Pub Date : 2026-02-02DOI: 10.1016/j.aim.2026.110828
Elena Cordero , Gianluca Giacchi , Edoardo Pucci , S. Ivan Trapasso
Motivated by the phase space analysis of Schrödinger evolution operators, in this paper we investigate how metaplectic operators are approximately diagonalized along the corresponding symplectic flows by exponentially localized Gabor wave packets. Quantitative bounds for the matrix coefficients arising in the Gabor wave packet decomposition of such operators are established, revealing precise exponential decay rates together with subtler dispersive and spreading phenomena. To this end, we present several novel results concerning the time-frequency analysis of functions with controlled Gelfand-Shilov regularity, which are of independent interest.
As a byproduct, we generalize Vemuri's Gaussian confinement results for the solutions of the quantum harmonic oscillator in two respects, namely by encompassing general exponential decay rates as well as arbitrary quadratic Schrödinger propagators. In particular, we extensively discuss some prominent models such as the harmonic oscillator, the free particle in a constant magnetic field and fractional Fourier transforms.
{"title":"Sparse Gabor representations of metaplectic operators: controlled exponential decay and Schrödinger confinement","authors":"Elena Cordero , Gianluca Giacchi , Edoardo Pucci , S. Ivan Trapasso","doi":"10.1016/j.aim.2026.110828","DOIUrl":"10.1016/j.aim.2026.110828","url":null,"abstract":"<div><div>Motivated by the phase space analysis of Schrödinger evolution operators, in this paper we investigate how metaplectic operators are approximately diagonalized along the corresponding symplectic flows by exponentially localized Gabor wave packets. Quantitative bounds for the matrix coefficients arising in the Gabor wave packet decomposition of such operators are established, revealing precise exponential decay rates together with subtler dispersive and spreading phenomena. To this end, we present several novel results concerning the time-frequency analysis of functions with controlled Gelfand-Shilov regularity, which are of independent interest.</div><div>As a byproduct, we generalize Vemuri's Gaussian confinement results for the solutions of the quantum harmonic oscillator in two respects, namely by encompassing general exponential decay rates as well as arbitrary quadratic Schrödinger propagators. In particular, we extensively discuss some prominent models such as the harmonic oscillator, the free particle in a constant magnetic field and fractional Fourier transforms.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110828"},"PeriodicalIF":1.5,"publicationDate":"2026-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-02-02DOI: 10.1016/j.aim.2026.110822
Feng Dai , Yinqin Li , Dachun Yang , Wen Yuan , Yirui Zhao
<div><div>In this article, we introduce a family of local lifted Hardy–Littlewood maximal operators <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>θ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></msub></math></span> on the upper half-plane and prove that, for any given <span><math><mi>p</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> and <span><math><mi>γ</mi><mo>∈</mo><mi>R</mi></math></span>, the estimate, with the implicit positive constant independent of <em>f</em>,<span><span><span><math><mrow><munder><mi>sup</mi><mrow><mi>θ</mi><mo>,</mo><mi>λ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></munder><mo></mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>p</mi></mrow></msup><munder><mrow><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></munder><munderover><mo>∫</mo><mrow><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></munderover></mrow><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>></mo><mi>λ</mi><msup><mrow><mi>t</mi></mrow><mrow><mfrac><mrow><mi>γ</mi></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></msup></mrow></munder><msup><mrow><mi>t</mi></mrow><mrow><mi>γ</mi><mo>−</mo><mn>1</mn></mrow></msup><mspace></mspace><mi>d</mi><mi>t</mi><mspace></mspace><mi>d</mi><mi>x</mi><mo>≲</mo><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></munder><mo>|</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup><mspace></mspace><mi>d</mi><mi>x</mi></mrow></math></span></span></span> holds for all <span><math><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> if and only if either <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> and <span><math><mi>γ</mi><mo>≠</mo><mn>0</mn></math></span> or <span><math><mi>p</mi><mo>=</mo><mn>1</mn></math></span> and <span><math><mi>γ</mi><mo>∉</mo><mo>[</mo><mo>−</mo><mi>n</mi><mo>,</mo><mn>0</mn><mo>]</mo></math></span>. Moreover, we use <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>θ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></msub></math></span> to refine the Lusin–Lipschitz inequality and the pointwise domination for the generalized approximation to the identity, which connect various differential operators with related one-parameter families of linear operators. As applications, we obtain several new weak-type representations for the norms of these differential operators, including endpoint estimates and extensions to spaces of homogeneous type, which gives an affirmative answer
{"title":"Sharp weak-type estimate for local lifted Hardy–Littlewood maximal operators with applications to generators of linear operator families and Hardy(–Sobolev) spaces","authors":"Feng Dai , Yinqin Li , Dachun Yang , Wen Yuan , Yirui Zhao","doi":"10.1016/j.aim.2026.110822","DOIUrl":"10.1016/j.aim.2026.110822","url":null,"abstract":"<div><div>In this article, we introduce a family of local lifted Hardy–Littlewood maximal operators <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>θ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></msub></math></span> on the upper half-plane and prove that, for any given <span><math><mi>p</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> and <span><math><mi>γ</mi><mo>∈</mo><mi>R</mi></math></span>, the estimate, with the implicit positive constant independent of <em>f</em>,<span><span><span><math><mrow><munder><mi>sup</mi><mrow><mi>θ</mi><mo>,</mo><mi>λ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></munder><mo></mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>p</mi></mrow></msup><munder><mrow><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></munder><munderover><mo>∫</mo><mrow><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></munderover></mrow><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>></mo><mi>λ</mi><msup><mrow><mi>t</mi></mrow><mrow><mfrac><mrow><mi>γ</mi></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></msup></mrow></munder><msup><mrow><mi>t</mi></mrow><mrow><mi>γ</mi><mo>−</mo><mn>1</mn></mrow></msup><mspace></mspace><mi>d</mi><mi>t</mi><mspace></mspace><mi>d</mi><mi>x</mi><mo>≲</mo><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></munder><mo>|</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msup><mspace></mspace><mi>d</mi><mi>x</mi></mrow></math></span></span></span> holds for all <span><math><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> if and only if either <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span> and <span><math><mi>γ</mi><mo>≠</mo><mn>0</mn></math></span> or <span><math><mi>p</mi><mo>=</mo><mn>1</mn></math></span> and <span><math><mi>γ</mi><mo>∉</mo><mo>[</mo><mo>−</mo><mi>n</mi><mo>,</mo><mn>0</mn><mo>]</mo></math></span>. Moreover, we use <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>θ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></msub></math></span> to refine the Lusin–Lipschitz inequality and the pointwise domination for the generalized approximation to the identity, which connect various differential operators with related one-parameter families of linear operators. As applications, we obtain several new weak-type representations for the norms of these differential operators, including endpoint estimates and extensions to spaces of homogeneous type, which gives an affirmative answer","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110822"},"PeriodicalIF":1.5,"publicationDate":"2026-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146174565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-30DOI: 10.1016/j.aim.2026.110840
Frank Garvan, Zhumagali Shomanov
In 1967, Atkin and O'Brien conjectured congruences for the partition function involving Hecke operators modulo powers of 13. While they proved the conjecture modulo 13 and 132, a proof for all powers of 13 has remained open. In this paper we provide a simple and complete proof of the conjecture.
