B, but luckily it is of higher order, so it does influence the first terms A 0 and F 0 which get determined right away by the Riemann-Hilbert problem.The original equation (1.7.3) contains more information than (1.7.4) alone.Suppose for the moment we were able to solve the equation (1.7.4).Then we may solve for B using (1.7.3): (1.7.6) Since R vanishes only to degree 1 with ∂ R = 0 along Γ 1 , the division produces a smooth function, and if the numerator is real-analytic across Γ 1 , so is the ratio and hence B. The combination of the Riemann-Hilbert "jump" problem (1.7.4) and the smooth division problem (1.7.6) supplies the full algorithm.We may liken it to the Newton algorithm for finding the zeros of polynomials: once we are in the ballpark the algorithm gets us ever closer to the solution.We use (1.7.4) with B = 0 to get the initial A and F .Next, we apply (1.7.6) with the previous choices of A, F , and B to get an updated choice for B. This new B is then implemented into (1.7.4) to get improved A and F .Proceeding iteratively we obtain the full asymptotic expansion.
Our algorithm only gives the asymptotic expansion for the functions A, B, F , and not the functions themselves, with Gevrey class growth of the coefficients.To pass from asymptotic expansion to actual functions with the indicated asymptotic behavior, we may e.g.employ the methods of [16].This gives us smooth functions which are holomorphic whenever the asymptotic expansions are.
1.8.Main results.Apart from the approach based on the ad-hoc shape of the solution as outlined in the prior two subsections, which is interesting per se and may carry over in other situations, it is valuable that we may achieve better control of the error in our approximations than what was available earlier.If we write is appropriately chosen, and the coefficient functions F j come from the algorithm presented in Subsection 1.7, we have the following.Theorem 1.8.1.We assume that the properties (i)-(ii) of Subsection 1.4 hold, and put P ≈ := c m φ m e mQ F ≈ , where F ≈ is as above.Then each F j extends holomorphically to a fixed neighborhood of C \ int S 1 for each j = 0, . . ., κ * (m), while log F 0 is bounded and holomorphic in the same neighborhood.Then there exists a fixed C ∞ -smooth cut-off function χ 1,1 on C, with 0 ≤ χ 1,1 ≤ 1, which equals 1 in an open neighborhood of the closure of C \ S 1 , and vanishes off a slightly larger neighborhood, such that χ 1,1 P ≈ becomes globally well-defined.Moreover, it is close to the monic orthogonal polynomial P of degree n = m in L 2 mQ : Here, c m = c m,m is given by (1.6.2) and ǫ > 0 is a constant that only depends on Q.It follows that holds in the uniform norm as m → +∞, where D m is the union of C \ S 1 and a certain band of width ≍ m − 1 4 around the loop Γ 1 = ∂S 1 .
Remark 1.8.2.The algorithm also calculates the asymptotic expansion of the norm P 2 mQ with an analogously small error term, as we may see from the relations (4.3.6) and (4.3.2),together with the observation that we may replace χ 0,1 in (4.3.2) by its square without any changes in the estimates.
1.9.Organization of the paper.In Section 2, we present useful background material: the Bernstein-Walsh lemma, Hörmander estimates for the ∂-equation, as well as Szegő projections and the exterior Herglotz operator.Then in Section 3, we present the algorithm which produces the successive approximations, at first intuitively, and later with the inclusion of error terms that come from an approximate solution of a partial differential equation for a function we denote by B. The algorithm dismisses what happens inside a fixed compact part of interior of the droplet S 1 , so to justify that, we apply the method of C ∞ -smooth cut-off functions in Section 4, and solve ∂-equations with growth control to obtain polynomials.The proof of the main theorem is also supplied in Section 4, but it requires certain estimates that are only developed in the following section, Section 5.The method in Section 5 involves sequences of nested Banach spaces of real-analytic functions in neighborhoods of the unit circle T, and resembles the Nishida-Nirenberg approach to the Cauchy-Kovalevskaya theorem.
