Published since January 1, 1989

Forum Mathematicum

ISSN: 1435-5337

Edited by: Thomas Alazard, Chantal David, Manfred Droste, Siegfried Echterhoff, Jan Frahm, Maria Gordina, Philipp Habegger, Matthias Hieber, Clara Löh, Karin Melnick, Maksym Radziwill, Freydoon Shahidi, Christopher D. Sogge, Shigeharu Takayama

Managing Editor: Jan Hendrik Bruinier

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Forum Mathematicum (FORUM) is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.

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applied mathematics
mathematical physics

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On Barnes beta distributions, Selberg integral and Riemann xi
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Volume 35 Issue 3

May 2023

Publicly Available May 1, 2023

Frontmatter

Page range: i-iv

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March 31, 2023

The theta cycles for modular forms modulo prime powers

Jigu Kim, Yoonjin Lee

Page range: 591-613

Abstract

Recently, Chen and Kiming studied the theta operator on modular forms modulo prime powers p m {p^{m}} , where p ≥ 5 {p\geq 5} and m ≥ 2 {m\geq 2} . In this paper, we study mod p m {p^{m}} filtrations and mod p m {p^{m}} theta cycles. We give a bound on some elements in the mod p m {p^{m}} theta cycle ( m ≥ 2 {m\geq 2} ), and we exactly compute those values in the case that m = 2 {m=2} .

March 31, 2023

Relations among Ramanujan-type congruences II: Ramanujan-type congruences in half-integral weights

Martin Raum

Page range: 615-646

Abstract

We link Ramanujan-type congruences, which emerge abundantly in combinatorics, to the Galois- and geometric theory of modular forms. Specifically, we show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra, and discover a dichotomy between congruences originating in Hecke eigenvalues and congruences on arithmetic progressions with cube-free periods. The latter provide congruences among algebraic parts of twisted central L {\mathrm{L}} -values. We specialize our results to integer partitions, for which we investigate the landmark proofs of partition congruences by Atkin and by Ono. Based on a modulo ℓ {\ell} analogue of the Maeda conjecture for certain partition generating functions, we conclude that their approach by Hecke operators acting diagonally modulo ℓ {\ell} on modular forms is indeed close to optimal. This work is enabled by several structure results for Ramanujan-type congruences that we establish. In an extended example, we showcase how to employ them to also benefit experimental work.

February 28, 2023

A unified approach to Gelfand and de Vries dualities

Guram Bezhanishvili, Luca Carai, Patrick J. Morandi, Bruce Olberding

Page range: 647-676

Abstract

We develop a unified approach to Gelfand and de Vries dualities for compact Hausdorff spaces, which is based on appropriate modifications of the classic results of Dieudonné (analysis), Dilworth (lattice theory), and Katětov and Tong (topology)

March 31, 2023

The cardinality of orthogonal exponentials of planar self-affine measures with two-element digit set

Qi Wang, Dan Ai

Page range: 677-688

Abstract

Let μ M , D {{\mu_{M,D}}} be the planar self-affine measure determined by an expanding integer matrix M ∈ M 2 ⁢ ( ℤ ) {M\in M_{2}(\mathbb{Z})} and a two-element digit set D ⊂ ℤ 2 {D\subset\mathbb{Z}^{2}} . It has been shown that the spectral or non-spectral problem on μ M , D {\mu_{M,D}} is only related to trace ⁡ ( M ) := r 1 {\operatorname{trace}(M):=r_{1}} and det ⁡ ( M ) := r 2 {\det(M):=r_{2}} . In the case when M ∈ M 2 ⁢ ( ℤ ) {M\in M_{2}(\mathbb{Z})} is an expanding matrix and r 1 2 = 3 ⁢ r 2 {r_{1}^{2}=3{r_{2}}} , r 2 ∈ 2 ⁢ ℤ + 1 ∖ { ± 1 } {{r_{2}}\in 2\mathbb{Z}+1\setminus\{\pm 1\}} , the Hilbert space L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} contains at most a finite number of orthogonal exponentials, and μ M , D {{\mu_{M,D}}} is a non-spectral measure. The remaining problem in this case is to determine the best upper bound on the cardinality of orthogonal exponentials in the Hilbert space L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} . In the present paper, we further the above research to show that there are at most 16 mutually orthogonal exponentials in the corresponding Hilbert space, and the number 16 is the best upper bound. This completes the non-spectrality of self-affine measures in the above case.