{"title":"A simple proof of the Atkin-O'Brien partition Hecke congruence conjecture for powers of 13","authors":"Frank Garvan, Zhumagali Shomanov","doi":"10.1016/j.aim.2026.110840","DOIUrl":"10.1016/j.aim.2026.110840","url":null,"abstract":"<div><div>In 1967, Atkin and O'Brien conjectured congruences for the partition function involving Hecke operators modulo powers of 13. While they proved the conjecture modulo 13 and 13<sup>2</sup>, a proof for all powers of 13 has remained open. In this paper we provide a simple and complete proof of the conjecture.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110840"},"PeriodicalIF":1.5,"publicationDate":"2026-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-30DOI: 10.1016/j.aim.2026.110794
Anirban Bhaduri , Yael Davidov , Eleonore Faber , Katrina Honigs , Peter McDonald , C. Eric Overton-Walker , Dylan Spence
<div><div>In this paper, we explore the derived McKay correspondence for several reflection groups, namely reflection groups of rank two generated by reflections of order two. We prove that for each of the reflection groups <span><math><mi>G</mi><mo>=</mo><mi>G</mi><mo>(</mo><mn>2</mn><mi>m</mi><mo>,</mo><mi>m</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>12</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>13</mn></mrow></msub></math></span>, or <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>22</mn></mrow></msub></math></span>, there is a semiorthogonal decomposition of the following form, where <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> are the normalizations of the irreducible components of the branch divisor <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>→</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mi>G</mi></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are exceptional objects:<span><span><span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>G</mi></mrow></msup><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>≅</mo><mo>〈</mo><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>D</mi><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><mi>D</mi><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>)</mo><mo>,</mo><mi>D</mi><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mi>G</mi><mo>)</mo><mo>〉</mo><mo>.</mo></math></span></span></span> We verify that the pieces of this decomposition correspond to the irreducible representations of <em>G</em>, verifying the Orbifold Semiorthogonal Decomposition Conjecture of Polishchuk and Van den Bergh. Due to work of Potter on the group <span><math><mi>G</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>m</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, this conjecture is now proven for all finite groups <span><math><mi>G</mi><mo>≤</mo><mi>GL</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo>)</mo></math></span> that are generated by order 2 reflections. Each of these groups contains, as a subgroup of index 2, a distinct finite group <span><math><mi>H</mi><mo>≤</mo><mi>SL</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo>)</mo></math></span>. A key part of our work is an explicit computation of the action of <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> on the <em>H</em>-Hilbert scheme <span><math><mrow><mtext>H</mtext><mtext>-Hilb</mtext></mrow
{"title":"An explicit derived McKay correspondence for some complex reflection groups of rank two","authors":"Anirban Bhaduri , Yael Davidov , Eleonore Faber , Katrina Honigs , Peter McDonald , C. Eric Overton-Walker , Dylan Spence","doi":"10.1016/j.aim.2026.110794","DOIUrl":"10.1016/j.aim.2026.110794","url":null,"abstract":"<div><div>In this paper, we explore the derived McKay correspondence for several reflection groups, namely reflection groups of rank two generated by reflections of order two. We prove that for each of the reflection groups <span><math><mi>G</mi><mo>=</mo><mi>G</mi><mo>(</mo><mn>2</mn><mi>m</mi><mo>,</mo><mi>m</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>12</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>13</mn></mrow></msub></math></span>, or <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>22</mn></mrow></msub></math></span>, there is a semiorthogonal decomposition of the following form, where <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> are the normalizations of the irreducible components of the branch divisor <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>→</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mi>G</mi></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are exceptional objects:<span><span><span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>G</mi></mrow></msup><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>≅</mo><mo>〈</mo><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>D</mi><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><mi>D</mi><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>)</mo><mo>,</mo><mi>D</mi><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mi>G</mi><mo>)</mo><mo>〉</mo><mo>.</mo></math></span></span></span> We verify that the pieces of this decomposition correspond to the irreducible representations of <em>G</em>, verifying the Orbifold Semiorthogonal Decomposition Conjecture of Polishchuk and Van den Bergh. Due to work of Potter on the group <span><math><mi>G</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>m</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, this conjecture is now proven for all finite groups <span><math><mi>G</mi><mo>≤</mo><mi>GL</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo>)</mo></math></span> that are generated by order 2 reflections. Each of these groups contains, as a subgroup of index 2, a distinct finite group <span><math><mi>H</mi><mo>≤</mo><mi>SL</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo>)</mo></math></span>. A key part of our work is an explicit computation of the action of <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> on the <em>H</em>-Hilbert scheme <span><math><mrow><mtext>H</mtext><mtext>-Hilb</mtext></mrow","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110794"},"PeriodicalIF":1.5,"publicationDate":"2026-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-30DOI: 10.1016/j.aim.2026.110832
Rui Han , Wilhelm Schlag
We prove non-perturbative Anderson localization for quasi-periodic Jacobi block matrix operators assuming non-vanishing of all Lyapunov exponents. The base dynamics on tori is assumed to be a Diophantine rotation. Results on arithmetic localization are obtained for , and applications to skew shifts, stacked graphene, XY spin chains, and coupled Harper models are presented.