1.10.Remarks on other related work.Determinantal processes are used to model the repulsion of fermions.This applies to finite as well as infinite particle system, see, e.g., [26].The process is induced by a correlation kernel, often a Bergman kernel, or a polynomial Bergman kernel.These Bergman kernels enjoy local asymptotic expansions, following work of Tian, Zelditch, and Catlin (see, e.g., [33]).A version which relies on microlocal analysis is supplied by Berman, Berndtsson, and Sjöstrand [4] (compare with [7]), while a stronger expansion theorem with exponentially decaying error term was recently obtained by Rouby, Sjöstrand, and Vũ Ngo .c [29].It should be mentioned that while these earlier works are somewhat related to the theme of the present paper, the approaches are strictly local and cannot touch the nonlocal phenomena studied here.We also point out that the literature on local Bergman kernel expansion is extensive and that we only supply a sample of the possible references.
A finite particle determinantal process may be thought of in terms of a corresponding Gibbs model with a specific thermodynamical inverse temperature β.As the temperature freezes to 0, the determinantal property is lost, but we expect crystallization to energy minimizing Fekete point configurations.Here, we should mention the Abrikosov conjecture, which maintains that the energetically optimal configuration is achieved in the many particle limit by the hexagonal lattice (see, e.g., [30]; a complete solution is available in the higher dimensional spaces R 8 and R 24 , see [8]).We also mention that a higher-dimensional complex manifold notion of Fekete points was considered by Berman, Boucksom, and Witt Nyström [3].1.11.Acknowledgements.I should thank Aron Wennman for his interest in the approach described here.

Preliminaries
We need two basic estimates, the growth estimate which comes from a polynomial growth assumption, and an estimate of solutions to ∂-problems with polynomial growth, which comes from application of Hörmander's L 2 -method.We prefer to formulate both results for a flexible parameter τ rather than just for τ = 1.We also introduce the Herglotz operator, which solves a boundary value problem.

2.1.
A weighted Bernstein-Walsh lemma.We recall Proposition 2.2.2 in [20].The formulation needs the space L 2 mQ (Ω), which for an open subset Ω ⊂ C is the standard L 2 -space with finite norm Proposition 2.1.1.Let τ = n/m be fixed with 0 < τ < 1 + ε 0 , and suppose that K is a compact subset of S τ .Also, fix a positive real parameter δ.Then there exists a positive constant C such that for any u ∈ L 2 mQ,C\K which is holomorphic on 2.2.Hörmander's estimate for polynomial ∂-problems.As our method relies on neglecting the behavior of the function we are looking for in certain regions of the plane, it is important to be able to reconstruct a globally defined function.We do this by solving the appropriate ∂-problem, while appealing to the following basic estimate, which follows from Proposition 2.4.1 in [20].provided that the right-hand side is finite.

2.3.
Projections and the Herglotz operator.We shall need the Szegő projection It is the orthogonal projection L 2 (T) → H 2 , where H 2 is the standard Hardy space in the unit disk, which may be understood both as a space of holomorphic functions in the disk D and as a subspace of L 2 (T) via nontangential boundary values.For the details, see any book on Hardy spaces, e.g., [11], [14], [25].We also need the orthogonal projections to the Hardy subspaces H 2 0 , H 2 − , and H 2 −,0 , denoted by P H 2 0 , P H 2 − , and P H 2 −,0 .Here, the space H 2 0 is the codimension 1 subspace of H 2 consisting of all functions that vanish at the origin.The space H 2 − may be thought of as a space of conjugate-holomorphic functions in the disk, or, alternatively, by Schwarz reflection, as a space of holomorphic functions in the exterior disk D e .Then f ∈ H 2 − if and only if the reflected function f (1/z) is in H 2 .Moreover, the space H 2 −,0 has codimension 1 in H 2 − , and consists of all functions in H 2 − that vanish at the point at infinity.The space L 2 (T) splits orthogonally as a direct sum in two ways: The corresponding exterior Szegő projections P H 2 − and P H 2 −,0 may be expressed as integrals as well: for z ∈ D e .Analogously, we find that We shall also be interested in the exterior Herglotz transform it follows that where we introduce the notation for the mean value on the unit circle, (2.3.1) If f is a real-valued function on the circle, say in L 2 (T), then the Herglotz transform obtains a solution u = H De f to the problem Re u = f on the circle T, with u holomorphic in D e .As such, it is uniquely determined by the requirements that u ∈ H 2 − and Im u(∞) = 0. We may use the Herglotz operator (or the exterior Szegő projection) to solve jump problems.The version we need is Proposition 2.5.1 in [20], formulated below.Proposition 2.3.1.Suppose that ϑ ∈ L ∞ (T) is of the form ϑ = e u+v , where u, v ∈ H ∞ , and let G be a function in L 2 (T).Then f satisfies if and only if At times we shall also need the harmonic extension to the exterior disk D e given by the Poisson integral If f is continuous, then U De f extends continuously to De , while if f is real-analytic, then U De f extends harmonically across T.