February 28, 2023

Categorically closed countable semigroups

Taras Banakh, Serhii Bardyla

Page range: 689-711

Abstract

In this paper, we establish a connection between categorical closedness and nontopologizability of semigroups. In particular, for the class 𝖳 𝟣 ⁢ 𝖲 {\mathsf{T_{\!1}S}} of T 1 {T_{1}} topological semigroups we prove that a countable semigroup X with finite-to-one shifts is injectively 𝖳 𝟣 ⁢ 𝖲 {\mathsf{T_{\!1}S}} -closed if and only if X is 𝖳 𝟣 ⁢ 𝖲 {\mathsf{T_{\!1}S}} -discrete in the sense that every T 1 {T_{1}} semigroup topology on X is discrete. Moreover, a countable cancellative semigroup X is absolutely 𝖳 𝟣 ⁢ 𝖲 {\mathsf{T_{\!1}S}} -closed if and only if every homomorphic image of X is 𝖳 𝟣 ⁢ 𝖲 {\mathsf{T_{\!1}S}} -discrete. Also, we introduce and investigate a new notion of a polybounded semigroup. It is proved that a countable semigroup X with finite-to-one shifts is polybounded if and only if X is 𝖳 𝟣 ⁢ 𝖲 {\mathsf{T_{\!1}S}} -closed if and only if X is 𝖳 𝗓 ⁢ 𝖲 {\mathsf{T_{\!z}S}} -closed, where 𝖳 𝗓 ⁢ 𝖲 {\mathsf{T_{\!z}S}} is the class of Tychonoff zero-dimensional topological semigroups. We show that polybounded cancellative semigroups are groups, and polybounded T 1 {T_{1}} paratopological groups are topological groups.

February 28, 2023

Symmetric skew braces and brace systems

Valeriy G. Bardakov, Mikhail V. Neshchadim, Manoj K. Yadav

Page range: 713-738

Abstract

For a skew left brace ( G , ⋅ , ∘ ) {(G,\cdot\,,\circ)} , the map λ : ( G , ∘ ) → Aut ⁡ ( G , ⋅ ) {\lambda:(G,\circ)\to\operatorname{Aut}(G,\cdot\,)} , a ↦ λ a , {a\mapsto\lambda_{a},} where λ a ⁢ ( b ) = a - 1 ⋅ ( a ∘ b ) {\lambda_{a}(b)=a^{-1}\cdot(a\circ b)} for all a , b ∈ G {a,b\in G} , is a group homomorphism. Then λ can also be viewed as a map from ( G , ⋅ ) {(G,\cdot\,)} to Aut ⁡ ( G , ⋅ ) {\operatorname{Aut}(G,\cdot\,)} , which, in general, may not be a homomorphism. A skew left brace will be called λ-anti-homomorphic (λ-homomorphic) if λ : ( G , ⋅ ) → Aut ⁡ ( G , ⋅ ) {\lambda:(G,\cdot\,)\to\operatorname{Aut}(G,\cdot\,)} is an anti-homomorphism (a homomorphism). We mainly study such skew left braces. We device a method for constructing a class of binary operations on a given set so that the set with any two such operations constitutes a λ-homomorphic symmetric skew brace. Most of the constructions of symmetric skew braces dealt with in the literature fall in the framework of our construction. We then carry out various such constructions on specific infinite groups.

March 31, 2023

A new class of uncertainty principles for the k-Hankel wavelet transform

Hatem Mejjaoli, Firdous A. Shah

Page range: 739-762

Abstract

The k -Hankel wavelet transform is a novel addition to the class of wavelet transforms which relies on a pair of generalized translation and dilation operators governed by the well-known k -Hankel transform. The aim of this paper is to explore a class of new uncertainty principles associated with the k -Hankel wavelet transform, including the Benedick–Amrein–Berthier and Shapiro’s uncertainty inequalities. Nevertheless, we shall also establish certain local-type uncertainty principles abreast of the mean dispersion theorems for the k -Hankel wavelet transform.