{"title":"Non-perturbative localization for quasi-periodic Jacobi block matrices","authors":"Rui Han , Wilhelm Schlag","doi":"10.1016/j.aim.2026.110832","DOIUrl":"10.1016/j.aim.2026.110832","url":null,"abstract":"<div><div>We prove non-perturbative Anderson localization for quasi-periodic Jacobi block matrix operators assuming non-vanishing of all Lyapunov exponents. The base dynamics on tori <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>b</mi></mrow></msup></math></span> is assumed to be a Diophantine rotation. Results on arithmetic localization are obtained for <span><math><mi>b</mi><mo>=</mo><mn>1</mn></math></span>, and applications to skew shifts, stacked graphene, XY spin chains, and coupled Harper models are presented.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110832"},"PeriodicalIF":1.5,"publicationDate":"2026-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081817","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-30DOI: 10.1016/j.aim.2026.110827
Andrea Colesanti , Lei Qin , Paolo Salani
We prove a Brunn-Minkowski type inequality for the first (nontrivial) Dirichlet eigenvalue of the weighted p-operator where , in the class of bounded Lipschitz domains in . We also prove that the corresponding positive eigenfunctions are log-concave if the domain is convex.
{"title":"Geometric properties of solutions to elliptic PDE's in Gauss space and related Brunn-Minkowski type inequalities","authors":"Andrea Colesanti , Lei Qin , Paolo Salani","doi":"10.1016/j.aim.2026.110827","DOIUrl":"10.1016/j.aim.2026.110827","url":null,"abstract":"<div><div>We prove a Brunn-Minkowski type inequality for the first (nontrivial) Dirichlet eigenvalue of the weighted <em>p</em>-operator<span><span><span><math><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>γ</mi></mrow></msub><mi>u</mi><mo>=</mo><mo>−</mo><mtext>div</mtext><mo>(</mo><mo>|</mo><mi>∇</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>∇</mi><mi>u</mi><mo>)</mo><mo>+</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>∇</mi><mi>u</mi><mo>)</mo><mo>|</mo><mi>∇</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>,</mo></math></span></span></span> where <span><math><mi>p</mi><mo>></mo><mn>1</mn></math></span>, in the class of bounded Lipschitz domains in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. We also prove that the corresponding positive eigenfunctions are log-concave if the domain is convex.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110827"},"PeriodicalIF":1.5,"publicationDate":"2026-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-30DOI: 10.1016/j.aim.2026.110829
Daniel McGinnis , Nikola Sadovek
We solve a long-standing open problem posed by Goodman & Pollack in 1988 by establishing a necessary and sufficient condition for a family of convex sets in to admit a k-transversal for any . This result is a common generalization of Helly's theorem () and the Goodman-Pollack-Wenger theorem (). Additionally, we obtain an analogue in the complex setting by characterizing the existence of a complex k-transversal to a family of convex sets in , extending the work of McGinnis (). Our approach is topological and employs a Borsuk-Ulam-type theorem on Stiefel manifolds. Finally, we demonstrate how our results imply the central transversal theorems of Živaljević-Vrećica and Dol'nikov in the real case and of Sadovek-Soberón in the complex case.