Presentation of the the algorithm
We first use the conformal mapping φ = φ 1 : C \ S 1 → D e and its inverse ϕ = ϕ 1 = φ −1 1 : D e → C \ S 1 to turn our interface loop Γ 1 = ∂S 1 into the unit circle T. Since ϕ remains conformal on a slightly bigger external disk D e (0, ρ) := {z ∈ C : |z| > ρ} with 0 < ρ < 1, this means that we discard what happens inside some compact subset K contained in the interior of the droplet S 1 (we may think of K as the complement of ϕ(D e (0, ̺)).Later, we justify this by using cut-off functions to get a globally defined function, and then we apply Hörmander's ∂-estimate to get a polynomial solution which is close to the function we started with.
3.1.Transfer to the unit circle and the idea of the algorithm.We write ϕ = ϕ 1 := φ −1 1 : D e → C \ S 1 for the indicated conformal mapping, tacitly extended across T, and consider and the associated functions Here, A and F are holomorphic functions in a neighborhood of De with asymptotics in accordance with (1.7.5): as |z| → +∞.In particular, F is bounded with a positive value at infinity, while A is bounded and vanishes at infinity.In terms of these functions, the equation (1.7.3) reads F. Just as before, we split the equation in two separate steps: We will refer to these as steps I and II, which we analyze below in some detail.
Local analysis around the circle T shows that Here and in the sequel, ζ stands for the coordinate function ζ(z) = z.We recall the notation H De for the exterior Herglotz operator from Subsection 2.3.We introduce the function H R given by which is a bounded (and bounded away from 0) holomorphic function in D e with holomorphic extension across T.
2 holds on T, and the value at infinity equals It now follows from (3.1.5)that Step I: In view of (3.1.6),the equation (3.1.3)may be written as From the data (1.7.5), we see that in ( where a 1 is a constant.At the same time, while the constant a 1 is given by (3.1.10) since F(∞) = 1 is our assumed normalization and H R (∞) is known.This ends our analysis of step I.
Step II: We turn to the analysis of the next step, based on the the division formula (3.1.4).To this end, we apply first (3.1.6),to see that we may write where the expression W R is real-analytic near T. We recall that U De stands for the harmonic extension to D e of the restriction to T of a given smooth function.It is an immediate consequence of (3.1.7)that holds in a neighborhood of the closed exterior disk De , in the sense that each term extends harmonically across T. It now follows from (3.1.11)and (3.1.12)that

3.2.
The equation for the function B alone.In view of the expression for the solution to the Riemann-Hilbert problem (3.1.9)from Step I and the relation (3.1.10),the expression for B may be written as since a 1 > 0, which we simplify to Note that in (3.2.1), we have merged steps I and II in a single equation which only concerns the function B (the function A no longer appears).This is an operator equation of the type and the operator L is given by We may attempt to solve the equation (3.2.2) by iteration, that is, by finite Neumann series approximation.

