arXiv subjects

Johannes Testorf

Publications and source records attributed to Johannes Testorf.

4 recordsLinked to original sources

arXiv2026

Ross-Witt Nyström correspondence and Ohsawa-Takegoshi extension

This paper is an attempt to study a few \emph{effective} universality phenomenons in complex geometry using deformations and the Hörmander $L^2$-method. The main idea of the proof is to deform the general case to the toric (or convex) case and to apply two new Berndtsson-Lempert type Ohsawa-Takegoshi extension theorems. The first one is based on the Ross-Witt Nyström correspondence picture, Darvas-Xia-Zhang's asymptotic slope formula for Ding-type functionals and Berndtsson's monotonicity theorem. The second version is proved using an $S^1$-symmetrization method for PSH potentials based on Witt Nyström's canonical Kähler deformations and Berndtsson-P\u aun's positivity theorem for Stein fibrations associated to deformations to normal bundles. The resulting effective universality phenomenons include a sharp lower bound of the Bergman kernel for compact Riemann surfaces, an effective Okounkov body construction and a sharp Faber-Widom type Bergman approximation of the logarithmic capacity.

math.CV↗

arXiv2026

Explicit Estimates for the Bergman Kernel Form

Let $(L,e^{-ϕ})$ be a positive Hermitian holomorphic line bundle over a compact Riemann surface $X$, and let $ω=i\partial\overline{\partial}ϕ$. We obtain explicit pointwise estimates for the Bergman form of the tensor power $mL$. If $\mathrm{Ric}\,ω\leqω$ and the shortest nonconstant closed geodesic has length at least $2π$, then \[ K_{mϕ}\geq \frac{2m-1}{4π}\,ω, \] with sharpness holding for $(\mathbb P^1,\mathcal O_{\mathbb P^1}(2))$. We also obtain a local version, depending on an upper curvature bound and the injectivity radius, which recovers the first two terms of the Bergman expansion when the curvature is constant. We also find a higher dimensional version. Under the two-sided bound $-ω\leq\mathrm{Ric}\,ω\leqω$ and the same closed-geodesic hypothesis, we also prove \[ K_{mϕ}\leq \frac{mω}{2π} \left(1+\frac{3}{2m}\right). \] The lower estimates use the deformation to the tangent space version of the Ohsawa--Takegoshi theorem established by He, Wang, and the author, whereas the upper bound via Błocki--Zwonek and isoperimetric inequalities.

math.CV↗

arXiv2026

A Complex Geometric Approach to the Discrete Gabor Transform and Localization Operators on the Flat Torus

In a recent paper, the discrete Gabor transform was connected to a Gabor transform with a time frequency domain given by the flat torus. We show that the corresponding Bargmann-Fock spaces can be expressed as theta functions (or equivalently line bundles on Abelian varieties). We give applications of this viewpoint to frame results for the discrete Gabor transform. In particular, we get necessary conditions which hold in higher dimensions and can expand the known results in the one dimensional case, the primary tool being the theorem of the square. We also give an application to asymptotics of restriction operators which arises via the asymptotic behavior of Bergman kernels and Toeplitz operators for high tensor powers of line bundles and find that time frequency restriction operators on the flat torus will exhibit "plunge" behaviors similar to those of time frequency restriction operators in other contexts.

math.FA↗

arXiv2025

On the transcendentality condition for Gaussian Gabor frames and Hermite super/multiwindow frames

We give a criterion for higher-dimensional Gaussian Gabor frames, which is a reformulation of one of the main results in in a previous article by the first and last authors in more explicit terms. We use this formulation in order to extend a result of Romero, Ulanovskii, and Zlotnikov to lattices given by irrational rotations. We also show that this density criterion for Gaussian Gabor frames is generic in a certain sense. In addition, we also extend the methods of the first and last named authors to the pseudoeffective threshold which gives a condition for uniqueness in the Bargmann-Fock space. We also use this viewpoint to study super and multi-window Gabor frames with Hermitian windows. In particular, we find a density criterion for transcendental lattices.

math.FA↗