Mathematische Nachrichten最新文献

{"title":"Topology of a complex double-plane branching along a real line arrangement","authors":"Ichiro Shimada","doi":"10.1002/mana.12023","DOIUrl":"https://doi.org/10.1002/mana.12023","url":null,"abstract":"<p>We investigate the topology of the double cover of the complex affine plane branching along a nodal real line arrangement. We define certain topological 2-cycles in the double plane using the real structure of the arrangement, and calculate their intersection numbers.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 7","pages":"2204-2241"},"PeriodicalIF":0.8,"publicationDate":"2025-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144657567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Correlations of fractions whose denominators are products of primes","authors":"Meijie Lu","doi":"10.1002/mana.12027","DOIUrl":"https://doi.org/10.1002/mana.12027","url":null,"abstract":"<p>In this paper, we focus on the pair correlation of fractions whose denominators are products of primes. We show that the limiting pair correlation function of such fractions on any short interval <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>I</mi>\u0000 <mo>⊂</mo>\u0000 <mo>[</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <annotation>${bf I}subset [0,1]$</annotation>\u0000 </semantics></math> exists and is independent of <span></span><math>\u0000 <semantics>\u0000 <mi>I</mi>\u0000 <annotation>${bf I}$</annotation>\u0000 </semantics></math>. Furthermore, we use this result to compute the pair correlation function of the angles of elements in specific types of regions.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 7","pages":"2263-2281"},"PeriodicalIF":0.8,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144657595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"General infinitesimal variations of the Hodge structure of ample curves in surfaces","authors":"Víctor González-Alonso,&nbsp;Sara Torelli","doi":"10.1002/mana.12028","DOIUrl":"https://doi.org/10.1002/mana.12028","url":null,"abstract":"<p>Given a smooth projective complex curve inside a smooth projective surface, one can ask how its Hodge structure varies when the curve moves inside the surface. In this paper, we develop a general theory to study the infinitesimal version of this question in the case of ample curves. We can then apply the machinery to show that the infinitesimal variation of the Hodge structure of a general deformation of an ample curve in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <mo>×</mo>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$mathbb {P}^1times mathbb {P}^1$</annotation>\u0000 </semantics></math> is an isomorphism.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 7","pages":"2282-2308"},"PeriodicalIF":0.8,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.12028","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144657596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Substochastic operators in symmetric spaces","authors":"Maciej Ciesielski,&nbsp;Grzegorz Lewicki","doi":"10.1002/mana.12029","DOIUrl":"https://doi.org/10.1002/mana.12029","url":null,"abstract":"<p>First, we solve a crucial problem under which conditions increasing uniform <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>-monotonicity is equivalent to lower local uniform <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>-monotonicity. Next, we investigate properties of substochastic operators on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>∞</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$L^1+L^infty$</annotation>\u0000 </semantics></math> with applications. Namely, we show that a countable infinite combination of substochastic operators is also substochastic. Using <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>-monotonicity properties, we prove several theorems devoted to the convergence of the sequence of substochastic operators in the norm of a symmetric space <span></span><math>\u0000 <semantics>\u0000 <mi>E</mi>\u0000 <annotation>$E$</annotation>\u0000 </semantics></math> under addition assumption on <span></span><math>\u0000 <semantics>\u0000 <mi>E</mi>\u0000 <annotation>$E$</annotation>\u0000 </semantics></math>. In our final discussion, we focus on compactness of admissible operators for Banach couples under additional assumption.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 7","pages":"2309-2326"},"PeriodicalIF":0.8,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144657597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Differential norms and Rieffel algebras","authors":"Rodrigo A. H. M. Cabral,&nbsp;Michael Forger,&nbsp;Severino T. Melo","doi":"10.1002/mana.12019","DOIUrl":"https://doi.org/10.1002/mana.12019","url":null,"abstract":"&lt;p&gt;We develop criteria to guarantee uniqueness of the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;${rm C}^*$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-norm on a &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;annotation&gt;$*$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-algebra &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {B}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Nontrivial examples are provided by the noncommutative algebras of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {C}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-valued functions &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;mi&gt;J&lt;/mi&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {S}_J^mathcal {C}(mathbb {R}^n)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 &lt;mi&gt;J&lt;/mi&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathcal {B}_J^mathcal {C}(mathbb {R}^n)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; defined by M.A. Rieffel via a deformation quantization procedure, where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {C}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;${rm C}^*$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-algebra and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;J&lt;/mi&gt;\u0000 &lt;annotation&gt;$J$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a skew-symmetric linear transformation on &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathbb {R}^n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with respect to which the usual pointwise product is de","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 7","pages":"2177-2203"},"PeriodicalIF":0.8,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144657593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reiteration theorems for the interpolation of quasi-subadditive functions on Banach spaces","authors":"Ralph Chill,&nbsp;Praveen Sharma,&nbsp;Sachi Srivastava","doi":"10.1002/mana.12025","DOIUrl":"https://doi.org/10.1002/mana.12025","url":null,"abstract":"<p>We study reiteration theorems for quasi-subadditive functions on Banach spaces. We prove various reiteration theorems that are generalizations of classical reiteration theorems.