3.3.
Quantitative work: approximate solutions A ≈ , B ≈ , F ≈ .In the preceding subsection, we were a little naive, believing that the equation (3.2.2) would have an exact solution.As for now, we only expect the equation (3.2.2) to be approximately solvable.So, we suppose we have found an approximate solution B ≈ , with (3.3.1) where E m measures the error in solving the equation (3.2.2).Later on, in Section 5, we will show that there exists a B ≈ such E m decays rapidly as m → +∞.Naturally, B ≈ also depends on m, but is more important that the error term reflects this.Based on our previously derived equations (3.1.8)and (3.1.9),we put accordingly In particular, it follows that F ≈ is properly normalized at infinity, as We see from this and (3.1.11)that Finally, if we apply the relation (3.3.1) which defines the error E m , we find that Since ∂R = 2 R ∂ R, we read off the definition of T that where in the last step we used the definitions of B 0 and A ≈ .

The usefulness of cut-off functions
The method outlined in the previous subsection only supplies a tentative solution to the soft Riemann-Hilbert problem of (1.3.3) and (1.3.5).So far, however, we have not even been able to assert much rigorously.For instance, it is not clear that the approximate solution Y = (P , Ψ ) supplied in terms of the triple (A ≈ , B ≈ , F ≈ ) is even close to the actual solution.We will explain why the function F ≈ defines a good approximation P ≈ of the orthogonal polynomial P .4.1.Cut-off functions and a modification of the potential R. Let χ 0 be a radial cut-off function, which is C ∞ -smooth in the plane with 0 ≤ χ 0 ≤ 1 globally: χ 0 = 1 on a fixed (closed) annular neighborhood A 1 of T, while χ 0 = 0 off a slightly larger fixed open annular neighborhood A 0 .We ask of χ 0 furthermore that | ∂χ 0 | 2 ≤ C 1 χ 0 holds for some large positive constant C 1 .The closure clos A 0 is the support of χ 0 , and A 1 ⊂ A 0 ⊂ clos A 0 .We let V ≥ 0 be a smooth potential in the plane such that V = R holds on clos A 0 , the support of χ 0 , and that V = 0 holds only along the circle T, and V is Lipschitz continuous in the plane, with minimal growth V(z) ≥ log |z| for |z| ≫ 1.It is not difficult to see that (4.1.1) for some positive constants C 2 and ε 1 and big enough m.Here, V 2 − := 1 D V 2 equals V 2 in the disk D while it vanishes outside.The property (4.1.1)might not hold with V replaced by R, and it is the reason why we need to introduce V in the first place.It is not too difficult to construct a viable V; the necessary details are left to the interested reader.We put (4.1.2) and use the function Ψ ≈ to define the corresponding approximate function which is then well-defined and smooth in the whole exterior domain to the curve Γ 1 as well as across the interface Γ 1 (so that it is smooth in a neighborhood of Γ 1 ).We take a radius ρ 1 with 0 < ρ 1 < 1 and the associated exterior disk We require the support clos A 0 of the cut-off function χ 0 to be contained inside the exterior disk D e (0, ρ 1 ), and moreover, that ρ 1 is so close to 1 that R is smooth and well-defined in the annulus {z ∈ C : ρ 1 ≤ |z| ≤ 1}.In addition, we require of V that V = R holds there in addition to the annulus clos A 0 .