April 1, 2023

Homogeneous ACM bundles on isotropic Grassmannians

Rong Du, Xinyi Fang, Peng Ren

Page range: 763-782

Abstract

In this paper, we characterize homogeneous arithmetically Cohen–Macaulay (ACM) bundles over isotropic Grassmannians of types 𝐵, 𝐶 and 𝐷 in terms of step matrices. We show that there are only finitely many irreducible homogeneous ACM bundles by twisting line bundles over these isotropic Grassmannians. So we classify all homogeneous ACM bundles over isotropic Grassmannians combining the results on usual Grassmannians by Costa and Miró-Roig. Moreover, if the irreducible initialized homogeneous ACM bundles correspond to some special highest weights, then they can be characterized by succinct forms.

January 27, 2023

On the global L ²-boundedness of Fourier integral operators with rough amplitude and phase functions

Joachim Sindayigaya

Page range: 783-792

Abstract

In this paper, we study the L 2 {L^{2}} -boundedness of Fourier integral operators T ϕ , a {T_{\phi,a}} with rough amplitude a ⁢ ( x , η ) {a(x,\eta)} and rough phase functions ϕ ⁢ ( x , η ) {\phi(x,\eta)} , which satisfies the new class of rough non-degeneracy condition. In this study, under the conditions a ∈ L ∞ ⁢ S ρ m {a\in L^{\infty}S^{m}_{\rho}} , ϕ ∈ L ∞ ⁢ Φ 2 {{\phi}\in L^{\infty}\Phi^{2}} and some conditions of m , we show that T ϕ , a {T_{\phi,a}} is bounded from L 2 {L^{2}} to itself. Our main results extend and improve some known results about L 2 {L^{2}} , however, the method we use to prove these results is new.

March 31, 2023

A local converse theorem for Archimedean GL(n)

Moshe Adrian, Shuichiro Takeda

Page range: 793-808

Abstract

We prove a local converse theorem for GL n {\operatorname{GL}_{n}} over the Archimedean local fields which characterizes an infinitesimal equivalence class of irreducible admissible generic representations of GL n ⁡ ( ℝ ) {\operatorname{GL}_{n}(\mathbb{R})} or GL n ⁡ ( ℂ ) {\operatorname{GL}_{n}(\mathbb{C})} in terms of twisted local gamma factors.

March 31, 2023

Fractional Bloom boundedness and compactness of commutators

Tuomas Hytönen, Tuomas Oikari, Jaakko Sinko

Page range: 809-830

Abstract

Let T be a non-degenerate Calderón–Zygmund operator and let b : ℝ d → ℂ {b:\mathbb{R}^{d}\to\mathbb{C}} be locally integrable. Let 1 < p ≤ q < ∞ {1<p\leq q<\infty} and let μ p ∈ A p {\mu^{p}\in A_{p}} and λ q ∈ A q {\lambda^{q}\in A_{q}} , where A p {A_{p}} denotes the usual class of Muckenhoupt weights. We show that ∥ [ b , T ] ∥ L μ p → L λ q ∼ ∥ b ∥ BMO ν α , [ b , T ] ∈ 𝒦 ⁢ ( L μ p , L λ q ) iff b ∈ VMO ν α , \lVert[b,T]\rVert_{L^{p}_{\mu}\to L^{q}_{\lambda}}\sim\lVert b\rVert_{% \operatorname{BMO}_{\nu}^{\alpha}},\quad[b,T]\in\mathcal{K}(L^{p}_{\mu},L^{q}_% {\lambda})\quad\text{iff}\quad b\in\operatorname{VMO}_{\nu}^{\alpha}, where L μ p = L p ⁢ ( μ p ) {L^{p}_{\mu}=L^{p}(\mu^{p})} and α / d = 1 / p - 1 / q {\alpha/d=1/p-1/q} , the symbol 𝒦 {\mathcal{K}} stands for the class of compact operators between the given spaces, and the fractional weighted BMO ν α {\operatorname{BMO}_{\nu}^{\alpha}} and VMO ν α {\operatorname{VMO}_{\nu}^{\alpha}} spaces are defined through the following fractional oscillation and Bloom weight: 𝒪 ν α ⁢ ( b ; Q ) = ν ⁢ ( Q ) - α / d ⁢ ( 1 ν ⁢ ( Q ) ⁢ ∫ Q | b - 〈 b 〉 Q | ) , ν = ( μ λ ) β , β = ( 1 + α / d ) - 1 . \mathcal{O}_{\nu}^{\alpha}(b;Q)=\nu(Q)^{-\alpha/d}\biggl{(}\frac{1}{\nu(Q)}% \int_{Q}\lvert b-\langle b\rangle_{Q}\rvert\biggr{)},\quad\nu=\biggl{(}\frac{% \mu}{\lambda}\biggr{)}^{\beta},\quad\beta=(1+\alpha/d)^{-1}. The key novelty is dealing with the off-diagonal range p < q {p<q} , whereas the case p = q {p=q} was previously studied by Lacey and Li. However, another novelty in both cases is that our approach allows complex-valued functions b , while other arguments based on the median of b on a set are inherently real-valued.

