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Wolfram Bauer

Publications and source records attributed to Wolfram Bauer.

18 recordsLinked to original sources
arXiv2026

Sub-Laplacians on Compact Lie Groups: Heat Kernels, Distance, and Zeta Determinants

We study heat kernels, sub-Riemannian distances, and spectral zeta functions of sub-Laplacians determined by closed connected subgroups of compact Lie groups. Combining Hall's inversion formula with the affine lattice expansion of the compact group heat kernel, we derive a Cartan integral representation involving the group's exponential lattice. For two-step compact Lie pairs, the full algebraic small-time heat trace expansion is determined, up to an exponentially small remainder, by two explicit constants $C_{G,L}$ and $\beta_{G,L}$. This expansion determines all heat coefficients, the poles and residues of the reduced spectral zeta function, and its values at nonpositive integers. For the transvective symmetric subclass, we prove uniform vertical asymptotics for the Carnot-Carath\'eodory distance; the leading coefficient $\mathfrak F_{G,K}(Z)$ is the attained minimum of a finite-dimensional singular value problem. For simply connected two-step pairs, we obtain an exact decomposition of the zeta-regularized determinant into local, lattice, and spectral terms, with exponential truncation estimates. We specialize these results to block subgroups of $\mathrm{SU}(N)$, recovering the classical $\mathrm{SU}(2)$ and CR sphere spectra.

math.CA↗
arXiv2026

Orlicz Space Interpolation and Its Applications to Operator Convolution

We prove a strong-type interpolation result for noncommutative Orlicz spaces over semifinite von Neumann algebras. Based on this result, we obtain Young-type convolution estimates for the Weyl pseudodifferential symbols of operators in appropriate Orlicz-Schatten spaces. Equivalently, we prove convolution estimates of Young type for Werner's function-operator convolutions in quantum harmonic analysis.

math.FA↗
arXiv2025

Convolutions of Orlicz spaces and Orlicz Schatten classes, with applications to Toeplitz operators

Let $\Phi$ be a Young function. We study convolution properties for symbol classes $s_{A,\Phi}$, which consist of all $a$ such that the pseudo-differential operator $\operatorname{Op} _A(a)$ is in the Orlicz Schatten class $\mathscr I _\Phi (L^2(\mathbf R^d))$. Especially we prove Young type results for such classes. We apply the results on Toeplitz operators and prove Orlicz Schatten properties of such operators.

math.FA↗
arXiv2024

On a class of Nonlinear Grushin equations

In this paper, we study two kinds of nonlinear degenerate elliptic equations containing the Grushin operator. First, we prove radial symmetry and a decay rate at infinity of solutions to such a Grushin equation by using the moving plane method in combination with suitable integral inequalities. Applying similar methods, we obtain nonexistence results for solutions to a second type of Grushin equation in Euclidean half space. Finally, we derive a priori estimates and the existence for positive solutions to more general types of Grushin equations by employing blow up analysis and topological degree methods, respectively.

math.AP↗
arXiv2024

Operators in the Fock-Toeplitz algebra

We consider various classes of bounded operators on the Fock space $F^2$ of Gaussian square integrable entire functions over the complex plane. These include Toeplitz (type) operators, weighted composition operators, singular integral operators, Volterra-type operators and Hausdorff operators and range from classical objects in harmonic analysis to more recently introduced classes. As a leading problem and closely linked to well-known compactness characterizations we pursue the question of when these operators are contained in the Toeplitz algebra. This paper combines a (certainly in-complete) survey of the classical and more recent literature including new ideas for proofs from the perspective of quantum harmonic analysis (QHA). Moreover, we have added a number of new theorems and links between known results.

math.FA↗
arXiv2024

Complete integrability of subriemannian geodesic flows on $\mathbb{S}^7$

Four subriemannian (SR) structures over the Euclidean sphere $\mathbb{S}^7$ are considered in accordance to the previous literature. The defining bracket generating distribution is chosen as the horizontal space in the Hopf fibration, the quaternionic Hopf fibration or spanned by a suitable number of canonical vector fields. In all cases the induced SR geodesic flow on $T^*\mathbb{S}^7$ is studied. Adapting a method by A. Thimm, a maximal set of functionally independent and Poisson commuting first integrals are constructed, including the corresponding SR Hamiltonian. As a result, the complete integrability in the sense of Liouville is proved for the SR geodesic flow. It is observed that these first integrals arise as the symbols of commuting second order differential operators one of them being a (not necessarily intrinsic) sublaplacian. On the way one explicitly derives the Lie algebras of all SR isometry groups intersected with $O(8)$.