Application of the
Since V(z) approaches +∞ as |z| → +∞, the potential Ψ ≈ has a limit at infinity, which is given by (4.2.3) In view of the Cauchy-Green formula, we have De(0,ρ1) De(0,ρ1) De(0,ρ1) where in the last step we used that χ 0 = 0 and V = R on the circle T(0, ρ 1 ), as well as the formulae (4.1.2) and (4.2.2).We now turn to estimating the expressions on the right-hand side of (4.2.4).A first observation is that in the integral along the circle T(0, ρ 1 ), R < 0, so that erf(2m ) is small there.To obtain a precise estimate, we may apply integration by parts to see that for x > 0, This gives that (4.2.5) where the positive constant ε 2 may be taken as the minimum of R on the circle T(0, ρ 1 ), and the implicit "O" constant remains bounded as m → +∞.This offers an exponential decay estimate of one term on the right-hand side of (4.2.5).To handle the next term, we use the estimate (4.2.6) where the positive constant ε 3 may be taken as the infimum of R on the set A 0 \ A 1 , and the implicit "O" constant remains bounded as m → +∞.The following term is handled similarly: (4.2.7) in view of (4.1.1)and the maximum principle applied to the bounded holomorphic function A ≈ on D e (0, ρ 1 ).The last term on the right-hand side of (4.2.4) involves the error term E m , and we estimate it as follows: (4.2.8) which tells us about the need to estimate the error E m uniformly on the annulus A 0 .For the bookkeeping, we apply the obtained estimates ( in the context of (4.2.4), to obtain that (4.2.9) m 1 2 De(0,ρ1) where ε 4 stands for the least of the positive constants ε 1 , ε 2 , ε 3 , and we have applied the identity (4.2.3).If G(∞) = 0, and if the norm of E m decays quickly to 0 as m → +∞, it follows from (4.2.9) that χ 0 F ≈ is approximately orthogonal to G in the space L 2 (D e (0, ρ 1 ), e −2mR dA).Finally, we would like to turn this into an approximate orthogonality between the functions F ≈ and G, but involving a related smooth cut-off function which avoids making a smooth cut-off in the exterior disk D e .The cut-off function χ 0 factors χ 0 = χ 0,0 χ 0,1 , where χ 0,j are radial and C ∞smooth with 0 ≤ χ 0,j ≤ 1, and χ 0,0 decreases as the radius increases, whereas χ 0,1 instead increases.Then χ 0,0 = χ 0,1 = 1 on the annulus A 1 , so that in particular χ 0,1 = 1 holds on D e while χ 0,1 = 1 on D.Moreover, for j = 1, 2, | ∂χ 0,j | 2 ≤ C 1 χ 0,j holds with the same constant as for χ 0 .The identity (4.2.10) together with the estimate (4.2.11) , tells us that we may think instead of χ 0,1 F ≈ as being approximately orthogonal to G provided that G(∞) = 0.The estimate (4.2.11) holds for some ε 5 > 0 because R is bounded below by a positive constant on D e (0, ρ 1 ) \ A 1 , and because R has some minimal logarithmic growth at infinity.4.3.Control of the error terms.By a combination of (4.2.9), (4.2.10), and (4.2.11), we obtain that if ε 6 > 0 is the minimum of ε 4 and ε 5 .Here, we applied the formula for a 1 of (3.1.10).We may apply the identity with G = F ≈ , while recalling the normalization F ≈ (∞) = 1, and since, by the maximum principle, De(0,ρ1) , which tells us how to compute the left-hand side norm once we may control the error term E m .Incidentally, it follows that up to small error, the mean value ∂B ≈ HR T must be real-valued.If we write G := φ ′ G• φ and F ≈ = φ ′ F ≈ • φ, which means that we think that the two functions transform as (1, 0)-forms, and put χ 1,1 := χ 0,1 • φ, the change-of-variables formula gives that (4.3.3) and that (4.3.4) We note that the integrals on the right-hand sides of (4.3.3) and (4.3.4) may be thought to extend to all of C by declaring that the integrand vanishes in the complement of ϕ(D(0, ρ 1 )).The extended integrands are then C ∞ -smooth on C. We put F ≈ := φ ′ F ≈ • φ, and write in accordance with (1.6.3), which we think of as the approximate monic orthogonal polynomial of degree m and a general approximate polynomial of degree ≤ m, respectively.We find that (4.3.5) De(0,ρ1) Note that in (4.3.6)we square the cut-off function, which is all right since the square shares all the essential features with the cut-off function itself.Most importantly, (4.3.2) holds with χ 0,1 replaced by its square χ 2 0,1 .We shall apply