Real Analytic Boundary Research Articles

Riemann-Hilbert hierarchies for hard edge planar orthogonal polynomials

abstract: We obtain a full asymptotic expansion for orthogonal polynomials with respect to weighted area measure on a Jordan domain $\mathscr{D}$ with real-analytic boundary. The weight is fixed and assumed to be real-analytically smooth and strictly positive, and for any given precision $\varkappa$, the expansion holds with an $\mathrm{O}(N^{-\varkappa-1})$ error in $N$-dependent neighborhoods of the exterior region as the degree $N$ tends to infinity. The main ingredient is the derivation and analysis of Riemann-Hilbert hierarchies---sequences of scalar Riemann-Hilbert problems---which allows us to express all higher order correction terms in closed form. Indeed, the expansion may be understood as a Neumann series involving an explicit operator. The expansion theorem leads to a semiclassical asymptotic expansion of the corresponding hard edge probability wave function in terms of distributions supported on $\partial\mathscr{D}$.

Singularities almost always scatter: Regularity results for non‐scattering inhomogeneities

AbstractIn this paper we examine necessary conditions for an inhomogeneity to be non‐scattering, or equivalently, by negation, sufficient conditions for it to be scattering. These conditions are formulated in terms of the regularity of the boundary of the inhomogeneity. We examine broad classes of incident waves in both two and three dimensions. Our analysis is greatly influenced by the analysis carried out by Williams in order to establish that a domain, which does not possess the Pompeiu Property, has a real analytic boundary. That analysis, as well as ours, relies crucially on classical free boundary regularity results due to Kinderlehrer and Nirenberg, and Caffarelli.

Simultaneous recovery of attenuation and source density in SPECT

We show that under a certain non-cancellation condition the attenuated Radon transform uniquely determines piecewise constant attenuation $ a $ and piecewise $ C^2 $ source density $ f $ with jumps over real analytic boundaries possibly having corners. We also look at numerical examples in which the non-cancellation condition fails and show that unique reconstruction of multi-bang $ a $ and $ f $ still appears to be possible although not yet explained by theoretical results.

Nonabelian Ramified Coverings and $$L^p$$-boundedness of Bergman Projections in $${\mathbb {C}}^2$$

In this work we explore the theme of $L^p$-boundedness of Bergman projections of domains that can be covered, in the sense of ramified coverings, by "nice" domains (e.g. strictly pseudoconvex domains with real analytic boundary). In particular, we focus on two-dimensional normal ramified coverings whose covering group is a finite unitary reflection group. In an infinite family of examples we are able to prove $L^p$-boundedness of the Bergman projection for every $p\in(1,\infty)$.

On the Automorphism Group of Certain Short ℂ2’s

Abstract For a Hénon map of the form $H(x, y) = (y, p(y) - ax)$, where $p$ is a polynomial of degree at least two and $a \not = 0$, it is known that the sub-level sets of the Green’s function $G^+_H$ associated with $H$ are Short $\mathbb {C}^2$’s. For a given $c> 0$, we study the holomorphic automorphism group of such a Short $\mathbb {C}^2$, namely $\Omega _c = \{ G^+_H < c \}$. The unbounded domain $\Omega _c \subset \mathbb {C}^2$ is known to have smooth real analytic Levi-flat boundary. Despite the fact that $\Omega _c$ admits an exhaustion by biholomorphic images of the unit ball, it turns out that its automorphism group, $\textrm {Aut}(\Omega _c)$, cannot be too large. On the other hand, examples are provided to show that these automorphism groups are non-trivial in general. We also obtain necessary and sufficient conditions for such a pair of Short $\mathbb {C}^2$’s to be biholomorphic.

