Pub Date : 2026-01-27DOI: 10.1016/j.aim.2026.110802
Xenia Flamm
The main result of this article is that Hitchin representations over real closed field extensions of correspond precisely to those representations of the fundamental group of a closed surface into that are conjugate to -positive representations, i.e. representations that admit an equivariant limit map from the set of fixed points in the boundary of the universal cover of the surface into the set of full flags in satisfying specific positivity properties. As the theorem treats general real closed fields, and not only the reals, the tools of analysis are not available. Instead, our proof is based on the Tarski–Seidenberg transfer principle and a multiplicative version of the Bonahon–Dreyer coordinates.
We use this result to prove that -positive representations form semi-algebraically connected components of the space of all representations, that consist entirely of injective and discrete representations, which are positively hyperbolic and weakly dynamics-preserving over . Furthermore, we show how to associate intersection geodesic currents to -positive representations, and conclude with applications to the Weyl chamber length compactification and to dual spaces of geodesic currents.
{"title":"Real spectrum compactification of Hitchin components, Weyl chamber valued lengths, and dual spaces","authors":"Xenia Flamm","doi":"10.1016/j.aim.2026.110802","DOIUrl":"10.1016/j.aim.2026.110802","url":null,"abstract":"<div><div>The main result of this article is that Hitchin representations over real closed field extensions <span><math><mi>F</mi></math></span> of <span><math><mi>R</mi></math></span> correspond precisely to those representations of the fundamental group of a closed surface into <span><math><mtext>PSL</mtext><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> that are conjugate to <span><math><mi>F</mi></math></span>-positive representations, i.e. representations that admit an equivariant limit map from the set of fixed points in the boundary of the universal cover of the surface into the set of full flags in <span><math><msup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> satisfying specific positivity properties. As the theorem treats general real closed fields, and not only the reals, the tools of analysis are not available. Instead, our proof is based on the Tarski–Seidenberg transfer principle and a multiplicative version of the Bonahon–Dreyer coordinates.</div><div>We use this result to prove that <span><math><mi>F</mi></math></span>-positive representations form semi-algebraically connected components of the space of all representations, that consist entirely of injective and discrete representations, which are positively hyperbolic and weakly dynamics-preserving over <span><math><mi>F</mi></math></span>. Furthermore, we show how to associate intersection geodesic currents to <span><math><mi>F</mi></math></span>-positive representations, and conclude with applications to the Weyl chamber length compactification and to dual spaces of geodesic currents.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110802"},"PeriodicalIF":1.5,"publicationDate":"2026-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080758","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}
We characterize the limiting root distribution μ of a sequence of polynomials with nonnegative roots and degree d, in terms of their coefficients. Specifically, we relate the asymptotic behavior of the ratio of consecutive coefficients of to Voiculescu's S-transform of μ.
In the framework of finite free probability, we interpret these ratios of coefficients as a new notion of finite S-transform, which converges to in the large d limit. It also satisfies several analogous properties to those of the S-transform in free probability, including multiplicativity and monotonicity.
The proof of the main theorem is based on various ideas and new results relating finite free probability and free probability. In particular, we provide a simplified explanation of why free fractional convolution corresponds to the differentiation of polynomials, by finding how the finite free cumulants of a polynomial behave under differentiation.
This new insight has several applications that strengthen the connection between free and finite free probability. Most notably, we generalize the approximation of to ⊠ and prove a finite approximation of the Tucci–Haagerup–Möller limit theorem in free probability, conjectured by two of the authors. We also provide finite analogues of the free multiplicative Poisson law, the free max-convolution powers and some free stable laws.