the identity (4.3.5) to an actual polynomial g of degree ≤ m − 1.In this case, G(∞) = 0, so that the first term on the right-hand side of (4.3.1)vanishes and all that remains are the error terms.To simplify the further discussion we recall the notation Pol n = Pol mQ,n for the subspace of all polynomials with degree ≤ n in the Hilbert space L 2 mQ = L 2 (C, e −2mQ ).We normalize g to have norm 1 in the space Pol mQ,m−1 .The Bernstein-Walsh estimate of Proposition 2.1.1 applies in particular to polynomials of degree ≤ m, and reads for some positive constant C Q and every h ∈ Pol mQ,m .As we apply this estimate to g ∈ Pol mQ,m−1 , it follows that 2 ).By classical duality, we have that and if we combine this with (4.3.1),we arrive at 4.4.Application of Hörmander's ∂-estimate.The next step follows the lines of [20].We want to turn the approximate polynomial P ≈ into an actual polynomial.We put P * := χ 1,1 P ≈ − u 1 , where u 1 solves the ∂-problem ∂u 1 = P ≈ ∂χ 1,1 .
As before, χ 1,1 P ≈ is extended to vanish wherevever P ≈ gets undefined, and hence defines a C ∞ -smooth function in the complex plane C. Clearly, ∂P * = 0 holds throughout and hence P * defines an entire function.By Proposition 2.2.1, based on Hörmander's classical ∂-estimate (which in itself is a dualized version of the corresponding Carleman estimate), there exists a u 1 of growth u 1 (z) = O(|z| n−1 ) at infinity, such that (4.4.1) In view of the growth bound on u 1 , P * must be a polynomial of degree n, and its leading coefficient equals 1, since F ≈ (∞) = 1 and hence, by the definition of c m , After a little bit of rewriting, we find that This expression decays exponentially as m → +∞, as R is strictly positive on the support of ∂χ 0,1 .It now follows from (4.4.1) that (4.4.2) for a positive constant ε 7 > 0. This procedure gives us a monic polynomial P * of degree n with (4.4.3) inf if ε 8 is the least of ε 6 and ε 7 .The minimizing element h ∈ (Pol m−1 ) ⊥ must now be a constant multiple of the true monic orthogonal polynomial P , that is, inf h∈(Polm−1) ⊥ P * − h mQ = P * − αP mQ , where α = P −2 mQ P * , P mQ , simply as a result of the Pythagorean theorem.Later on, in Subsections 5.8 and 5.9, we shall see that it is possible to arrange that as m → +∞, for some constant ǫ > 0 which only depends on R.This assertion is contained in Theorem 5.9.1, and also expressed explicitly in (5.8.4), (5.8.5), and (5.9.6), provided that the annulus A 0 is thin enough and ρ 1 < 1 is sufficiently close to 1 (the control of the norm in H 1 2 σ0 implies uniform control on an annulus about T).Since the rate of decay e −ǫ √ m is slower than exponential decay, we conclude from (4. uniformly in z as m → +∞.Since both P * and P are monic, we find that which means that α is very close to 1. Finally, to appreciate the value of (4.4.5), we should have some general idea of the size of the norm P * mQ .In view of (3.3.3),we have that , a fact which relies on the properties of B ≈ .Actually, this is asserted by Theorem 5.9.1 in the Subsection 5.9 below (cf.equation (5.9.5)).Using the norm identity (4.3.6), this information gives that (4.4.6) and if we use the estimate (4.4.2), it follows that The formula as m → +∞.We can easily turn this into a pointwise estimate by applying the Bernstein-Walsh inequality (4.3.7).However, it is more appealing to relate P directly to the approximate polynomial P ≈ which the algorithm provides.In view of the exponential decay estimate (4.4.2) together with the basic estimate (4.4.4), we have as well (4.4.7) as m → +∞.For instance, this means that in the sense of the L 2 mQ -norm, the polynomial P is small where χ 1,1 = 0.The Bernstein-Walsh estimate of Proposition 2.2.2 in [20] applies here as well, although P − χ 1,1 P ≈ is not a polynomial.It gives that holds uniformly as m → +∞ over the whole region where χ 1,1 = 1 (this is the image under ϕ of a closed exterior disk with a radius ρ 2 such that ρ 1 < ρ 2 < 1), except that we would need to take a step away from the boundary of size at least m − 1 2 .We rewrite (4.4.8) in the form , so that asymptotically as σ → 0 + , we have that ρ(σ) = 1 − σ + O(σ 2 ).For this case, we simplify the notation for the polarized domain: Â(σ) := Â(ρ(σ), σ).
