James Cruickshank , Bill Jackson , Shin-ichi Tanigawa
{"title":"Global rigidity of triangulated manifolds","authors":"James Cruickshank , Bill Jackson , Shin-ichi Tanigawa","doi":"10.1016/j.aim.2024.109953","DOIUrl":"10.1016/j.aim.2024.109953","url":null,"abstract":"<div><div>We prove that if <em>G</em> is the graph of a connected triangulated <span><math><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-manifold, for <span><math><mi>d</mi><mo>≥</mo><mn>3</mn></math></span>, then <em>G</em> is generically globally rigid in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> if and only if it is <span><math><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-connected and, if <span><math><mi>d</mi><mo>=</mo><mn>3</mn></math></span>, <em>G</em> is not planar. The special case <span><math><mi>d</mi><mo>=</mo><mn>3</mn></math></span> verifies a conjecture of Connelly. Our results actually apply to a much larger class of simplicial complexes, namely the circuits of the simplicial matroid. We also give two significant applications of our main theorems. We show that the characterisation of pseudomanifolds with extremal edge numbers given by the Lower Bound Theorem extends to circuits of the simplicial matroid. We also prove the generic case of a conjecture of Kalai concerning the reconstructability of a polytope from its space of stresses. The proofs of our main results adapt earlier ideas of Fogelsanger and Whiteley to the setting of global rigidity. In particular we verify a special case of Whiteley's vertex splitting conjecture for global rigidity.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"458 ","pages":"Article 109953"},"PeriodicalIF":1.5,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315774","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}
{"title":"Correlation inequalities for linear extensions","authors":"Swee Hong Chan , Igor Pak","doi":"10.1016/j.aim.2024.109954","DOIUrl":"10.1016/j.aim.2024.109954","url":null,"abstract":"<div><div>We employ the combinatorial atlas technology to prove new correlation inequalities for the number of linear extensions of finite posets. These include the approximate independence of probabilities and expectations of values of random linear extensions, closely related to Stanley's inequality. We also give applications to the numbers of standard Young tableaux and to Euler numbers.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"458 ","pages":"Article 109954"},"PeriodicalIF":1.5,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0001870824004699/pdfft?md5=616cbaef7e6c6cceb3d8a76287c928b8&pid=1-s2.0-S0001870824004699-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generalized Frank characterizations of Muckenhoupt weights and homogeneous ball Banach Sobolev spaces","authors":"Yirui Zhao, Yinqin Li, Dachun Yang, Wen Yuan, Yangyang Zhang","doi":"10.1016/j.aim.2024.109957","DOIUrl":"10.1016/j.aim.2024.109957","url":null,"abstract":"<div><div>In this article, the authors first establish a new characterization of Muckenhoupt weights in terms of oscillations. As an application, the authors give a new characterization of homogeneous ball Banach Sobolev spaces, which extends the elegant characterization of Sobolev spaces obtained by R. L. Frank in 2024 and is a variant of the famous formula obtained by H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung in 2024 with difference quotients replaced by oscillations. Moreover, the authors also obtain new representation formulae of gradients in terms of oscillations in ball Banach function spaces, which even include the critical case where Frank did not consider. Furthermore, via some counterexamples, we prove that all the main results are sharp. Applying these results, the authors further reveal the mutual equivalences among Muckenhoupt weights, the weighted upper estimate of the characterization of Frank, and the weighted upper estimate of the formula of Brezis et al.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"458 ","pages":"Article 109957"},"PeriodicalIF":1.5,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315773","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}