{"title":"S-transform in finite free probability","authors":"Octavio Arizmendi , Katsunori Fujie , Daniel Perales , Yuki Ueda","doi":"10.1016/j.aim.2026.110803","DOIUrl":"10.1016/j.aim.2026.110803","url":null,"abstract":"<div><div>We characterize the limiting root distribution <em>μ</em> of a sequence of polynomials <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> with nonnegative roots and degree <em>d</em>, in terms of their coefficients. Specifically, we relate the asymptotic behavior of the ratio of consecutive coefficients of <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> to Voiculescu's <em>S</em>-transform <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>μ</mi></mrow></msub></math></span> of <em>μ</em>.</div><div>In the framework of finite free probability, we interpret these ratios of coefficients as a new notion of finite <em>S</em>-transform, which converges to <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>μ</mi></mrow></msub></math></span> in the large <em>d</em> limit. It also satisfies several analogous properties to those of the <em>S</em>-transform in free probability, including multiplicativity and monotonicity.</div><div>The proof of the main theorem is based on various ideas and new results relating finite free probability and free probability. In particular, we provide a simplified explanation of why free fractional convolution corresponds to the differentiation of polynomials, by finding how the finite free cumulants of a polynomial behave under differentiation.</div><div>This new insight has several applications that strengthen the connection between free and finite free probability. Most notably, we generalize the approximation of <span><math><msub><mrow><mo>⊠</mo></mrow><mrow><mi>d</mi></mrow></msub></math></span> to ⊠ and prove a finite approximation of the Tucci–Haagerup–Möller limit theorem in free probability, conjectured by two of the authors. We also provide finite analogues of the free multiplicative Poisson law, the free max-convolution powers and some free stable laws.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110803"},"PeriodicalIF":1.5,"publicationDate":"2026-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146080764","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 : 2026-01-23DOI: 10.1016/j.aim.2026.110808
Luis Bernal-González , Daniel L. Rodríguez-Vidanes , Juan B. Seoane-Sepúlveda , Hyung-Joon Tag
In this article, we investigate the existence of closed vector subspaces (i.e. spaceability) in various nonlinear subsets of Orlicz-Lorentz spaces equipped with the Luxemburg norm. If a family of Orlicz functions satisfies certain order relations with respect to a given Orlicz function φ, the subset of the order-continuous subspace whose elements do not belong to is spaceable, and even maximal-spaceable when φ satisfies the -condition. We also show that this subset is either residual or empty. In addition, sufficient conditions for this subset not being -spaceable are provided. A similar analysis is also performed on the subset when φ does not satisfy the -condition.
The comparison between different Orlicz-Lorentz spaces is characterized via the generating pairs . For a fixed Orlicz function that satisfies the -condition, we provide a characterization of disjointly strictly singular inclusion operators between Orlicz-Lorentz spaces with different weights. As a consequence, there are certain subsets of Orlicz-Lorentz spaces on for which the lineability problem is not valid. Moreover, various types of -lineability and pointwise lineability properties on other nonlinear subsets of Orlicz-Lorentz spaces are examined. These results extend a number of previously known results in Orlicz and Lorentz spaces.
{"title":"New results in analysis of Orlicz-Lorentz spaces","authors":"Luis Bernal-González , Daniel L. Rodríguez-Vidanes , Juan B. Seoane-Sepúlveda , Hyung-Joon Tag","doi":"10.1016/j.aim.2026.110808","DOIUrl":"10.1016/j.aim.2026.110808","url":null,"abstract":"<div><div>In this article, we investigate the existence of closed vector subspaces (i.e. spaceability) in various nonlinear subsets of Orlicz-Lorentz spaces <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mi>φ</mi><mo>,</mo><mi>w</mi></mrow></msub></math></span> equipped with the Luxemburg norm. If a family of Orlicz functions <span><math><msubsup><mrow><mo>(</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span> satisfies certain order relations with respect to a given Orlicz function <em>φ</em>, the subset of the order-continuous subspace <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mi>φ</mi><mo>,</mo><mi>w</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>a</mi></mrow></msub></math></span> whose elements do not belong to <span><math><msubsup><mrow><mo>⋃</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mrow><mi>Λ</mi></mrow><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>w</mi></mrow></msub></math></span> is spaceable, and even maximal-spaceable when <em>φ</em> satisfies the <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-condition. We also show that this subset is either residual or empty. In addition, sufficient conditions for this subset not being <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span>-spaceable are provided. A similar analysis is also performed on the subset <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mi>φ</mi><mo>,</mo><mi>w</mi></mrow></msub><mo>∖</mo><msub><mrow><mo>(</mo><msub><mrow><mi>Λ</mi></mrow><mrow><mi>φ</mi><mo>,</mo><mi>w</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>a</mi></mrow></msub></math></span> when <em>φ</em> does not satisfy the <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-condition.