We consider a suitable scale of Banach spaces H ∞ σ indexed by σ.To be precise, the space H ∞ σ consists of all L ∞ functions on Â(σ) that are holomorphic in the variables (z, w), supplied with the In terms of unique analytic continuation, it is clear that the spaces H ∞ σ get smaller as σ increases, that is, We begin with a sufficiently small σ 0 with 0 < σ 0 < 1 such that the functions ζH R appearing in the definitions of L and T may be thought of as elements of H ∞ σ0 .How small σ 0 will need to be then naturally will depend on R. In view of (3.2.2), the operator T consists of first taking the ∂-derivative, then multiplying by H−1 R , and finally applying the operator L. In its turn, the main ingredients for the operator L are P H 2 − and the operator M given by Indeed, the direct relationship reads (5.2.2)

5.3.
The basic growth estimate.The basic result in this section consists of the following couple of estimates.
Theorem 5.3.1.If σ 0 > 0 is as in the preceding subsection, then there exists a positive constant M 1 , depending on R, such that the following estimates hold for f ∈ H ∞ σ0 : In the lemma, we understand the value of k 2k to be equal to 1 for k = 0. We proceed with the necessary steps to obtain the lemma.
5.4.Control of the ∂-derivative.In terms of polarizations, the ∂-derivative is the operator ∂w , the holomorphic differentiation with respect to the variable w.
For a given function f ∈ H ∞ σ , we may think of z as fixed in the smaller annulus A(ρ ′ ), where ρ ′ = ρ(σ ′ ) is given by (5.2.1) and 0 < σ ′ < σ ≤ σ 0 < 1.The condition that (z, w) ∈ Â(σ ′ ) entails that w is at distance at least ρ ′ − ρ to the boundary of the w-slice of Â(σ) where we know that f (z, w) is w-holomorphic and bounded.Moreover, a simple calculation gives that (5.4.1) and hence the standard Cauchy estimates for the derivative show that for f ∈ H 5.5.Control of the operator L. For f with polarization f ⋄ ∈ H ∞ σ , the restriction of f to T has a bounded holomorphic extension to the annulus ρ < |z| < ρ −1 with ρ = ρ(σ) (see (5.2.1)) provided by the formula The Cauchy integral formula allows us to split uniquely is bounded and holomorphic in the exterior disk |z| > ρ.Indeed, we may estimate the norms: (5.5.1) Next, we identify (5.5.2) and, moreover, we see that the harmonic extension U De [f ] is given by , which is bounded and harmonic for |z| > ρ 0 .Hence the polarization of the harmonic extension is given by σ and vanishes on the complex variety 1 − z w = 0.In view of (5.5.1), its norm is easily controlled: (5.5.3) It follows from the Weierstrass division theorem that the function In view of (5.5.3), (5.4.1), and the maximum principle, then, we obtain that This is the main expression to control in the definition of the operator L, the other being controlled by (5.5.1) in view of the identity (5.5.2).Together, these estimates show that (5.5.4) for some positive constant L 1 .
