Mathematische Nachrichten最新文献

{"title":"Weighted Bourgain–Morrey-Besov–Triebel–Lizorkin spaces associated with operators","authors":"Tengfei Bai,&nbsp;Jingshi Xu","doi":"10.1002/mana.202400223","DOIUrl":"https://doi.org/10.1002/mana.202400223","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> be a space of homogeneous type and <span></span><math>\u0000 <semantics>\u0000 <mi>L</mi>\u0000 <annotation>$L$</annotation>\u0000 </semantics></math> be a nonnegative self-adjoint operator on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$L^2(X)$</annotation>\u0000 </semantics></math> satisfying a Gaussian upper bound on its heat kernel. First, we obtain the boundedness of the Hardy–Littlewood maximal function and its variant on weighted Bourgain–Morrey spaces. The Hardy-type inequality on sequence Bourgain–Morrey spaces are also given. Then, we introduce the weighted homogeneous Bourgain–Morrey Besov spaces and Triebel–Lizorkin spaces associated with the operator <span></span><math>\u0000 <semantics>\u0000 <mi>L</mi>\u0000 <annotation>$L$</annotation>\u0000 </semantics></math>. We obtain characterizations of these spaces in terms of Peetre maximal functions, noncompactly supported functional calculus, and heat kernel. Atomic decompositions and molecular decompositions of weighted homogeneous Bourgain–Morrey Besov spaces and Triebel–Lizorkin spaces are also proved. Finally, we apply our results to prove the boundedness of the fractional power of <span></span><math>\u0000 <semantics>\u0000 <mi>L</mi>\u0000 <annotation>$L$</annotation>\u0000 </semantics></math> and the spectral multiplier of <span></span><math>\u0000 <semantics>\u0000 <mi>L</mi>\u0000 <annotation>$L$</annotation>\u0000 </semantics></math> on Bourgain–Morrey Besov and Triebel–Lizorkin spaces.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 3","pages":"886-924"},"PeriodicalIF":0.8,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143594853","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":"Simultaneous approximation by neural network operators with applications to Voronovskaja formulas","authors":"Marco Cantarini,&nbsp;Danilo Costarelli","doi":"10.1002/mana.202400281","DOIUrl":"https://doi.org/10.1002/mana.202400281","url":null,"abstract":"<p>In this paper, we considered the problem of the simultaneous approximation of a function and its derivatives by means of the well-known neural network (NN) operators activated by the sigmoidal function. Other than a uniform convergence theorem for the derivatives of NN operators, we also provide a quantitative estimate for the order of approximation based on the modulus of continuity of the approximated derivative. Furthermore, a qualitative and quantitative Voronovskaja-type formula is established, which provides information about the high order of approximation that can be achieved by NN operators. To prove the above theorems, several auxiliary results involving sigmoidal functions have been established. At the end of the paper, noteworthy examples have been discussed in detail.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 3","pages":"871-885"},"PeriodicalIF":0.8,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202400281","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143595082","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":"L\u0000 p\u0000 \u0000 $L^p$\u0000 -boundedness properties for some harmonic analysis operators defined by resolvents for a Laplacian with drift in Euclidean spaces","authors":"Jorge J. Betancor,&nbsp;Juan C. Fariña,&nbsp;Lourdes Rodríguez-Mesa","doi":"10.1002/mana.202400212","DOIUrl":"https://doi.org/10.1002/mana.202400212","url":null,"abstract":"&lt;p&gt;We consider the Laplacian with drift in &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; defined by &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;Δ&lt;/mi&gt;\u0000 &lt;mi&gt;ν&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mo&gt;∑&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ν&lt;/mi&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mi&gt;i&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Delta _nu = sum _{i=1}^n(frac{partial ^2}{partial x_i^2} + 2 nu _ifrac{partial }{partial {x_i}})$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ν&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ν&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mtext&gt;…&lt;/mtext&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ν&lt;/mi&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;)&lt;/mo&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;mo&gt;{&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 ","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 3","pages":"849-870"},"PeriodicalIF":0.8,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143594855","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":"Approximation of skew Brownian motion by snapping-out Brownian motions","authors":"Adam Bobrowski,&nbsp;Elżbieta Ratajczyk","doi":"10.1002/mana.202400179","DOIUrl":"https://doi.org/10.1002/mana.202400179","url":null,"abstract":"<p>We elaborate on the theorem saying that as permeability coefficients of snapping-out Brownian motions tend to infinity in such a way that their ratio remains constant, these processes converge to a skew Brownian motion. In particular, convergence of the related semigroups, cosine families, and projections is discussed.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 3","pages":"829-848"},"PeriodicalIF":0.8,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143594854","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":"Normal forms and Tyurin degenerations of K3 surfaces polarized by a rank 18 lattice","authors":"Charles F. Doran,&nbsp;Joseph Prebble,&nbsp;Alan Thompson","doi":"10.1002/mana.202400021","DOIUrl":"https://doi.org/10.1002/mana.202400021","url":null,"abstract":"<p>We study projective Type II degenerations of K3 surfaces polarized by a certain rank 18 lattice, where the central fiber consists of a pair of rational surfaces glued along a smooth elliptic curve. Given such a degeneration, one may construct other degenerations of the same kind by flopping curves on the central fiber, but the degenerations obtained from this process are not usually projective. We construct a series of examples which are all projective and which are all related by flopping single curves from one component of the central fiber to the other. Moreover, we show that this list is complete, in the sense that no other flops are possible. The components of the central fibers obtained include weak del Pezzo surfaces of all possible degrees. This shows that projectivity need not impose any meaningful constraints on the surfaces that can arise as components in Type II degenerations.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 3","pages":"806-828"},"PeriodicalIF":0.8,"publicationDate":"2025-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.202400021","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143594880","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":"Liouville property and quasi-isometries on non-positively curved Riemannian surfaces","authors":"Ana Granados,&nbsp;Ana Portilla,&nbsp;José M. Rodríguez-García,&nbsp;Eva Tourís","doi":"10.1002/mana.202400121","DOIUrl":"https://doi.org/10.1002/mana.202400121","url":null,"abstract":"<p>Kanai proved the stability under quasi-isometries of several global properties for Riemannian manifolds with the restriction of having positive injectivity radius. This work shows the stability of the Liouville property for Riemannian surfaces with non-positive curvature, where the restriction on the injectivity radius has been removed.