{"title":"Deformations of Lagrangian NQ-submanifolds","authors":"Miquel Cueca , Jonas Schnitzer","doi":"10.1016/j.aim.2024.109952","DOIUrl":"10.1016/j.aim.2024.109952","url":null,"abstract":"<div><div>In this paper we prove graded versions of the Darboux Theorem and Weinstein's Lagrangian tubular neighbourhood Theorem in order to study the deformation theory of Lagrangian <em>NQ</em>-submanifolds of degree <em>n</em> symplectic <em>NQ</em>-manifolds. Using Weinstein's Lagrangian tubular neighbourhood Theorem, we attach to every Lagrangian <em>NQ</em>-submanifold an <span><math><msub><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-algebra, which controls its deformation theory. The main examples are coisotropic submanifolds of Poisson manifolds and (higher) Dirac structures with support in (higher) Courant algebroids.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"458 ","pages":"Article 109952"},"PeriodicalIF":1.5,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0001870824004675/pdfft?md5=5940eacdc2d09ddd157ad0d959322302&pid=1-s2.0-S0001870824004675-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142310988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Mating quadratic maps with the modular group III: The modular Mandelbrot set","authors":"Shaun Bullett , Luna Lomonaco","doi":"10.1016/j.aim.2024.109956","DOIUrl":"10.1016/j.aim.2024.109956","url":null,"abstract":"<div><div>We prove that there exists a homeomorphism <em>χ</em> between the connectedness locus <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>Γ</mi></mrow></msub></math></span> for the family <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span> of <span><math><mo>(</mo><mn>2</mn><mo>:</mo><mn>2</mn><mo>)</mo></math></span> holomorphic correspondences introduced by Bullett and Penrose, and the parabolic Mandelbrot set <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. The homeomorphism <em>χ</em> is dynamical (<span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span> is a mating between <span><math><mi>P</mi><mi>S</mi><mi>L</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>Z</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>χ</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msub></math></span>), it is conformal on the interior of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>Γ</mi></mrow></msub></math></span>, and it extends to a homeomorphism between suitably defined neighbourhoods in the respective one parameter moduli spaces.</div><div>Following the recent proof by Petersen and Roesch that <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is homeomorphic to the classical Mandelbrot set <span><math><mi>M</mi></math></span>, we deduce that <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>Γ</mi></mrow></msub></math></span> is homeomorphic to <span><math><mi>M</mi></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"458 ","pages":"Article 109956"},"PeriodicalIF":1.5,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142310989","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}
Christian Carrick , Michael A. Hill , Douglas C. Ravenel
{"title":"The homological slice spectral sequence in motivic and Real bordism","authors":"Christian Carrick , Michael A. Hill , Douglas C. Ravenel","doi":"10.1016/j.aim.2024.109955","DOIUrl":"10.1016/j.aim.2024.109955","url":null,"abstract":"<div><p>For a motivic spectrum <span><math><mi>E</mi><mo>∈</mo><mrow><mi>SH</mi></mrow><mo>(</mo><mi>k</mi><mo>)</mo></math></span>, let <span><math><mi>Γ</mi><mo>(</mo><mi>E</mi><mo>)</mo></math></span> denote the global sections spectrum, where <em>E</em> is viewed as a sheaf of spectra on <span><math><msub><mrow><mi>Sm</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>. Voevodsky's slice filtration determines a spectral sequence converging to the homotopy groups of <span><math><mi>Γ</mi><mo>(</mo><mi>E</mi><mo>)</mo></math></span>. In this paper, we introduce a spectral sequence converging instead to the mod 2 homology of <span><math><mi>Γ</mi><mo>(</mo><mi>E</mi><mo>)</mo></math></span> and study the case <span><math><mi>E</mi><mo>=</mo><mi>B</mi><mi>P</mi><mi>G</mi><mi>L</mi><mo>〈</mo><mi>m</mi><mo>〉</mo></math></span> for <span><math><mi>k</mi><mo>=</mo><mi>R</mi></math></span> in detail. We show that this spectral sequence contains the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span>-comodule algebra <span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msub><msub><mrow><mo>□</mo></mrow><mrow><mi>A</mi><msub><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow><mrow><mo>⁎</mo></mrow></msub></mrow></msub><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> as permanent cycles, and we determine a family of differentials interpolating between <span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msub><msub><mrow><mo>□</mo></mrow><mrow><mi>A</mi><msub><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mrow><mo>⁎</mo></mrow></msub></mrow></msub><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msub><msub><mrow><mo>□</mo></mrow><mrow><mi>A</mi><msub><mrow><mo>(</mo><mi>m</mi><mo>)</mo></mrow><mrow><mo>⁎</mo></mrow></msub></mrow></msub><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Using this, we compute the spectral sequence completely for <span><math><mi>m</mi><mo>≤</mo><mn>3</mn></math></span>.