math.DG↗
arXiv2022

Self-Adjointness of Toeplitz Operators on the Segal-Bargmann Space

We prove a new criterion that guarantees self-adjointness of Toeplitz operator with unbounded operator-valued symbols. Our criterion applies, in particular, to symbols with Lipschitz continuous derivatives, which is the natural class of Hamiltonian functions for classical mechanics. For this we extend the Berger-Coburn estimate to the case of vector-valued Segal-Bargmann spaces. Finally, we apply our result to prove self-adjointness for a class of (operator-valued) quadratic forms on the space of Schwartz functions in the Schrödinger representation.

math-ph↗
arXiv2022

Local Invariants and Geometry of the sub-Laplacian on H-type Foliations

$H$-type foliations $(\mathbb{M},\mathcal{H},g_{\mathcal{H}})$ are studied in the framework of sub-Riemannian geometry with bracket generating distribution defined as the bundle transversal to the fibers. Equipping $\mathbb{M}$ with the Bott connection we consider the scalar horizontal curvature $κ_{\mathcal{H}}$ as well as a new local invariant $τ_{\mathcal{V}}$ induced from the vertical distribution. We extend recent results on the small-time asymptotics of the sub-Riemannanian heat kernel on quaternion-contact (qc-)manifolds due to A. Laaroussi and we express the second heat invariant in sub-Riemannian geometry as a linear combination of $κ_{\mathcal{H}}$ and $τ_{\mathcal{V}}$. The use of an analog to normal coordinates in Riemannian geometry that are well-adapted to the geometric structure of $H$-type foliations allows us to consider the pull-back of Korányi balls to $\mathbb{M}$. We explicitly obtain the first three terms in the asymptotic expansion of their Popp volume for small radii. Finally, we address the question of when $\mathbb{M}$ is locally isometric as a sub-Riemannian manifold to its $H$-type tangent group.

math.DG↗
arXiv2022

Resolvent algebra in Fock-Bargmann representation

The resolvent algebra $\mathcal{R}(X, σ)$ associated to a symplectic space $(X, σ)$ was introduced by D. Buchholz and H. Grundling as a convenient model of the canonical commutation relation (CCR) in quantum mechanics. We first study a representation of $\mathcal{R}(\mathbb{C}^n, σ)$ with the standard symplectic form $σ$ inside the full Toeplitz algebra over the Fock-Bargmann space. We prove that $\mathcal{R}(\mathbb{C}^n, σ)$ itself is a Toeplitz algebra. In the sense of R. Werner's correspondence theory we determine its corresponding shift-invariant and closed space of symbols. Finally, we discuss a representation of the resolvent algebra $\mathcal{R}(\mathcal{H}, \tildeσ)$ for an infinite dimensional symplectic separable Hilbert space $(\mathcal{H}, \tildeσ)$. More precisely, we find a representation of $\mathcal{R}(\mathcal{H}, \tildeσ)$ inside the full Toeplitz algebra over the Fock-Bargmann space in infinitely many variables.

math.FA↗
arXiv2021

Trivializable and quaternionic subriemannian structure on $\mathbb{S}^7$ and subelliptic heat kernel

On the seven dimensional Euclidean sphere $\mathbb{S}^7$ we compare two subriemannian structures with regards to various geometric and analytical properties. The first structure is called trivializable and the underlying distribution $\mathcal{H}_T$ is induced by a Clifford module structure of $\mathbb{R}^8$. More precisely, $\mathcal{H}_T$ is rank $4$, bracket generating of step two and generated by globally defined vector fields. The distribution $\mathcal{H}_{Q}$ of the second structure is of rank 4 and step two as well and obtained as the horizontal distribution in the quaternionic Hopf fibration $\mathbb{S}^3\hookrightarrow\mathbb{S}^7\rightarrow\mathbb{S}^4$. Answering a question in arXiv:0901.1406 we first show that $\mathcal{H}_{Q}$ does not admit a global nowhere vanishing smooth section. In both cases we determine the Popp measures, the intrinsic sublaplacians $Δ_{sub}^T$ and $Δ_{sub}^{Q}$ and the nilpotent approximations. We conclude that both subriemannian structures are not locally isometric and we discuss properties of the isometry group. By determining the first heat invariant of the sublaplacians it is shown that both structures are also not isospectral in the subriemannian sense.