Direct proof of termination of the Kohn algorithm in the real-analytic case

In 1979 J.J. Kohn gave an indirect argument via the Diederich-Fornaess Theorem showing that finite D'Angelo type implies termination of the Kohn algorithm for a pseudoconvex domain with real-analytic boundary. We give here a direct argument for this same implication using the stratification coming from Catlin's notion of a boundary system as well as algebraic geometry on the ring of real-analytic functions. We also indicate how this argument could be used in order to compute an effective lower bound for the subelliptic gain in the $\bar\partial$-Neumann problem in terms of the D'Angelo type, the dimension of the space, and the level of forms provided that an effective Łojasiewicz inequality can be proven in the real-analytic case and slightly more information obtained about the behavior of the sheaves of multipliers in the Kohn algorithm.

Free boundary methods and non-scattering phenomena

We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from the theory of free boundary problems.

Algorithms for subelliptic multipliers in ℂ²

We give examples of pseudoconvex domains of finite type in C 2 \mathbb {C}^2 where the Kohn algorithm for subelliptic estimates fails to yield an effective lower bound for the order of subellipticity in terms of the type. We show how to modify the algorithm to obtain an effective procedure to prove subellipticity on domains of finite type in C 2 \mathbb {C}^2 with real analytic boundary satisfying a condition slightly stronger than pseudoconvexity. We close with a generalization to higher dimensions.

Fourier transforms of indicator functions, lattice point discrepancy, and the stability of integrals

We prove sharp estimates for Fourier transforms of indicator functions of bounded open sets in \({\mathbb {R}}^n\) with real analytic boundary, as well as nontrivial lattice point discrepancy results. Both are derived from estimates on Fourier transforms of hypersurface measures. Relations with maximal averages are discussed, connecting two conjectures of Iosevich and Sawyer (Adv Math 132(1):46–119, 1997). We also prove a theorem concerning the stability under function perturbations of the growth rate of a real analytic function near a zero. This result is sharp in an appropriate sense. It implies a corresponding stability result for the local integrablity of negative powers of a real analytic function near a zero.

REGULARITY OF THE WEIGHTED BERGMAN PROJECTION ON THE FOCK–BARGMANN–HARTOGS DOMAIN

The Fock–Bargmann–Hartogs domain $D_{n,m}(\,\unicode[STIX]{x1D707}):=\{(z,w)\in \mathbb{C}^{n}\times \mathbb{C}^{m}:\Vert w\Vert ^{2}<e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}\}$, where $\unicode[STIX]{x1D707}>0$, is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. We compute the weighted Bergman kernel of $D_{n,m}(\,\unicode[STIX]{x1D707})$ with respect to the weight $(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}$, where $\unicode[STIX]{x1D70C}(z,w):=\Vert w\Vert ^{2}-e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}$ and $\unicode[STIX]{x1D6FC}>-1$. Then, for $p\in [1,\infty ),$ we show that the corresponding weighted Bergman projection $P_{D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}}$ is unbounded on $L^{p}(D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}})$, except for the trivial case $p=2$. This gives an example of an unbounded strongly pseudoconvex domain whose ordinary Bergman projection is $L^{p}$ irregular when $p\in [1,\infty )\setminus \{2\}$, in contrast to the well-known positive $L^{p}$ regularity result on a bounded strongly pseudoconvex domain.

Flat Bundles Over Some Compact Complex Manifolds

We construct examples of flat fiber bundles over the Hopf surface such that the total spaces have no pseudoconvex neighborhood basis, admit a complete Kahler metric, or are hyperconvex but have no nonconstant holomorphic functions. For any compact Riemannian surface of positive genus, we construct a flat $${\mathbb {P}}^1$$ bundle over it and a Stein domain with real analytic boundary in it whose closure does not have pseudoconvex neighborhood basis. For a compact complex manifold with positive first Betti number, we construct a flat bundle over it such that the total space is hyperconvex but admits no nonconstant holomorphic functions.