</div><div>The comparison between different Orlicz-Lorentz spaces is characterized via the generating pairs <span><math><mo>(</mo><mi>φ</mi><mo>,</mo><mi>w</mi><mo>)</mo></math></span>. For a fixed Orlicz function that satisfies the <span><math><msubsup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>∞</mo></mrow></msubsup></math></span>-condition, we provide a characterization of disjointly strictly singular inclusion operators between Orlicz-Lorentz spaces with different weights. As a consequence, there are certain subsets of Orlicz-Lorentz spaces on <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> for which the lineability problem is not valid. Moreover, various types of <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span>-lineability and pointwise lineability properties on other nonlinear subsets of Orlicz-Lorentz spaces are examined. These results extend a number of previously known results in Orlicz and Lorentz spaces.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110808"},"PeriodicalIF":1.5,"publicationDate":"2026-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025722","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 : 2026-01-23DOI: 10.1016/j.aim.2026.110800
Tong Yang , Mingying Zhong
We study the diffusion limit of the classical solution to the Vlasov-Maxwell-Boltzmann (VMB) system with initial data near a global Maxwellian. By introducing a new decomposition of the solution to identify the essential components for generating the initial layer, we prove the convergence and establish the optimal convergence rate of the classical solution to the VMB system to the solution of the Navier-Stokes-Maxwell system based on the spectral analysis.
{"title":"Diffusion limit with optimal convergence rate of classical solutions to the Vlasov-Maxwell-Boltzmann system","authors":"Tong Yang , Mingying Zhong","doi":"10.1016/j.aim.2026.110800","DOIUrl":"10.1016/j.aim.2026.110800","url":null,"abstract":"<div><div>We study the diffusion limit of the classical solution to the Vlasov-Maxwell-Boltzmann (VMB) system with initial data near a global Maxwellian. By introducing a new decomposition of the solution to identify the essential components for generating the initial layer, we prove the convergence and establish the optimal convergence rate of the classical solution to the VMB system to the solution of the Navier-Stokes-Maxwell system based on the spectral analysis.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110800"},"PeriodicalIF":1.5,"publicationDate":"2026-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025725","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 : 2026-01-23DOI: 10.1016/j.aim.2026.110806
Utsav Choudhury, Biman Roy
In this article we prove that any -connected smooth k-variety is -uniruled for any algebraically closed field k. We establish that if a non-empty open subscheme X of a smooth affine k-scheme is -weakly equivalent to , then as k-varieties for any field k of characteristic 0.
{"title":"A1-homotopy type of A2∖{(0,0)}","authors":"Utsav Choudhury, Biman Roy","doi":"10.1016/j.aim.2026.110806","DOIUrl":"10.1016/j.aim.2026.110806","url":null,"abstract":"<div><div>In this article we prove that any <span><math><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-connected smooth <em>k</em>-variety is <span><math><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-uniruled for any algebraically closed field <em>k</em>. We establish that if a non-empty open subscheme <em>X</em> of a smooth affine <em>k</em>-scheme is <span><math><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-weakly equivalent to <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>∖</mo><mrow><mo>{</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>}</mo></mrow></math></span>, then <span><math><mi>X</mi><mo>≅</mo><msubsup><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>∖</mo><mrow><mo>{</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>}</mo></mrow></math></span> as <em>k</em>-varieties for any field <em>k</em> of characteristic 0.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110806"},"PeriodicalIF":1.5,"publicationDate":"2026-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025720","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 : 2026-01-23DOI: 10.1016/j.aim.2026.110789
Alexis Michelat , Andrea Mondino
We show that the quantization of energy for Willmore spheres into closed Riemannian manifolds holds provided that the Willmore energy and the area be uniformly bounded. The analogous energy quantization result holds for Willmore surfaces of arbitrary genus, under the additional assumptions that the immersion maps weakly converge to a limiting (possibly branched, weak immersion) map from the same surface, and that the conformal structures stay within a compact domain of the moduli space.