</p><p>In the height 2 case, the Betti realization of <span><math><mi>B</mi><mi>P</mi><mi>G</mi><mi>L</mi><mo>〈</mo><mn>2</mn><mo>〉</mo></math></span> is the <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-spectrum <span><math><mi>B</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>〈</mo><mn>2</mn><mo>〉</mo></math></span>, a form of which was shown by Hill and Meier to be an equivariant model for <span><math><msub><mrow><mi>tmf</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mn>3</mn><mo>)</mo></math></span>. Our spectral sequence therefore gives a computation of the comodule algebra <span><math><msub><mrow><mi>H</mi></mrow><mrow><mo>⁎</mo></mrow></msub><msub><mrow><mi>tmf</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mn>3</mn><mo>)</mo></math></span>. As a consequence, we deduce a new (2-local) Wood-t","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"458 ","pages":"Article 109955"},"PeriodicalIF":1.5,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0001870824004705/pdfft?md5=5881a17b5ae2bf26359dfa18561bd41c&pid=1-s2.0-S0001870824004705-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142272013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Growth of polynomials on arcs in the complex plane","authors":"Annie R. Wei","doi":"10.1016/j.aim.2024.109940","DOIUrl":"10.1016/j.aim.2024.109940","url":null,"abstract":"<div><p>We prove that the growth rate of polynomials on an arc in the complex plane is exponential in its degree and can be computed by a linear program.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"458 ","pages":"Article 109940"},"PeriodicalIF":1.5,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0001870824004559/pdfft?md5=b7a46e228e4403bdfcbf5b4f2b083235&pid=1-s2.0-S0001870824004559-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142272223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Integral p-adic non-abelian Hodge theory for small representations","authors":"Yu Min , Yupeng Wang","doi":"10.1016/j.aim.2024.109950","DOIUrl":"10.1016/j.aim.2024.109950","url":null,"abstract":"<div><p>Let <span><math><mi>X</mi></math></span> be a smooth <em>p</em>-adic formal scheme over <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>C</mi></mrow></msub></math></span> with rigid generic fiber <em>X</em>. In this paper, we construct a new integral period sheaf <span><math><mi>O</mi><msubsup><mrow><mover><mrow><mi>C</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>pd</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> on <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>pro</mi><mover><mrow><mi>e</mi></mrow><mrow><mo>´</mo></mrow></mover><mi>t</mi></mrow></msub></math></span> and use it to establish an integral <em>p</em>-adic Simpson correspondence for small <span><math><msubsup><mrow><mover><mrow><mi>O</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>X</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>-representations on <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>pro</mi><mover><mrow><mi>e</mi></mrow><mrow><mo>´</mo></mrow></mover><mi>t</mi></mrow></msub></math></span> and small Higgs bundles on <span><math><msub><mrow><mi>X</mi></mrow><mrow><mover><mrow><mi>e</mi></mrow><mrow><mo>´</mo></mrow></mover><mi>t</mi></mrow></msub></math></span>, which recovers rational <em>p</em>-adic Simpson correspondence for small coefficients after inverting <em>p</em> (at least in the good reduction case). Moreover, for a small <span><math><msubsup><mrow><mover><mrow><mi>O</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>X</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>-representation <span><math><mi>L</mi></math></span> with induced Higgs bundle <span><math><mo>(</mo><mi>H</mi><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>)</mo></math></span>, we provide a natural morphism <span><math><mrow><mi>HIG</mi></mrow><mo>(</mo><mi>H</mi><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>)</mo><mo>→</mo><mi>R</mi><msub><mrow><mi>ν</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mi>L</mi></math></span> with a bounded <span><math><msup><mrow><mi>p</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>-torsion cofiber. Finally, we shall use this natural map to study an analogue of Deligne–Illusie decomposition with coefficients in small <span><math><msubsup><mrow><mover><mrow><mi>O</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>X</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>-representations.