arXiv subjects

Wolfgang Staubach

Publications and source records attributed to Wolfgang Staubach.

At least 19 recordsLinked to original sources

arXiv2026

A unified approach to conformal and modular invariants

In this paper we give a general family of conformal invariants associated to bordered Riemann surfaces endowed with boundary parametrizations, or equivalently compact surfaces endowed with conformal maps. Each invariant is specified by a field of one-forms over a Teichmüller space of infinite conformal type. The invariants are positive, and under certain conditions monotonic. It is shown that these conformal invariants can be viewed as generalized modular invariants on Teichmüller space and as functions on the rigged moduli space of Segal and Vafa. The construction uses an identification of Teichmüller space and the rigged moduli space, as well as analytic work of the authors showing that the transfer or ``overfare'' of harmonic functions sharing boundary values on a quasicircle is bounded. Demanding invariance under various subgroups of the modular group -- equivalently, under the group of quasisymmetric reparametrizations of a sub-collection of borders -- generates conformal invariants. We show that a wide variety of conformal invariants can be obtained through various choices of the field of one-forms. These include modules of doubly-connected domains, period mappings obtained from harmonic measures, inequalities for higher-order conformal invariants, and the Grunsky inequalities and their recent generalizations to Riemann surfaces.

math.DG↗

arXiv2025

Scattering theory on Riemann surfaces I: Schiffer operators, cohomology, and index theorems

We consider a compact Riemann surface $\mathscr{R}$ with a complex of non-intersecting Jordan curves, whose complement is a pair of Riemann surfaces with boundary, each of which may be possibly disconnected. We investigate conformally invariant integral operators of Schiffer, which act on $L^{2}$ anti-holomorphic one-forms on one of these surfaces with boundary and produce holomorphic one-forms on the disjoint union. These operators arise in potential theory, boundary value problems, approximation theory, and conformal field theory, and are closely related to a kind of Cauchy operator. We develop an extensive calculus for the Schiffer and Cauchy operators, including a number of adjoint identities for the Schiffer operators. In the case that the Jordan curves are quasicircles, we derive a Plemelj-Sokhotski jump formula for Dirichlet-bounded functions. We generalize a theorem of Napalkov and Yulmukhametov, which shows that a certain Schiffer operator is an isomorphism for quasicircles. Finally, we characterize the kernels and images, and derive index theorems for the Schiffer operators, which will in turn connect conformal invariants to topological invariants.

math.DG↗

arXiv2025

Scattering theory on Riemann surfaces II: The scattering matrix and generalized period mappings

We construct a scattering theory for harmonic one-forms on Riemann surfaces, obtained from boundary value problems involving systems of curves and the jump problem. We obtain an explicit expression for the scattering matrix in terms of integral operators which we call Schiffer operators, and show that the matrix is unitary. We also obtain a general association of positive polarizing Lagrangian spaces to bordered Riemann surfaces, which unifies the classical polarizations for compact surfaces of algebraic geometry with the infinite-dimensional period map of the universal Teichmüller space.

math.DG↗

arXiv2023

A Fermionic Grunsky operator

To a conformal map $f$ from the disk $\mathbb{D}$ into the complex plane onto a domain with rectifiable Ahlfors-regular boundary, we associate a new kind of Grunsky operator on the Hardy space of the unit disk. This is analogous to the classical Grunsky operator, which itself can be viewed as an operator on Bergman or Dirichlet space. We show that the pull-back of the Smirnov space of the complement of $f(\mathbb{D})$ by $f$ is the graph of the Grunsky operator. We also characterize those domains with rectifiable Ahlfors-regular boundaries such that the Grunsky operator is Hilbert-Schmidt. In particular, we show that if the Grunsky operator is Hilbert-Schmidt, then $f(\mathbb{D})$ is a Weil-Petersson quasidisk. The formulations of the results and proofs make essential use of a geometric treatment of Smirnov space as a space of half-order differentials.

math.CV↗

arXiv2023

Regularity of oscillatory integral operators

In this paper, we establish the global boundedness of oscillatory integral operators on Besov-Lipschitz and Triebel-Lizorkin spaces, with amplitudes in general $S^m_{ρ,δ}(\mathbb{R}^n)$-classes and non-degenerate phase functions in the class $\textart F^k$. Our results hold for a wide range of parameters $0\leqρ\leq1$, $0\leqδ<1$, $0 0$. We also provide a sufficient condition for the boundedness of operators with amplitudes in the forbidden class $S^m_{1,1}(\mathbb{R}^n)$ in Triebel-Lizorkin spaces.