{"title":"Quantization of the Willmore energy in Riemannian manifolds","authors":"Alexis Michelat , Andrea Mondino","doi":"10.1016/j.aim.2026.110789","DOIUrl":"10.1016/j.aim.2026.110789","url":null,"abstract":"<div><div>We show that the quantization of energy for Willmore spheres into closed Riemannian manifolds holds provided that the Willmore energy and the area be uniformly bounded. The analogous energy quantization result holds for Willmore surfaces of arbitrary genus, under the additional assumptions that the immersion maps weakly converge to a limiting (possibly branched, weak immersion) map from the same surface, and that the conformal structures stay within a compact domain of the moduli space.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110789"},"PeriodicalIF":1.5,"publicationDate":"2026-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025723","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 : 2026-01-23DOI: 10.1016/j.aim.2026.110799
Di Wu, Xi Zhang
Given a flat vector bundle over a compact Riemannian manifold, the Corlette-Donaldson theorem indicates that it admits harmonic metrics if and only if it is semi-simple. We extend this equivalence to arbitrary vector bundles without any additional hypotheses, it can be viewed as a Riemannian Donaldson-Uhlenbeck-Yau correspondence. Furthermore, we prove an equivalence of categories in Sasakian geometry, relating projective flat vector bundles to Higgs bundles. Along the way, a transparent proof is also provided for the Reeb invariance of harmonic metrics in Sasakian geometry that had required Sasakian curvature theory and spinorial trick before, the Reeb invariance plays a crucial role in defining stability of basic Higgs bundles and establishing Sasakian Corlette-Simpson correspondence.
{"title":"Harmonic metrics and semi-simpleness","authors":"Di Wu, Xi Zhang","doi":"10.1016/j.aim.2026.110799","DOIUrl":"10.1016/j.aim.2026.110799","url":null,"abstract":"<div><div>Given a flat vector bundle over a compact Riemannian manifold, the Corlette-Donaldson theorem indicates that it admits harmonic metrics if and only if it is semi-simple. We extend this equivalence to arbitrary vector bundles without any additional hypotheses, it can be viewed as a Riemannian Donaldson-Uhlenbeck-Yau correspondence. Furthermore, we prove an equivalence of categories in Sasakian geometry, relating projective flat vector bundles to Higgs bundles. Along the way, a transparent proof is also provided for the Reeb invariance of harmonic metrics in Sasakian geometry that had required Sasakian curvature theory and spinorial trick before, the Reeb invariance plays a crucial role in defining stability of basic Higgs bundles and establishing Sasakian Corlette-Simpson correspondence.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110799"},"PeriodicalIF":1.5,"publicationDate":"2026-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146025721","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 : 2026-01-22DOI: 10.1016/j.aim.2026.110807
Sam Chow , Péter P. Varjú , Han Yu
We establish a new upper bound for the number of rationals up to a given height in a missing-digit set, making progress towards a conjecture of Broderick, Fishman, and Reich. This enables us to make novel progress towards another conjecture of those authors about the corresponding intrinsic diophantine approximation problem. Moreover, we make further progress towards conjectures of Bugeaud–Durand and Levesley–Salp–Velani on the distribution of diophantine exponents in missing-digit sets.
A key tool in our study is Fourier dimension introduced by the last named author in Yu (2021) [12]. An important technical contribution of the paper is a method to compute this quantity.