March 31, 2023

A Mikhlin-type multiplier theorem for the partial harmonic oscillator

Xiaoyan Su, Ying Wang, Guixiang Xu

Page range: 831-841

Abstract

We prove a Mikhlin-type multiplier theorem for the partial harmonic oscillator H par = - ∂ ρ 2 - Δ x + | x | 2 {H_{\mathrm{par}}=-\partial_{\rho}^{2}-\Delta_{x}+\lvert x\rvert^{2}} for ( ρ , x ) ∈ ℝ × ℝ d {(\rho,x)\in\mathbb{R}\times\mathbb{R}^{d}} by using the Littlewood–Paley g and g ∗ {g^{\ast}} functions and the associated heat kernel estimates. The multipliers we consider are continuous in one direction and discrete in the other directions.

February 28, 2023

Rational pullbacks of toric foliations

Javier Gargiulo Acea, Ariel Molinuevo, Sebastián Velazquez

Page range: 843-861

Abstract

This article is dedicated to the study of singular codimension-one foliations ℱ {\mathcal{F}} on a simplicial complete toric variety X and their pullbacks by dominant rational maps φ : ℙ n ⇢ X {\varphi:\mathbb{P}^{n}\dashrightarrow X} . First, we describe the singularities of ℱ {\mathcal{F}} and φ * ⁢ ℱ {\varphi^{*}\mathcal{F}} for a generic pair ( φ , ℱ ) {(\varphi,\mathcal{F})} . Then we show that the first-order deformations of φ * ⁢ ℱ {\varphi^{*}\mathcal{F}} arising from first-order unfoldings are the families of the form φ ε * ⁢ ℱ {\varphi_{\varepsilon}^{*}\mathcal{F}} , where φ ε {\varphi_{\varepsilon}} is a perturbation of φ. We also prove that the deformations of the form φ * ⁢ ℱ ε {\varphi^{*}\mathcal{F}_{\varepsilon}} consist exactly of the families which are tangent to the fibers of φ. In order to do so, we state some results of independent interest regarding the Kupka singularities of these foliations.

February 28, 2023

On fractional integrals generated by Radon transforms over paraboloids

Boris Rubin

Page range: 863-881

Abstract

We apply the Fourier transform technique and a modified version of E. Stein’s interpolation theorem communicated by L. Grafakos, to obtain sharp L p {L^{p}} - L q {L^{q}} estimates for the Radon transform and more general convolution-type fractional integrals with the kernels having singularity on the paraboloids.

Ahead of Print / Just Accepted

Ahead of Print / Just Accepted

Volume 35 (2023)

Volume 34 (2022)

Volume 33 (2021)

Volume 32 (2020)

Volume 31 (2019)

Volume 30 (2018)

Volume 29 (2017)

Volume 28 (2016)

Volume 27 (2015)

Volume 26 (2014)

Volume 25 (2013)

Volume 24 (2012)

Volume 23 (2011)

Volume 22 (2010)

Volume 21 (2009)

Volume 20 (2008)

Volume 19 (2007)

Volume 18 (2006)

Volume 17 (2005)

Volume 16 (2004)

Volume 15 (2003)

Volume 14 (2002)

Volume 13 (2001)

Volume 12 (2000)

Volume 11 (1999)

Volume 10 (1998)

5-year Journal Impact Factor	0.921
CiteScore	1.6
Journal Impact Factor	0.943
Journal Citation Indicator	0.70
Mathematical Citation Quotient	0.86
SCImago Journal Rank	0.853
Source Normalized Impact per Paper	1.230

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Editorial

Managing Editor

Jan H. Bruinier
Department of Mathematics
Technische Universität Darmstadt
bruinier@mathematik.tu-darmstadt.de
Primary research areas: Number theory, automorphic forms, algebraic geometry.