{"title":"A necessary and sufficient condition for k-transversals","authors":"Daniel McGinnis , Nikola Sadovek","doi":"10.1016/j.aim.2026.110829","DOIUrl":"10.1016/j.aim.2026.110829","url":null,"abstract":"<div><div>We solve a long-standing open problem posed by Goodman & Pollack in 1988 by establishing a necessary and sufficient condition for a family of convex sets in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> to admit a <em>k</em>-transversal for any <span><math><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>d</mi><mo>−</mo><mn>1</mn></math></span>. This result is a common generalization of Helly's theorem (<span><math><mi>k</mi><mo>=</mo><mn>0</mn></math></span>) and the Goodman-Pollack-Wenger theorem (<span><math><mi>k</mi><mo>=</mo><mi>d</mi><mo>−</mo><mn>1</mn></math></span>). Additionally, we obtain an analogue in the complex setting by characterizing the existence of a complex <em>k</em>-transversal to a family of convex sets in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, extending the work of McGinnis (<span><math><mi>k</mi><mo>=</mo><mi>d</mi><mo>−</mo><mn>1</mn></math></span>). Our approach is topological and employs a Borsuk-Ulam-type theorem on Stiefel manifolds. Finally, we demonstrate how our results imply the central transversal theorems of Živaljević-Vrećica and Dol'nikov in the real case and of Sadovek-Soberón in the complex case.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"490 ","pages":"Article 110829"},"PeriodicalIF":1.5,"publicationDate":"2026-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-29DOI: 10.1016/j.aim.2026.110823
Andrii Ilienko , Ilya Molchanov , Tommaso Visonà
We obtain a complete characterization of planar monotone σ-continuous valuations taking integer values, without assuming invariance under any group of transformations. We further investigate the consequences of dropping monotonicity or σ-continuity and give a full classification of line valuations. We also introduce a construction of the product for valuations of this type.
{"title":"Integer-valued valuations","authors":"Andrii Ilienko , Ilya Molchanov , Tommaso Visonà","doi":"10.1016/j.aim.2026.110823","DOIUrl":"10.1016/j.aim.2026.110823","url":null,"abstract":"<div><div>We obtain a complete characterization of planar monotone <em>σ</em>-continuous valuations taking integer values, without assuming invariance under any group of transformations. We further investigate the consequences of dropping monotonicity or <em>σ</em>-continuity and give a full classification of line valuations. We also introduce a construction of the product for valuations of this type.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110823"},"PeriodicalIF":1.5,"publicationDate":"2026-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080759","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-28DOI: 10.1016/j.aim.2026.110821
Jinlei Dong, Fang Li
One of the key points in Galois theory via field extensions is to build up a correspondence between subfields of a field and subgroups of its automorphism group, so as to study fields via methods of groups. As an analogue of the Galois theory, we want to discuss the relations between cluster subalgebras of a cluster algebra and subgroups of its automorphism group and then set up the Galois-like method.
In the first part, we build up a Galois map from a skew-symmetrizable cluster algebra to its cluster automorphism group, and introduce notions of Galois-like extensions and Galois extensions. A necessary condition for Galois extensions of a cluster algebra is given, which is also a sufficient condition if has a -stable basis or stable monomial basis with unique expression. Some properties for Galois-like extensions are discussed. It is shown that two subgroups and of the automorphism group are conjugate to each other if and only if there exists and two Galois-like extension subalgebras , corresponding to and such that f is an isomorphism between and .
In the second part, as the answers of two conjectures proposed in the first part, for a cluster algebra from a feasible surface, we prove that Galois-like extension subalgebras corresponding to a subgroup of a cluster automorphism group have the same rank. Moreover, it is shown that there are order-preserving reverse Galois maps for these cluster algebras. We also give examples of -stable bases and some discussions on the Galois inverse problem in this part.