</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"458 ","pages":"Article 109950"},"PeriodicalIF":1.5,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142272014","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}
{"title":"Goerss–Hopkins obstruction theory for ∞-categories","authors":"Aaron Mazel-Gee","doi":"10.1016/j.aim.2024.109951","DOIUrl":"10.1016/j.aim.2024.109951","url":null,"abstract":"<div><p>Goerss–Hopkins obstruction theory is a powerful tool for constructing structured ring spectra from purely algebraic data. Using the formalism of model ∞-categories, we provide a generalization that applies in an arbitrary presentably symmetric monoidal stable ∞-category (such as that of equivariant spectra or of motivic spectra).</p></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"458 ","pages":"Article 109951"},"PeriodicalIF":1.5,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142238509","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}
{"title":"Weighted anisotropic isoperimetric inequalities and existence of extremals for singular anisotropic Trudinger-Moser inequalities","authors":"Guozhen Lu , Yansheng Shen , Jianwei Xue , Maochun Zhu","doi":"10.1016/j.aim.2024.109949","DOIUrl":"10.1016/j.aim.2024.109949","url":null,"abstract":"<div><p>In this paper, we establish a class of isoperimetric inequalities on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with respect to weights which are negative powers of the distance to the origin associated with the Finsler metric. (See <span><span>Theorem 1.1</span></span>.) Based on these weighted anisotropic isoperimetric inequalities, we can classify a class of singular Liouville's equation associated with the <em>n</em>-Finsler-Laplacian <span><span>(1.10)</span></span> and construct a blow-up sequence to show the existence of extremals for the singular Trudinger-Moser inequality involving the anisotropic Dirichlet norm:<span><span><span><math><munder><mi>sup</mi><mrow><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msubsup><mo>(</mo><mi>Ω</mi><mo>)</mo><mo>,</mo><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>F</mi><msup><mrow><mo>(</mo><mi>∇</mi><mi>u</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mi>d</mi><mi>x</mi><mo>≤</mo><mn>1</mn></mrow></munder><mo></mo><munder><mo>∫</mo><mrow><mi>Ω</mi></mrow></munder><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>β</mi></mrow></msub><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup></mrow></msup><msup><mrow><mi>F</mi></mrow><mrow><mo>∘</mo></mrow></msup><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>−</mo><mi>β</mi></mrow></msup><mi>d</mi><mi>x</mi><mo><</mo><mo>∞</mo></math></span></span></span> for any <span><math><mi>β</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span>, where <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is a smooth and bounded domain containing the origin, and <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>β</mi></mrow></msub><mo>:</mo><mo>=</mo><mo>(</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mi>β</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mo>)</mo><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup><msubsup><mrow><mi>κ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msubsup></math></span>. Here <em>F</em> is a convex function, which is even and positively homogeneous of degree 1, and its polar <span><math><msup><mrow><mi>F</mi></mrow><mrow><mo>∘</mo></mrow></msup></math></span> represents a Finsler metric on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><msub><mrow><mi>κ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is the Lebesgue measure of the unit Wulff ball.</p><p>The presence of the weight in <span><span>Theorem 1.1</span></span> adds significant difficulties because","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"458 ","pages":"Article 109949"},"PeriodicalIF":1.5,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142238507","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}