Mathematische Annalen最新文献

{"title":"Abelian varieties of prescribed order over finite fields.","authors":"Raymond van Bommel, Edgar Costa, Wanlin Li, Bjorn Poonen, Alexander Smith","doi":"10.1007/s00208-024-03084-4","DOIUrl":"https://doi.org/10.1007/s00208-024-03084-4","url":null,"abstract":"<p><p>Given a prime power <i>q</i> and <math><mrow><mi>n</mi> <mo>≫</mo> <mn>1</mn></mrow> </math> , we prove that every integer in a large subinterval of the Hasse-Weil interval <math><mrow><mo>[</mo> <msup><mrow><mo>(</mo> <msqrt><mi>q</mi></msqrt> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> <mrow><mn>2</mn> <mi>n</mi></mrow> </msup> <mo>,</mo> <msup><mrow><mo>(</mo> <msqrt><mi>q</mi></msqrt> <mo>+</mo> <mn>1</mn> <mo>)</mo></mrow> <mrow><mn>2</mn> <mi>n</mi></mrow> </msup> <mo>]</mo></mrow> </math> is <math><mrow><mo>#</mo> <mi>A</mi> <mo>(</mo> <msub><mi>F</mi> <mi>q</mi></msub> <mo>)</mo></mrow> </math> for some ordinary geometrically simple principally polarized abelian variety <i>A</i> of dimension <i>n</i> over <math><msub><mi>F</mi> <mi>q</mi></msub> </math> . As a consequence, we generalize a result of Howe and Kedlaya for <math><msub><mi>F</mi> <mn>2</mn></msub> </math> to show that for each prime power <i>q</i>, every sufficiently large positive integer is realizable, i.e., <math><mrow><mo>#</mo> <mi>A</mi> <mo>(</mo> <msub><mi>F</mi> <mi>q</mi></msub> <mo>)</mo></mrow> </math> for some abelian variety <i>A</i> over <math><msub><mi>F</mi> <mi>q</mi></msub> </math> . Our result also improves upon the best known constructions of sequences of simple abelian varieties with point counts towards the extremes of the Hasse-Weil interval. A separate argument determines, for fixed <i>n</i>, the largest subinterval of the Hasse-Weil interval consisting of realizable integers, asymptotically as <math><mrow><mi>q</mi> <mo>→</mo> <mi>∞</mi></mrow> </math> ; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if <math><mrow><mi>q</mi> <mo>≤</mo> <mn>5</mn></mrow> </math> , then every positive integer is realizable, and for arbitrary <i>q</i>, every positive integer <math><mrow><mo>≥</mo> <msup><mi>q</mi> <mrow><mn>3</mn> <msqrt><mi>q</mi></msqrt> <mo>log</mo> <mi>q</mi></mrow> </msup> </mrow> </math> is realizable.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"392 1","pages":"1167-1202"},"PeriodicalIF":1.3,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11971235/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143795736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Regular logarithmic connections.","authors":"Piotr Achinger","doi":"10.1007/s00208-024-03047-9","DOIUrl":"https://doi.org/10.1007/s00208-024-03047-9","url":null,"abstract":"<p><p>We introduce the notion of a regular integrable connection on a smooth log scheme over <math><mi>C</mi></math> and construct an equivalence between the category of such connections and the category of integrable connections on its analytification, compatible with de Rham cohomology. This extends the work of Deligne (when the log structure is trivial), and combined with the work of Ogus yields a topological description of the category of regular connections in terms of certain constructible sheaves on the Kato-Nakayama space. The key ingredients are the notion of a canonical extension in this context and the existence of good compactifications of log schemes obtained recently by Włodarczyk.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"391 4","pages":"5293-5339"},"PeriodicalIF":1.3,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11954731/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143753555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Transversality of holomorphic maps into hyperquadrics.","authors":"Xiaojun Huang, Weixia Zhu","doi":"10.1007/s00208-025-03134-5","DOIUrl":"https://doi.org/10.1007/s00208-025-03134-5","url":null,"abstract":"<p><p>We study holomorphic maps <i>F</i> from a smooth Levi non-degenerate real hypersurface <math> <mrow><msub><mi>M</mi> <mi>ℓ</mi></msub> <mo>⊂</mo> <msup><mrow><mi>C</mi></mrow> <mi>n</mi></msup> </mrow> </math> into a hyperquadric <math><msubsup><mi>H</mi> <mrow><msup><mi>ℓ</mi> <mo>'</mo></msup> </mrow> <mi>N</mi></msubsup> </math> with signatures <math><mrow><mi>ℓ</mi> <mo>≤</mo> <mo>(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> <mo>/</mo> <mn>2</mn></mrow> </math> and <math> <mrow><msup><mi>ℓ</mi> <mo>'</mo></msup> <mo>≤</mo> <mrow><mo>(</mo> <mi>N</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> <mo>/</mo> <mn>2</mn> <mo>,</mo></mrow> </math> respectively. Assuming that <math><mrow><mi>N</mi> <mo>-</mo> <mi>n</mi> <mo><</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo></mrow> </math> we prove that if <math><mrow><mi>ℓ</mi> <mo>=</mo> <msup><mi>ℓ</mi> <mo>'</mo></msup> <mo>,</mo></mrow> </math> then <i>F</i> is either CR transversal to <math><msubsup><mi>H</mi> <mrow><mi>ℓ</mi></mrow> <mi>N</mi></msubsup> </math> at every point of <math> <mrow><msub><mi>M</mi> <mi>ℓ</mi></msub> <mo>,</mo></mrow> </math> or it maps a neighborhood of <math><msub><mi>M</mi> <mi>ℓ</mi></msub> </math> in <math> <msup><mrow><mi>C</mi></mrow> <mi>n</mi></msup> </math> into <math> <mrow><msubsup><mi>H</mi> <mrow><mi>ℓ</mi></mrow> <mi>N</mi></msubsup> <mo>.</mo></mrow> </math> Furthermore, in the case where <math> <mrow><msup><mi>ℓ</mi> <mo>'</mo></msup> <mo>></mo> <mi>ℓ</mi> <mo>,</mo></mrow> </math> we show that if <i>F</i> is not CR transversal at <math><mrow><mn>0</mn> <mo>∈</mo> <msub><mi>M</mi> <mi>ℓ</mi></msub> <mo>,</mo></mrow> </math> then it must be transversally flat. The latter is best possible.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"392 2","pages":"1731-1746"},"PeriodicalIF":1.3,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12084281/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144094324","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Foliation adjunction.","authors":"Paolo Cascini, Calum Spicer","doi":"10.1007/s00208-024-03067-5","DOIUrl":"https://doi.org/10.1007/s00208-024-03067-5","url":null,"abstract":"<p><p>We present an adjunction formula for foliations on varieties and we consider applications of the adjunction formula to the cone theorem for rank one foliations and the study of foliation singularities.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"391 4","pages":"5695-5727"},"PeriodicalIF":1.3,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11954749/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143753515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Generic solutions of equations involving the modular <i>j</i> function.","authors":"Sebastian Eterović","doi":"10.1007/s00208-024-03082-6","DOIUrl":"https://doi.org/10.1007/s00208-024-03082-6","url":null,"abstract":"<p><p>Assuming a modular version of Schanuel's conjecture and the modular Zilber-Pink conjecture, we show that the existence of generic solutions of certain families of equations involving the modular <i>j</i> function can be reduced to the problem of finding a Zariski dense set of solutions. By imposing some conditions on the field of definition of the variety, we are also able to obtain versions of this result without relying on these conjectures, and even a result including the derivatives of <i>j</i>.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"391 4","pages":"6401-6449"},"PeriodicalIF":1.3,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11954734/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143753525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hyperfiniteness and Borel asymptotic dimension of boundary actions of hyperbolic groups.","authors":"Petr Naryshkin, Andrea Vaccaro","doi":"10.1007/s00208-024-03075-5","DOIUrl":"https://doi.org/10.1007/s00208-024-03075-5","url":null,"abstract":"<p><p>We give a new short proof of the theorem due to Marquis and Sabok, which states that the orbit equivalence relation induced by the action of a finitely generated hyperbolic group on its Gromov boundary is hyperfinite. Our methods permit moreover to show that every such action has finite Borel asymptotic dimension.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"392 1","pages":"197-208"},"PeriodicalIF":1.3,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11971200/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143795739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Splitting unramified Brauer classes by abelian torsors and the period-index problem.","authors":"Daniel Huybrechts, Dominique Mattei","doi":"10.1007/s00208-025-03161-2","DOIUrl":"10.1007/s00208-025-03161-2","url":null,"abstract":"<p><p>We use twisted relative Picard varieties to split Brauer classes on projective varieties over algebraically closed fields by torsors for a fixed abelian scheme independent of the Brauer class. The construction is also used to prove that the index of an unramified Brauer class divides a fixed power of its period.