{"title":"Counting rationals and diophantine approximation in missing-digit Cantor sets","authors":"Sam Chow , Péter P. Varjú , Han Yu","doi":"10.1016/j.aim.2026.110807","DOIUrl":"10.1016/j.aim.2026.110807","url":null,"abstract":"<div><div>We establish a new upper bound for the number of rationals up to a given height in a missing-digit set, making progress towards a conjecture of Broderick, Fishman, and Reich. This enables us to make novel progress towards another conjecture of those authors about the corresponding intrinsic diophantine approximation problem. Moreover, we make further progress towards conjectures of Bugeaud–Durand and Levesley–Salp–Velani on the distribution of diophantine exponents in missing-digit sets.</div><div>A key tool in our study is Fourier <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> dimension introduced by the last named author in Yu (2021) <span><span>[12]</span></span>. An important technical contribution of the paper is a method to compute this quantity.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110807"},"PeriodicalIF":1.5,"publicationDate":"2026-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039196","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 : 2026-01-22DOI: 10.1016/j.aim.2026.110788
Ulrich Bunke , Daniel Kasprowski , Christoph Winges
We prove the Farrell–Jones conjecture for finitary localising invariants with coefficients in left-exact ∞-categories for finitely -amenable groups and, more generally, Dress-Farrell-Hsiang-Jones groups. Our result subsumes and unifies arguments for the K-theory of additive categories and spherical group rings and extends it for example to categories of perfect modules over -ring spectra.
{"title":"On the Farrell–Jones conjecture for localising invariants","authors":"Ulrich Bunke , Daniel Kasprowski , Christoph Winges","doi":"10.1016/j.aim.2026.110788","DOIUrl":"10.1016/j.aim.2026.110788","url":null,"abstract":"<div><div>We prove the Farrell–Jones conjecture for finitary localising invariants with coefficients in left-exact ∞-categories for finitely <span><math><mi>F</mi></math></span>-amenable groups and, more generally, Dress-Farrell-Hsiang-Jones groups. Our result subsumes and unifies arguments for the K-theory of additive categories and spherical group rings and extends it for example to categories of perfect modules over <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-ring spectra.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"489 ","pages":"Article 110788"},"PeriodicalIF":1.5,"publicationDate":"2026-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146006653","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 : 2026-01-22DOI: 10.1016/j.aim.2026.110797
Jin Gao , Meng Yang
We introduce a Hölder regularity condition for harmonic functions on metric measure spaces and prove that, under a slow volume regular condition and an upper heat kernel estimate, the Hölder regularity condition, the weak Bakry-Émery non-negative curvature condition, Hölder continuity of the heat kernel (with or without exponential terms), and the near-diagonal lower bound for the heat kernel are equivalent. As applications, first, we establish the validity of the so-called generalized reverse Hölder inequality on the Sierpiński carpet cable system, resolving an open problem left by Devyver et al. (2023) [26]. Second, we prove that two-sided heat kernel estimates alone imply gradient estimates for the heat kernel on strongly recurrent fractal-like cable systems, improving the main results of the aforementioned paper. Third, we obtain Hölder (Lipschitz) estimates for the heat kernel on strongly recurrent metric measure spaces, extending the classical Li-Yau gradient estimate for the heat kernel on Riemannian manifolds.
{"title":"Hölder regularity of harmonic functions on metric measure spaces","authors":"Jin Gao , Meng Yang","doi":"10.1016/j.aim.2026.110797","DOIUrl":"10.1016/j.aim.2026.110797","url":null,"abstract":"<div><div>We introduce a Hölder regularity condition for harmonic functions on metric measure spaces and prove that, under a <em>slow</em> volume regular condition and an upper heat kernel estimate, the Hölder regularity condition, the weak Bakry-Émery non-negative curvature condition, Hölder continuity of the heat kernel (with or without exponential terms), and the near-diagonal lower bound for the heat kernel are equivalent. As applications, first, we establish the validity of the so-called generalized reverse Hölder inequality on the Sierpiński <em>carpet</em> cable system, resolving an open problem left by Devyver et al. (2023) <span><span>[26]</span></span>. Second, we prove that two-sided heat kernel estimates <em>alone</em> imply gradient estimates for the heat kernel on strongly recurrent fractal-like cable systems, improving the main results of the aforementioned paper. Third, we obtain Hölder (Lipschitz) estimates for the heat kernel on <em>strongly recurrent</em> metric measure spaces, extending the classical Li-Yau gradient estimate for the heat kernel on Riemannian manifolds.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"488 ","pages":"Article 110797"},"PeriodicalIF":1.5,"publicationDate":"2026-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146039202","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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