The Cauchy Integral Formula, Quadrature Domains, and Riemann Mapping Theorems

It is well known that a domain in the plane is a quadrature domain with respect to area measure if and only if the function z extends meromorphically to the double, and it is a quadrature domain with respect to boundary arc length measure if and only if the complex unit tangent vector function T(z) extends meromorphically to the double. By applying the Cauchy integral formula to $$\bar{z}$$ , we will shed light on the density of area quadrature domains among smooth domains with real analytic boundary. By extending $$\bar{z}$$ and T(z) and applying the Cauchy integral formula to the Szegő kernel, we will obtain conformal mappings to nearby arc length quadrature domains and even domains that are like the unit disc in that they are simultaneously area and arc length quadrature domains. These “double quadrature domains” can be thought of as analogs of the unit disc in the multiply connected setting and the mappings so obtained as generalized Riemann mappings. The main theorems of this paper are not new, but the methods used in their proofs are new and more constructive than previous methods. The new computational methods give rise to numerical methods for computing generalized Riemann maps to nearby quadrature domains.

Pointwise Bounds for Steklov Eigenfunctions

Let $$(\Omega ,g)$$ be a compact, real-analytic Riemannian manifold with real-analytic boundary $$\partial \Omega .$$ The harmonic extensions of the boundary Dirichlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfunctions decay exponentially into the interior in terms of the Dirichlet-to-Neumann eigenvalues and give a sharp rate of decay to first order at the boundary. The proof uses the Poisson representation for the Steklov eigenfunctions combined with sharp h-microlocal concentration estimates for the boundary Dirichlet-to-Neumann eigenfunctions near the cosphere bundle $$S^*\partial \Omega .$$ These estimates follow from sharp estimates on the concentration of the FBI transforms of solutions to analytic pseudodifferential equations $$Pu=0$$ near the characteristic set $$\{\sigma (P)=0\}$$ .

Sup norms of Cauchy data of eigenfunctions on manifolds with concave boundary

ABSTRACTWe prove that the Cauchy data of Dirichlet or Neumann Δ- eigenfunctions of Riemannian manifolds with concave (diffractive) boundary can only achieve maximal sup norm bounds if there exists a self-focal point on the boundary, i.e., a point at which a positive measure of geodesics leaving the point return to the point. In the case of real analytic Riemannian manifolds with real analytic boundary, maximal sup norm bounds on boundary traces of eigenfunctions can only be achieved if there exists a point on the boundary at which all geodesics loop back. As an application, the Dirichlet or Neumann eigenfunctions of Riemannian manifolds with concave boundary and non-positive curvature never have eigenfunctions whose boundary traces achieve maximal sup norm bounds. The key new ingredient is the Melrose–Taylor diffractive parametrix and Melrose’s analysis of the Weyl law.

The $$L^p$$ L p CR Hartogs separate analyticity theorem for convex domains

In this note, a very general theorem of the CR Hartogs type is proved for almost generic strictly convex domains in \(\mathbf{C}^{\mathbf{n}}\) with real analytic boundary. Given such a domain D, and given an \(L^p\) function f on \(\partial D\) which has holomorphic extensions on the slices of D by complex lines parallel to the coordinate axes, f must be CR—i.e. f has a holomorphic extension to D which is in the Hardy space \(H^p(D)\). This is the first general result of CR Hartogs type which is not for the ball, or some other domain with symmetries, and holds for \(L^1\) functions. As a corollary, the Szegő kernel is shown to be the strong operator limit of \((\pi _1 \pi _2)^n\), where \(\pi _i\) is the projection onto \(z_i\) holomorphically extendible \(L^2(\partial D)\) functions (in \(\mathbf{C}^{\mathbf{2}}\), with a slightly more complicated formula in \(\mathbf{C}^{\mathbf{n}}\)).