math.AP↗

arXiv2023

Weil-Petersson Teichmüller theory of surfaces of infinite conformal type

Over the past two decades the theory of the Weil-Petersson metric has been extended to general Teichmüller spaces of infinite type, including for example the universal Teichmüller space. In this paper we give a survey of the main results in the Weil-Petersson geometry of infinite-dimensional Teichmüller spaces. This includes the rigorous definition of complex Hilbert manifold structures, Kähler geometry and global analysis, and generalizations of the period mapping. We also discuss the motivations of the theory in representation theory and physics beginning in the 1980s. Some examples of the appearance of Weil-Petersson Teichmüller space in other fields such as fluid mechanics and two-dimensional conformal field theory are also provided.

math.CV↗

arXiv2023

Boundedness of Fourier integral operators on classical function spaces

We investigate the global boundedness of Fourier integral operators with amplitudes in the general Hörmander classes $S^{m}_{ρ, δ}(\mathbb{R}^n)$, $ρ, δ\in [0,1]$ and non-degenerate phase functions of arbitrary rank $κ\in \{0,1,\dots, n-1\}$ on Besov-Lipschitz $B^{s}_{p,q}(\mathbb{R}^n)$ and Triebel-Lizorkin $F^{s}_{p,q}(\mathbb{R}^n)$ of order $s$ and $0<p\leq\infty$, $0<q\leq\infty$. The results that are obtained are all up to the end-point and sharp and are also applied to the regularity of Klein-Gordon-type oscillatory integrals in the aforementioned function spaces.

math.AP↗

arXiv2023

Multilinear oscillatory integrals and estimates for coupled systems of dispersive PDEs

We establish sharp global regularity of a class of multilinear oscillatory integral operators that are associated to nonlinear dispersive equations with both Banach and quasi-Banach target spaces. As a consequence we also prove the (local in time) continuous dependence on the initial data for solutions of a large class of coupled systems of dispersive partial differential equations.

math.AP↗

arXiv2021

A scattering theory of harmonic one-forms on Riemann surfaces

We construct a scattering theory for harmonic one-forms on Riemann surfaces, obtained from boundary value problems through systems of curves and the jump problem. We obtain an explicit expression for the scattering matrix in terms of integral operators which we call Schiffer operators, and show that the matrix is unitary. As a consequence of this scattering theory, we prove index theorems relating these conformally invariant integral operators to topological invariants. We also obtain a general association of positive polarizing Lagrangian spaces to bordered Riemann surfaces, which unifies the classical polarizations for compact surfaces of algebraic geometry with the infinite-dimensional period map of the universal Teichmueller space.

math.DG↗

arXiv2020

Local and global estimates for hyperbolic equations in Besov-Lipschitz and Triebel-Lizorkin spaces

In this paper we establish optimal local and global Besov-Lipschitz and Triebel-Lizorkin estimates for the solutions to linear hyperbolic partial differential equations. These estimates are based on local and global estimates for Fourier integral operators that span all possible scales (and in particular both Banach and quasi-Banach scales) of Besov-Lipschitz spaces $B^s_{p,q}(\R^n)$, and certain Banach and quasi-Banach scales of Triebel-Lizorkin spaces $F^s_{p,q}(\R^n)$

math.AP↗

arXiv2020

Analysis on quasidisks; a unified approach through transmission and jump problems

We give an exposition of results from a crossroad between geometric function theory, harmonic analysis, boundary value problems and approximation theory, which characterize quasicircles. We will specifically expose the interplay between the jump decomposition, singular integral operators and approximation by Faber series. Our unified point of view is made possible by the the concept of transmission.

math.CV↗

arXiv2020

Regularity of Fourier integral operators with amplitudes in general Hörmander classes

We prove the global $L^p$-boundedness of Fourier integral operators that model the parametrices for hyperbolic partial differential equations, with amplitudes in classical Hörmander classes $S^{m}_{ρ, δ}(\mathbb{R}^n)$ for parameters $0<ρ\leq 1$, $0\leq δ<1$. We also consider the regularity of operators with amplitudes in the exotic class $S^{m}_{0, δ}(\mathbb{R}^n)$, $0\leq δ< 1$ and the forbidden class $S^{m}_{ρ, 1}(\mathbb{R}^n)$, $0\leqρ\leq 1.$ Furthermore we show that despite the failure of the $L^2$-boundedness of operators with amplitudes in the forbidden class $S^{0}_{1, 1}(\mathbb{R}^n)$, the operators in question are bounded on Sobolev spaces $H^s(\mathbb{R}^n)$ with $s>0.$ This result extends those of Y. Meyer and E. M. Stein to the setting of Fourier integral operators.