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 7","pages":"2242-2262"},"PeriodicalIF":0.8,"publicationDate":"2025-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144657594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Convergence results and optimal control problems via gap functions for n-player generalized multiobjective games with applications","authors":"Nguyen Van Hung,&nbsp;Andre A. Keller","doi":"10.1002/mana.12026","DOIUrl":"https://doi.org/10.1002/mana.12026","url":null,"abstract":"<p>The aim of this paper is to study some new results on the convergence of solutions for controlled systems driven by generalized multiobjective games, optimal control problems where the systems are governed by generalized multiobjective games and controlled systems of traffic networks. First, we recall the controlled systems of generalized multiobjective games proposed by Hung and Keller (Math. Nachr. <b>296</b> (2023), 3676–3698). Second, we introduce gap functions and a key Assumption 3.6 using nonlinear scalarization functions for these games. Results on the lower convergence and convergence of the solutions for such problems using the key Assumption 3.6 are established. Third, we revisit optimal control problems governed by generalized multiobjective games. We investigate necessary and sufficient conditions for the convergence of solutions to optimal control problems. Finally, as a real-world application, we consider the special case of controlled systems of traffic networks. The necessary and sufficient conditions for the convergence of solutions for these problems are also obtained. Many examples are given for the illustration of our results.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 6","pages":"1989-2013"},"PeriodicalIF":0.8,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144281418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The 2-divisibility of divisors on K3 surfaces in characteristic 2","authors":"Toshiyuki Katsura,&nbsp;Shigeyuki Kondō,&nbsp;Matthias Schütt","doi":"10.1002/mana.12024","DOIUrl":"https://doi.org/10.1002/mana.12024","url":null,"abstract":"<p>We show that K3 surfaces in characteristic 2 can admit sets of <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math> disjoint smooth rational curves whose sum is divisible by 2 in the Picard group, for each <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>=</mo>\u0000 <mn>8</mn>\u0000 <mo>,</mo>\u0000 <mn>12</mn>\u0000 <mo>,</mo>\u0000 <mn>16</mn>\u0000 <mo>,</mo>\u0000 <mn>20</mn>\u0000 </mrow>\u0000 <annotation>$n=8,12,16,20$</annotation>\u0000 </semantics></math>. More precisely, all values occur on supersingular K3 surfaces, with exceptions only at Artin invariants 1 and 10, while on K3 surfaces of finite height, only <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>=</mo>\u0000 <mn>8</mn>\u0000 </mrow>\u0000 <annotation>$n=8$</annotation>\u0000 </semantics></math> is possible.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 6","pages":"1964-1988"},"PeriodicalIF":0.8,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.12024","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144281417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On birational automorphisms of double EPW-cubes","authors":"Simone Billi,&nbsp;Stevell Muller,&nbsp;Tomasz Wawak","doi":"10.1002/mana.12022","DOIUrl":"https://doi.org/10.1002/mana.12022","url":null,"abstract":"<p>We give a classification of finite groups of symplectic birational automorphisms on manifolds of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>K</mi>\u0000 <msup>\u0000 <mn>3</mn>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mn>3</mn>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$textnormal {K3}^{[3]}$</annotation>\u0000 </semantics></math>-type with stable cohomological action. We describe the group of polarized automorphisms of a smooth double EPW-cube. Using this description, we exhibit examples of projective hyperkähler manifolds of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>K</mi>\u0000 <msup>\u0000 <mn>3</mn>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mn>3</mn>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$textnormal {K3}^{[3]}$</annotation>\u0000 </semantics></math>–type of maximal Picard rank with a symplectic action of a large group.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 6","pages":"1943-1963"},"PeriodicalIF":0.8,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144281591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Calderón reproducing formulae on product spaces of homogeneous type and their applications","authors":"Ziyi He,&nbsp;Xianjie Yan,&nbsp;Dachun Yang","doi":"10.1002/mana.12014","DOIUrl":"https://doi.org/10.1002/mana.12014","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;μ&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(X_1,d_1,mu _1)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;μ&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$(X_2,d_2,mu _2)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be two spaces of homogeneous type in the sense of R. R. Coifman and G. Weiss. In this article, the authors first introduce spaces of product test functions and product approximations of the identity with exponential decay on the product space &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$X_1times X_2$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Using these, the authors establish product continuous/discrete Calderón reproducing formulae. As applications, the Littlewood–Paley characterizations, respectively, in terms of the Lusin area function, the Littlewood–Paley &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 &lt;annotation&gt;$g$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-function, and the Littlewood–Paley &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 &lt;mi&gt;λ&lt;/mi&gt;\u0000 &lt;mo&gt;∗&lt;/mo&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;annotation&gt;$g^*_{lambda }$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-function, of the Lebesgue space &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;X&lt;/mi&gt;\u0000 ","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 6","pages":"1839-1921"},"PeriodicalIF":0.8,"publicationDate":"2025-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144281437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}