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"392 2","pages":"2913-2932"},"PeriodicalIF":1.3,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12084267/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144094321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sampling in quasi shift-invariant spaces and Gabor frames generated by ratios of exponential polynomials.","authors":"Alexander Ulanovskii, Ilya Zlotnikov","doi":"10.1007/s00208-024-03011-7","DOIUrl":"https://doi.org/10.1007/s00208-024-03011-7","url":null,"abstract":"<p><p>We introduce two families of generators (functions) <math><mi>G</mi></math> that consist of entire and meromorphic functions enjoying a certain periodicity property and contain the classical Gaussian and hyperbolic secant generators. Sharp results are proved on the density of separated sets that provide non-uniform sampling for the shift-invariant and quasi shift-invariant spaces generated by elements of these families. As an application, new sharp results are obtained on the density of semi-regular lattices for the Gabor frames generated by elements from these families.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"391 3","pages":"3429-3456"},"PeriodicalIF":1.3,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11829847/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143441152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sequential topological complexity of aspherical spaces and sectional categories of subgroup inclusions.","authors":"Arturo Espinosa Baro, Michael Farber, Stephan Mescher, John Oprea","doi":"10.1007/s00208-024-03033-1","DOIUrl":"https://doi.org/10.1007/s00208-024-03033-1","url":null,"abstract":"<p><p>We generalize results from topological robotics on the topological complexity (TC) of aspherical spaces to sectional categories of fibrations inducing subgroup inclusions on the level of fundamental groups. In doing so, we establish new lower bounds on sequential TCs of aspherical spaces as well as the parametrized TC of epimorphisms. Moreover, we generalize the Costa-Farber canonical class for TC to classes for sequential TCs and explore their properties. We combine them with the results on sequential TCs of aspherical spaces to obtain results on spaces that are not necessarily aspherical.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"391 3","pages":"4555-4605"},"PeriodicalIF":1.3,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11829864/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143441159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the best constants of Schur multipliers of second order divided difference functions.","authors":"Martijn Caspers, Jesse Reimann","doi":"10.1007/s00208-025-03111-y","DOIUrl":"https://doi.org/10.1007/s00208-025-03111-y","url":null,"abstract":"&lt;p&gt;&lt;p&gt;We give a new proof of the boundedness of bilinear Schur multipliers of second order divided difference functions, as obtained earlier by Potapov, Skripka and Sukochev in their proof of Koplienko's conjecture on the existence of higher order spectral shift functions. Our proof is based on recent methods involving bilinear transference and the Hörmander-Mikhlin-Schur multiplier theorem. Our approach provides a significant sharpening of the known asymptotic bounds of bilinear Schur multipliers of second order divided difference functions. Furthermore, we give a new lower bound of these bilinear Schur multipliers, giving again a fundamental improvement on the best known bounds obtained by Coine, Le Merdy, Potapov, Sukochev and Tomskova. More precisely, we prove that for &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt; &lt;mo&gt;∈&lt;/mo&gt; &lt;msup&gt;&lt;mi&gt;C&lt;/mi&gt; &lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;R&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;/mrow&gt; &lt;/math&gt; and &lt;math&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt; &lt;mo&gt;&lt;&lt;/mo&gt; &lt;mi&gt;p&lt;/mi&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt; &lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt; &lt;mo&gt;&lt;&lt;/mo&gt; &lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt; &lt;/math&gt; with &lt;math&gt; &lt;mrow&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt; &lt;mi&gt;p&lt;/mi&gt;&lt;/mfrac&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt; &lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt; &lt;/mfrac&gt; &lt;mo&gt;+&lt;/mo&gt; &lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt; &lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt; &lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt; &lt;/mfrac&gt; &lt;/mrow&gt; &lt;/math&gt; we have &lt;dispformula&gt; &lt;math&gt; &lt;mrow&gt; &lt;mtable&gt; &lt;mtr&gt; &lt;mtd&gt; &lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt; &lt;msub&gt;&lt;mi&gt;M&lt;/mi&gt; &lt;msup&gt;&lt;mi&gt;f&lt;/mi&gt; &lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt; &lt;mn&gt;2&lt;/mn&gt; &lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt; &lt;/msup&gt; &lt;/msub&gt; &lt;mo&gt;:&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;S&lt;/mi&gt; &lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt; &lt;/msub&gt; &lt;mo&gt;×&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;S&lt;/mi&gt; &lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt; &lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt; &lt;/msub&gt; &lt;mo&gt;→&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;S&lt;/mi&gt; &lt;mi&gt;p&lt;/mi&gt;&lt;/msub&gt; &lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt; &lt;mo&gt;≲&lt;/mo&gt; &lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt; &lt;msup&gt;&lt;mi&gt;f&lt;/mi&gt; &lt;mrow&gt;&lt;mo&gt;'&lt;/mo&gt; &lt;mo&gt;'&lt;/mo&gt;&lt;/mrow&gt; &lt;/msup&gt; &lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;‖&lt;/mo&gt;&lt;/mrow&gt; &lt;mi&gt;∞&lt;/mi&gt;&lt;/msub&gt; &lt;mi&gt;D&lt;/mi&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;p&lt;/mi&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt; &lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;/mtr&gt; &lt;/mtable&gt; &lt;/mrow&gt; &lt;/math&gt; &lt;/dispformula&gt; where the constant &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt; &lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;p&lt;/mi&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt; &lt;mn&gt;1&lt;/mn&gt;&lt;/msub&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;p&lt;/mi&gt; &lt;mn&gt;2&lt;/mn&gt;&lt;/msub&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;/math&gt; is specified in Theorem 7.1 and &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;p&lt;/mi&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;mn&gt;2&lt;/mn&gt; &lt;mi&gt;p&lt;/mi&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;mn&gt;2&lt;/mn&gt; &lt;mi&gt;p&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;mo&gt;≈&lt;/mo&gt; &lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt; &lt;mn&gt;4&lt;/mn&gt;&lt;/msup&gt; &lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt; &lt;mo&gt;∗&lt;/mo&gt;&lt;/msup&gt; &lt;/mrow&gt; &lt;/math&gt; with &lt;math&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt; &lt;mo&gt;∗&lt;/mo&gt;&lt;/msup&gt; &lt;/math&gt; the Hölder conjugate of &lt;i&gt;p&lt;/i&gt;. We further show that for &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt; &lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;λ&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;mi&gt;λ&lt;/mi&gt; &lt;mo&gt;|&lt;/mo&gt; &lt;mi&gt;λ&lt;/mi&gt; &lt;mo&gt;|&lt;/mo&gt; &lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt; &lt;/math&gt; &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt; &lt;mo&gt;∈&lt;/mo&gt; &lt;mi&gt;R&lt;/mi&gt; &lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt; &lt;/math&gt; for every &lt;math&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt; &lt;mo&gt;&lt;&lt;/mo&gt; &lt;mi&gt;p&lt;/mi&gt; &lt;mo&gt;&lt;&lt;/mo&gt; &lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt; &lt;/math&gt; we have &lt;dispformula&gt; &lt;math&gt; &lt;mrow&gt; &lt;mtable&gt; &lt;mtr&gt; &lt;mtd&gt; &lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;p&lt;/mi&gt; &lt;mn&gt;","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"392 1","pages":"1119-1166"},"PeriodicalIF":1.3,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11971180/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143795741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}