math.AP↗

arXiv2020

Transmission of harmonic functions through quasicircles on compact Riemann surfaces

Let $R$ be a compact surface and let $Γ$ be a Jordan curve which separates $R$ into two connected components $Σ_1$ and $Σ_2$. A harmonic function $h_1$ on $Σ_1$ of bounded Dirichlet norm has boundary values $H$ in a certain conformally invariant non-tangential sense on $Γ$. We show that if $Γ$ is a quasicircle, then there is a unique harmonic function $h_2$ of bounded Dirichlet norm on $Σ_2$ whose boundary values agree with those of $h_1$. Furthermore, the resulting map from the Dirichlet space of $Σ_1$ into $Σ_2$ is bounded with respect to the Dirichlet semi-norm.

math.CV↗

arXiv2019

Regularity properties of Schrödinger integral operators and general oscillatory integrals

We introduce the notion of Schrödinger integral operators and prove sharp local and global regularity results for these (including propagators for the quantum mechanical harmonic oscillator). Furthermore we introduce general classes of oscillatory integral operators with inhomogeneous phase functions, whose local and global regularity are also established in classical function spaces (both in the Banach and quasi-Banach scales). The results are then applied to obtain optimal (local in time) estimates for the solution to the Cauchy problem for variable-coefficient Schrödinger equations as well as other evolutionary partial differential equations.

math.AP↗

arXiv2019

Schiffer comparison operators and approximations on Riemann surfaces bordered by quasicircles

We consider a compact Riemann surface $R$ of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate $R$ into two subsets: a connected Riemann surface $Σ$, and the union $\mathcal{O}$ of a finite collection of simply-connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on $\mathcal{O}$ to the Bergman space of holomorphic forms on $Σ$ is an isomorphism. We then apply this to prove versions of the Plemelj-Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on $Σ$ by elements of Bergman space and Dirichlet space on fixed regions in $R$ containing $Σ$.

math.CV↗

arXiv2019

Global boundedness of multilinear Fourier integral operators

We establish global regularity of multilinear Fourier integral operators that are associated to nonlinear wave equations on product of $L^p$ spaces by proving endpoint boundedness on suitable products spaces containing combinations of the local Hardy space, the local BMO and the $L^2$ spaces.

math.AP↗

arXiv2018

Plemelj-Sokhotski isomorphism for quasicircles in Riemann surfaces and the Schiffer operator

Let $R$ be a compact Riemann surface and $Γ$ be a Jordan curve separating $R$ into connected components $Σ_1$ and $Σ_2$. We consider Calderón-Zygmund type operators $T(Σ_1,Σ_k)$ taking the space of $L^2$ anti-holomorphic one-forms on $Σ_1$ to the space of $L^2$ holomorphic one-forms on $Σ_k$, which we call the Schiffer operators. We extend results of Menahem M. Schiffer and others, which where confined to analytic Jordan curves $Γ$, to general quasicircles in a characterizing manner, and prove new identities for adjoints of the Schiffer operators. Furthermore, we show that if $V$ is the space of anti-holomorphic one-forms orthogonal to $L^2$ forms on $R$ with respect to the inner product on $Σ_1$, then the Schiffer operator $T(Σ_1,Σ_2)$ is an isomorphism onto the set of exact one-forms on $Σ_2$. Using the relation between the Schiffer operator and a Cauchy-type integral involving Green's function, we also derive a jump decomposition (on arbitrary Riemann surfaces) for quasicircles and initial data which are boundary values of Dirichlet-bounded harmonic functions and satisfy the classical algebraic constraints. In particular we show that the jump operator is an isomorphism on the subspace determined by these constraints.

math.CV↗

arXiv2018

Dirichlet space of domains bounded by quasicircles

Consider a multiply-connected domain $Σ$ in the sphere bounded by $n$ non-intersecting quasicircles. We characterize the Dirichlet space of $Σ$ as an isomorphic image of a direct sum of Dirichlet spaces of the disk under a generalized Faber operator. This Faber operator is constructed using a jump formula for quasicircles and certain spaces of boundary values. Thereafter, we define a Grunsky operator on direct sums of Dirichlet spaces of the disk, and give a second characterization of the Dirichlet space of $Σ$ as the graph of the generalized Grunsky operator in direct sums of the space $\mathcal{H}^{1/2}(\mathbb{S}^1)$ on the circle. This has an interpretation in terms of Fourier decompositions of Dirichlet space functions on the circle.

math.CV↗