math.DG↗
arXiv2019

Spectral theory of a class of nilmanifolds attached to clifford modules

We determine the spectrum of the sub-Laplacian on pseudo H-type nilmanifolds and present pairs of isospectral but non-diffeomorphic nilmanifolds with respect to the sub-Laplacian. We observe that these pairs are also isospectral with respect to the Laplacian. More generally, our method allows us to construct an arbitrary number of isospectral but mutually non-diffeomorphic nilmanifolds. Finally, we present two nilmanifolds of different dimensions such that the short time heat trace expansions of the corresponding sub-Laplace operators coincide up to a term which vanishes to infinite order as time tends to zero.

math.SP↗
arXiv2019

Berger-Coburn theorem, localized operators, and the Toeplitz algebra

We give a simplified proof of the Berger-Coburn theorem on the boundedness of Toeplitz operators and extend this theorem to the setting of $p$-Fock spaces $(1\leq p \leq \infty)$. We present an overview of recent results by various authors on the compactness characterization via the Berezin transform for certain operators acting on the Fock space. Based on these results we present three new characterizations of the Toeplitz $C^*$ algebra generated by Toeplitz operators with bounded symbols.

math.OA↗
arXiv2019

The fundamental solution of a class of ultra-hyperbolic operators on Pseudo $H$-type groups

Pseudo $H$-type Lie groups $G_{r,s}$ of signature $(r,s)$ are defined via a module action of the Clifford algebra $C\ell_{r,s}$ on a vector space $V \cong \mathbb{R}^{2n}$. They form a subclass of all 2-step nilpotent Lie groups and based on their algebraic structure they can be equipped with a left-invariant pseudo-Riemannian metric. Let $\mathcal{N}_{r,s}$ denote the Lie algebra corresponding to $G_{r,s}$. A choice of left-invariant vector fields $[X_1, \ldots, X_{2n}]$ which generate a complement of the center of $\mathcal{N}_{r,s}$ gives rise to a second order operator \begin{equation*} Δ_{r,s}:= \big{(}X_1^2+ \ldots + X_n^2\big{)}- \big{(}X_{n+1}^2+ \ldots + X_{2n}^2 \big{)}, \end{equation*} which we call ultra-hyperbolic. In terms of classical special functions we present families of fundamental solutions of $Δ_{r,s}$ in the case $r=0$, $s>0$ and study their properties. In the case of $r>0$ we prove that $Δ_{r,s}$ admits no fundamental solution in the space of tempered distributions. Finally we discuss the local solvability of $Δ_{r,s}$ and the existence of a fundamental solution in the space of Schwartz distributions.

math.AP↗
arXiv2018

Algebras of Toeplitz operators on the $n$-dimensional unit ball

We study $C^*$-algebras generated by Toeplitz operators acting on the standard weighted Bergman space $\mathcal{A}_λ^2(\mathbb{B}^n)$ over the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$. The symbols $f_{ac}$ of generating operators are assumed to be of a certain product type. By choosing $a$ and $c$ in different function algebras $\mathcal{S}_a$ and $\mathcal{S}_c$ over lower dimensional unit balls $\mathbb{B}^{\ell}$ and $\mathbb{B}^{n-\ell}$, respectively, and by assuming the invariance of $a\in \mathcal{S}_a$ under some torus action we obtain $C^*$-algebras $\boldsymbol{\mathcal{T}}_λ(\mathcal{S}_a, \mathcal{S}_c)$ whose structural properties can be described. In the case of $k$-quasi-radial functions $\mathcal{S}_a$ and bounded uniformly continuous or vanishing oscillation symbols $\mathcal{S}_c$ we describe the structure of elements from the algebra $\boldsymbol{\mathcal{T}}_λ(\mathcal{S}_a, \mathcal{S}_c)$, derive a list of irreducible representations of $\boldsymbol{\mathcal{T}}_λ(\mathcal{S}_a, \mathcal{S}_c)$, and prove completeness of this list in some cases. Some of these representations originate from a `quantization effect', induced by the representation of $\mathcal{A}_λ^2(\mathbb{B}^n)$ as the direct sum of Bergman spaces over a lower dimensional unit ball with growing weight parameter. As an application we derive the essential spectrum and index formulas for matrix-valued operators.