Levi problem in complex manifolds

Let U be a pseudoconvex open set in a complex manifold M. When is U a Stein manifold? There are classical counter examples due to Grauert, even when U has real-analytic boundary or has strictly pseudoconvex points. We give new criteria for the Steinness of U and we analyze the obstructions. The main tool is the notion of Levi-currents. They are positive \({\partial \overline{\partial }}\)-closed currents T of bidimension (1, 1) and of mass 1 directed by the directions where all continuous psh functions in U have vanishing Levi-form. The extremal ones, are supported on the sets where all continuous psh functions are constant. We also construct under geometric conditions, bounded strictly psh exhaustion functions, and hence we obtain Donnelly–Fefferman weights. To any infinitesimally homogeneous manifold, we associate a foliation. The dynamics of the foliation determines the solution of the Levi-problem. Some of the results can be extended to the context of pseudoconvexity with respect to a Pfaff-system.

Codimension Two CR Singular Submanifolds and Extensions of CR Functions

Let $$M \subset {\mathbb {C}}^{n+1}$$ , $$n \ge 2$$ , be a real codimension two CR singular real analytic submanifold that is nondegenerate and holomorphically flat. We prove that every real analytic function on M that is CR outside the CR singularities extends to a holomorphic function in a neighbourhood of M. Our motivation is to prove the following analogue of the Hartogs–Bochner theorem. Let $$\Omega \subset {\mathbb {C}}^n \times {\mathbb {R}}$$ , $$n \ge 2$$ , be a bounded domain with a connected real analytic boundary such that $$\partial \Omega $$ has only nondegenerate CR singularities. We prove that if $$f :\partial \Omega \rightarrow {\mathbb {C}}$$ is a real analytic function that is CR at CR points of $$\partial \Omega $$ , then f extends to a holomorphic function on a neighbourhood of $${\overline{\Omega }}$$ in $${\mathbb {C}}^n \times {\mathbb {C}}$$ .

Almost strictly pseudo-convex domains. Examples and applications

In this work we introduce a class of smoothly bounded domains Ω in Cn with few non strictly pseudo-convex points in @ Ω with respect to a certain Minkowski dimension. We call them almost strictly pseudo-convex, aspc. For these domains we prove that a canonical measure associated to a separated se- quence of points in Ω which projects on the set of weakly pseudo-convex points is automatically a geometric Carleson measure. This class of aspc domains con- tains of course strictly pseudo-convex domains but also pseudo-convex domains of finite type in C2 , domains locally diagonalizable, convex domains of finite type 2 in Cn , domains with real analytic boundary and domains like |z1 | |z2 | 2 } < 1, which are not of finite type. + exp{1 As an application we study interpolating sequences for convex domains of finite type in Cn . After proving a Carleson-type embedding theorem, we get that if Ω is a convex domain of finite type in Cn and if S ⇢ Ω is a dual bounded sequence of points in H p (Ω), if p = 1 then for any q < 1, S is H q (Ω) interpolating with the linear extension property and if p < 1 then S is H q () interpolating with the linear extension property, provided that q < min( p, 2).

Inverse obstacle problems with backscattering or generalized backscattering data in one or two directions

This paper is concerned with scattering by a smooth compact obstacle in R. The main results are that for certain classes of obstacles knowledge of the the scattering amplitude in only one or two pairs of incident and reflected directions suffices to recover the Taylor series of a boundary defining function for the obstacle at a point. Thus certain obstacles with real analytic boundaries can be determined.

Compactness estimate for the -Neumann problem on a Q-pseudoconvex domain

The purpose of this article is to discuss compactness estimate for the -Neumann problem at a boundary with mixed Levi signature. We consider a domain D ⊂⊂ ℂ n which is q-pseudoconvex and introduce the ‘(q − P) property’ which is the natural variant of the classical ‘P property’ by Catlin adapted to the new class of domains. In Section 1, we prove that (q − P) property implies compactness estimate. Next, in Section 2, we introduce the notion of ‘weak q regularity’ of ∂D, the natural variant of the classical ‘weak regularity’ by Catlin and prove that it implies (q − P) property. In Section 3, we recall how compactness yields Sobolev estimates. In Section 4, we give a criterion for weak q regularity of a real-analytic boundary and finally, in Section 5, we exhibit a class of weakly q regular domains.