{"title":"On Galois theory of cluster algebras: general and that from Riemann surfaces","authors":"Jinlei Dong, Fang Li","doi":"10.1016/j.aim.2026.110821","DOIUrl":"10.1016/j.aim.2026.110821","url":null,"abstract":"<div><div>One of the key points in Galois theory via field extensions is to build up a correspondence between subfields of a field and subgroups of its automorphism group, so as to study fields via methods of groups. As an analogue of the Galois theory, we want to discuss the relations between cluster subalgebras of a cluster algebra and subgroups of its automorphism group and then set up the Galois-like method.</div><div>In the first part, we build up a Galois map from a skew-symmetrizable cluster algebra <span><math><mi>A</mi></math></span> to its cluster automorphism group, and introduce notions of Galois-like extensions and Galois extensions. A necessary condition for Galois extensions of a cluster algebra <span><math><mi>A</mi></math></span> is given, which is also a sufficient condition if <span><math><mi>A</mi></math></span> has a <span><math><mi>D</mi></math></span>-stable basis or stable monomial basis with unique expression. Some properties for Galois-like extensions are discussed. It is shown that two subgroups <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> of the automorphism group <span><math><mtext>Aut</mtext><mi>A</mi></math></span> are conjugate to each other if and only if there exists <span><math><mi>f</mi><mo>∈</mo><mtext>Aut</mtext><mi>A</mi></math></span> and two Galois-like extension subalgebras <span><math><mi>A</mi><mo>(</mo><msub><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span>, <span><math><mi>A</mi><mo>(</mo><msub><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> corresponding to <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> such that <em>f</em> is an isomorphism between <span><math><mi>A</mi><mo>(</mo><msub><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> and <span><math><mi>A</mi><mo>(</mo><msub><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>.</div><div>In the second part, as the answers of two conjectures proposed in the first part, for a cluster algebra from a feasible surface, we prove that Galois-like extension subalgebras corresponding to a subgroup of a cluster automorphism group have the same rank. Moreover, it is shown that there are order-preserving reverse Galois maps for these cluster algebras. We also give examples of <span><math><mi>D</mi></math></span>-stable bases and some discussions on the Galois inverse problem in this part.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110821"},"PeriodicalIF":1.5,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-28DOI: 10.1016/j.aim.2026.110820
Noah Kravitz , Borys Kuca , James Leng
Using PET and quantitative concatenation techniques, we establish box-norm control with the “expected” directions for counting operators for general multidimensional polynomial progressions, with at most polynomial losses in the parameters. Such results are often useful first steps towards obtaining explicit upper bounds on sets lacking instances of given such progressions. In the companion paper [20], we complete this program for sets in lacking nondegenerate progressions of the form , where is any fixed polynomial with an integer root of multiplicity 1.
{"title":"Quantitative concatenation for polynomial box norms","authors":"Noah Kravitz , Borys Kuca , James Leng","doi":"10.1016/j.aim.2026.110820","DOIUrl":"10.1016/j.aim.2026.110820","url":null,"abstract":"<div><div>Using PET and quantitative concatenation techniques, we establish box-norm control with the “expected” directions for counting operators for general multidimensional polynomial progressions, with at most polynomial losses in the parameters. Such results are often useful first steps towards obtaining explicit upper bounds on sets lacking instances of given such progressions. In the companion paper <span><span>[20]</span></span>, we complete this program for sets in <span><math><msup><mrow><mo>[</mo><mi>N</mi><mo>]</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> lacking nondegenerate progressions of the form <span><math><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>,</mo><mo>(</mo><mi>x</mi><mo>+</mo><mi>P</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>,</mo><mi>y</mi><mo>)</mo><mo>,</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>+</mo><mi>P</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>)</mo></math></span>, where <span><math><mi>P</mi><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>z</mi><mo>]</mo></math></span> is any fixed polynomial with an integer root of multiplicity 1.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110820"},"PeriodicalIF":1.5,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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