math.OA↗
arXiv2017

Toeplitz Quantization on Fock Space

For Toeplitz operators $T_f^{(t)}$ acting on the weighted Fock space $H_t^2$, we consider the semi-commutator $T_f^{(t)}T_g^{(t)}-T_{fg}^{(t)}$, where $t>0$ is a certain weight parameter that may be interpreted as Planck's constant $\hbar$ in Rieffel's deformation quantization. In particular, we are interested in the semi-classical limit \tag{$*$}\lim\limits_{t\to 0}\|T_f^{(t)}T_g^{(t)}-T_{fg}^{(t)}\|_t. It is well-known that $\|T_f^{(t)}T_g^{(t)}-T_{fg}^{(t)}\|_t$ tends to $0$ under certain smoothness assumptions imposed on $f$ and $g$. This result was extended to $f,g \in \mathrm{BUC}(\mathbb{C}^n)$ in a recent paper by Bauer and Coburn. We now further generalize this result to (not necessarily bounded) uniformly continuous functions and symbols in the algebra $\mathrm{VMO} \cap L^{\infty}$ of bounded functions having vanishing mean oscillation on $\mathbb{C}^n$. Our approach is based on the algebraic identity $T_f^{(t)}T_g^{(t)}-T_{fg}^{(t)}=-(H_{\bar{f}}^{(t)})^*H_g^{(t)}$, where $H_g^{(t)}$ denotes the Hankel operator corresponding to the symbol $g$, and norm estimates in terms of the (weighted) heat transform. As a consequence, only $f$ (or likewise only $g$) has to be contained in one of the above classes for $(*)$ to vanish. For $g$ we only have to impose $\limsup_{t \to 0}\|H_g^{(t)}\|_t<\infty$, e.g. $g \in L^{\infty}(\mathbb{C}^n)$. We prove that the set of all symbols $f\in L^{\infty}(\mathbb{C}^n)$ with the property that $\lim_{t \rightarrow 0}\|T^{(t)}_fT^{(t)}_g-T^{(t)}_{fg}\|_t=\lim_{t\to 0}\|T_g^{(t)}T_f^{(t)}-T_{gf}^{(t)}\|_t=0$ for all $g\in L^{\infty}(\mathbb{C}^n)$ coincides with $\mathrm{VMO}\cap L^{\infty}$. Additionally, we show that $\lim_{t\to 0}\|T_f^{(t)}\|_t=\|f\|_{\infty}$ holds for all $f\in L^{\infty}(\mathbb{C}^n)$. Finally, we present new examples, including bounded smooth functions, where $(*)$ does not vanish.

math.FA↗
arXiv2017

Uniform Continuity and Quantization on Bounded Symmetric Domains

We consider Toeplitz operators $T_f^λ$ with symbol $f$ acting on the standard weighted Bergman spaces over a bounded symmetric domain $Ω\subset \mathbb{C}^n$. Here $λ> genus-1$ is the weight parameter. The classical asymptotic semi-commutator relation $\lim_{λ\rightarrow \infty} \big{\|}T_f^λ T_g^λ -T_{fg}^λ \big{\|}=0$ with $f,g \in C(\overline{\mathbb{B}^n})$, where $Ω=\mathbb{B}^n$ denotes the complex unit ball, is extended to larger classes of bounded and unbounded operator symbol-functions and to more general domains. We deal with operator symbols that generically are neither continuous inside $Ω$ (Section 4) nor admit a continuous extension to the boundary (Section 3 and 4). Let $β$ denote the Bergman metric distance function on $Ω$. We prove that the semi-commutator relation remains true for $f$ and $g$ in the space ${\rm UC}(Ω)$ of all $β$-uniformly continuous functions on $Ω$. Note that this space contains also unbounded functions. In case of the complex unit ball $Ω=\mathbb{B}^n \subset \mathbb{C}^n$ we show that the semi-commutator relation holds true for bounded symbols in ${\rm VMO}(\mathbb{B}^n)$, where the vanishing oscillation inside $\mathbb{B}^n$ is measured with respect to $β$. At the same time the semi-commutator relation fails for generic bounded measurable symbols. We construct a corresponding counterexample using oscillating symbols that are continuous outside of a single point in $Ω$.

math.FA↗
arXiv2016

A co-dimensional 3 sub-Riemannian structure on Gromoll-Meyer exotic sphere

We construct a co-dimension $3$ completely non-holonomic sub-bundle on the Gromoll-Meyer exotic $7$ sphere based on its realization as a base space of a Sp(2)-principal bundle with the structure group Sp(1). The same method is valid for constructing a co-dimension 3 completely non-holonomic sub-bundle on the standard 7 sphere (or more general on a $4n+3$ dimensional standard sphere). In the latter case such a construction based on the Hopf bundle is well-known. Our method provides an alternated simple proof for the standard sphere $\mathbb{S}^7$.

math.DG↗