5.6.Estimation of the operator T. We think of ∂ as acting A combination of the estimates (5.4.2) and (5.5.4) shows that for some positive constant L 2 .We may insert the choice σ ′′ = 1 2 (σ + σ ′ ) and use the fact that σ ′′ ≥ σ ′ , to obtain the estimate (5.6.1)

5.7.
The proof of the basic growth estimate.We may now obtain Theorem 5.3.1.
By combining further with the Cauchy estimate (5.4.2), we find that (5.7.2) .
We make the simplest choice possible, all the σ j in an arithmetic progression: With this choice, we read off from (5.7.1) that as claimed.This tells us how to pick M 1 in terms of L 2 and σ 0 , and each depends only on the properties of R. Finally, as for (5.7.2), we obtain which completes the proof.is such that as a function of the parameter s = m −1 , the function B behaves as a if it were from the Gevrey class with exponent 3. We now make an effort to find a universal representative B. We recall from (5.1.2) that the asymptotic expansion for B should be where s = m −1 .The approach in [16] is based on the idea to work with the convergent series where the multiplier functions µ k (s) are chosen appropriately to ensure convergence for any s, while for an individual index k, µ k (s) = 1 + O(s N ) holds for any finite positive integer N as s → 0. This is natural when we would like to have some degree of smoothness in the parameter s = m −1 , for instance when considering Gevrey classes, but in our context, m is a positive integer, and s therefore discrete.A simpler approach which we employ here is is to consider finite cut-offs ("abschnitts") B κ−1 given by (5.8.1) where the positive integer parameter κ will be taken to depend on s = m −1 .We then calculate the error term E m from the equality (3.
It is now a calculus exercise to minimize over the parameter κ.To this end, we provide a lemma.
In particular, it follows that η(t) is convex.We look for critical points, and find the unique critical point t 0 = e − 1 2 β−1 with value η(t 0 ) = −2t 0 = −2e − 1 2 β−1 .At the point t = t 0 , the function η(t) has a global minimum.We attempt to estimate the growth of η(t) for t > t 0 .Since η ′′ (t) is decreasing, the integration-by-parts so the norm of B ≈ is uniformly bounded.To control ∂B ≈ as well, we may rely instead on the second estimate of Theorem 5.3.1.We find that (5.8.6) ∂w (B ≈ ) ⋄ where in the middle we used the elementary estimate (j + 1) j ≤ e j j which we recognize as (1 + j −1 ) j ≤ e.This estimate holds trivially for j = 0 since we interpret 0 0 as 1.
5.9.The approximate functions A ≈ and F ≈ .The equations (3.3.2) and (3.3.3)give us A ≈ and F ≈ in terms of B ≈ , which in turn is given by (5.8.1) with κ = κ * (m), where κ * (m) is as in the preceding subsection.The function ∂B ≈ gets estimated by (5.8.6), and division by HR is fine since its polarization is in H ∞ σ0 by assumption and it is bounded away from 0 in a fixed polarized neighborhood of T (so taking σ 0 > 0 small enough we may safely divide).In view of (5.In the same manner, we obtain from (5.5.1) that (5.9.2) , where we recall from Subsection 2.3 that P H 2 0 denotes the orthogonal projection L 2 (T) → H 2 0 .The identity where "conj" denote complex conjugation, shows that (5.9.2) gives the equivalent estimate (5.9.3) .
We obtain as a consequence of (5.9.2) that (5.9.4)

5. 8 . 1 2
The choice of an approximate function B. The first estimate in Theorem 5.3.1 says that in the Taylor expansion (5.1.2),the norm growth of the coefficients in H ∞ σ0

( 1 −
e −2 ) 2 , 3.1.7),F/H R s bounded and holomorphic in De , so that F/H R ∈ H 2 , whereas ζAH R extends by Schwarz reflection to a bounded holomorphic function on D, so that ζAH R ∈ H 2 − .This means that (3.1.7)is a Riemann-Hilbert problem with jump (2π) − 1 2 m −1 ∂ B/H R , a situation covered by Proposition 2.3.1.By that result, we may solve for F/H R ,