Editors

Thomas Alazard
CNRS and École Normale Supérieure Paris-Saclay
thomas.alazard@ens-paris-saclay.fr
Main research areas : partial differential equations,microlocal analysis, mathematical problems in fluid dynamics

Chantal David
Department of Mathematics and Statistics
Concordia University
chantal.david@concordia.ca
Primary research areas:　Analytic number theory, L-functions, arithmetic statistics.

Manfred Droste
Institut für Informatik
Universität Leipzig
droste@informatik.uni-leipzig.de
Primary research areas: Group theory, logic, ordered structures, automata theory.

Siegfried Echterhoff
Mathematisches Institut
Universität Münster
echters@uni-muenster.de
Primary research areas: Functional analysis, operator algebras, K-theory

Jan Frahm
Department of Mathematics
Aarhus University
frahm@math.au.dk
Primary research areas: Lie theory, harmonic analysis, homogeneous spaces

Maria Gordina
Department of Mathematics
University of Connecticut
maria.gordina@uconn.edu
Primary research areas: Probability theory and stochastic processes, global analysis, analysis on manifolds

Philipp Habegger
Universität Basel
Departement Mathematik und Informatik
Basel, Schweiz
philipp.habegger@unibas.ch
Research areas: Number Theory, Diophantine Geometry

Matthias Hieber
Department of Mathematics
TU Darmstadt, Germany
hieber@mathematik.tu-darmstadt.de
Primary research areas are: partial differential equations, fluid dynamics

Clara Löh
Faculty of Mathematics
University of Regensburg
clara.loeh@mathematik.uni-regensburg.de
Primary research areas are: Geometric Topology, Geometric Group Theory, Measured Group Theory

Karin Melnick
Department of Mathematics
University of Maryland
karin@math.umd.edu
Primary research areas: differential geometry, transformation groups

Maksym Radziwill
Department of Mathematics
Caltech, Pasadena
maksym.radziwill@gmail.com
Primary research areas: Analytic Number Theory

Freydoon Shahidi
Department of Mathematics
Purdue University
shahidi@math.purdue.edu
Primary research areas: Automorphic forms, group representations, number theory and arithmetic geometry.

Christopher D. Sogge
Department of Mathematics
Johns Hopkins University
sogge@jhu.edu
Primary research areas: Fourier analysis, partial differential equations.

Shigeharu Takayama
Department of Mathematics
University of Tokyo
taka@ms.u-tokyo.ac.jp
Primary research areas: complex geometry, Kähler geometry and algebraic geometry

Editorial Office at De Gruyter

Masha Dorogova
Journal Coordinator Forum Mathematicum
masha.dorogova@degruyter.com

(Deutsch)

Online ISSN: 1435-5337
Print ISSN: 0933-7741
Type: Journal
Language: English
Publisher: De Gruyter
First published: January 1, 1989
Publication Frequency: 6 Issues per Year
Audience: researchers in the field of mathematics

Forum Mathematicum

Frontmatter

The theta cycles for modular forms modulo prime powers

Abstract

Relations among Ramanujan-type congruences II: Ramanujan-type congruences in half-integral weights

Abstract

A unified approach to Gelfand and de Vries dualities

Abstract

The cardinality of orthogonal exponentials of planar self-affine measures with two-element digit set

Abstract

Categorically closed countable semigroups

Abstract

Symmetric skew braces and brace systems

Abstract

A new class of uncertainty principles for the k-Hankel wavelet transform

Abstract

Homogeneous ACM bundles on isotropic Grassmannians

Abstract

On the global L 2-boundedness of Fourier integral operators with rough amplitude and phase functions

Abstract

A local converse theorem for Archimedean GL(n)

Abstract

Fractional Bloom boundedness and compactness of commutators

Abstract

A Mikhlin-type multiplier theorem for the partial harmonic oscillator

Abstract

Rational pullbacks of toric foliations

Abstract

On fractional integrals generated by Radon transforms over paraboloids

Abstract

On the global L ²-boundedness of Fourier integral operators with rough amplitude and phase functions