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 3","pages":"794-805"},"PeriodicalIF":0.8,"publicationDate":"2025-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143594879","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":"Rank stability of elliptic curves in certain non-abelian extensions","authors":"Siddhi Pathak,&nbsp;Anwesh Ray","doi":"10.1002/mana.202400357","DOIUrl":"https://doi.org/10.1002/mana.202400357","url":null,"abstract":"&lt;p&gt;Let &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mi&gt;Q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$E_{/mathbb {Q}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be an elliptic curve with rank &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;Q&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$E(mathbb {Q})=0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Fix an odd prime &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, a positive integer &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;annotation&gt;$n$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, and a finite abelian extension &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mi&gt;Q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$K/mathbb {Q}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with rank &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;E&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$E(K) = 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In this paper, we show that there exist infinitely many extensions &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$L/K$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; such that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mi&gt;Q&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$L/mathbb {Q}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is Galois with &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;Gal&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mi&gt;Q&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;≃&lt;/mo&gt;\u0000 &lt;mo&gt;Gal&lt;/mo&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;K&lt;/mi&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mi&gt;Q&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;⋉&lt;/mo&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 ","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 2","pages":"730-753"},"PeriodicalIF":0.8,"publicationDate":"2025-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143397048","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":"On space-like class \u0000 \u0000 A\u0000 $mathcal {A}$\u0000 surfaces in Robertson–Walker spacetimes","authors":"Burcu Bektaş Demirci,&nbsp;Nurettin Cenk Turgay,&nbsp;Rüya Yeğin Şen","doi":"10.1002/mana.202400374","DOIUrl":"https://doi.org/10.1002/mana.202400374","url":null,"abstract":"&lt;p&gt;In this paper, we consider space-like surfaces in Robertson–Walker spacetimes &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;L&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;f&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;c&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$L^4_1(f,c)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; with the comoving observer field &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;annotation&gt;$frac{partial }{partial t}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. We study some problems related to such surfaces satisfying the geometric conditions imposed on the tangential and normal parts of the unit vector field &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;annotation&gt;$frac{partial }{partial t}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, as naturally defined. First, we investigate space-like surfaces in &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;L&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mn&gt;4&lt;/mn&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;f&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;c&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$L^4_1(f,c)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; satisfying that the tangent component of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;annotation&gt;$frac{partial }{partial t}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is an eigenvector of all shape operators, called class &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; surfaces. Then, we get a classification theorem for space-like class &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {A}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; surfaces in &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msubsup&gt;\u0000 ","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 2","pages":"718-729"},"PeriodicalIF":0.8,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143396841","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":"A version of Hilbert's 16th problem for 3D polynomial vector fields: Counting isolated invariant tori","authors":"D. D. Novaes,&nbsp;P. C. C. R. Pereira","doi":"10.1002/mana.202300568","DOIUrl":"https://doi.org/10.1002/mana.202300568","url":null,"abstract":"&lt;p&gt;Hilbert's 16th problem, about the maximum number of limit cycles of planar polynomial vector fields of a given degree &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;annotation&gt;$m$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, has been one of the most important driving forces for new developments in the qualitative theory of vector fields. Increasing the dimension, one cannot expect the existence of a finite upper bound for the number of limit cycles of, for instance, 3D polynomial vector fields of a given degree &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;annotation&gt;$m$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Here, as an extension of such a problem in the 3D space, we investigate the number of isolated invariant tori in 3D polynomial vector fields. In this context, given a natural number &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;annotation&gt;$m$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, we denote by &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$N(m)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; the upper bound for the number of isolated invariant tori of 3D polynomial vector fields of degree &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;annotation&gt;$m$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. Based on a recently developed averaging method for detecting invariant tori, our first main result provides a mechanism for constructing 3D differential vector fields with a number &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;annotation&gt;$H$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of normally hyperbolic invariant tori from a given planar differential vector field with &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;H&lt;/mi&gt;\u0000 &lt;annotation&gt;$H$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; hyperbolic limit cycles. The strength of our mechanism in studying the number &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$N(m)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; lies in the fact that the constructed 3D differential vector field is polynomial provided that the given planar differential vector field is polynomial. Accordingly, our second main result establishes a lower bound for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;N&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;m&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$N(m)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 2","pages":"709-717"},"PeriodicalIF":0.8,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143397232","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":"Two-parameter B-valued martingale Hardy–Lorentz–Karamata spaces: Inequalities and dualities","authors":"Zhiwei Hao,&nbsp;Lin Wang,&nbsp;Ferenc Weisz","doi":"10.1002/mana.202400204","DOIUrl":"https://doi.org/10.1002/mana.202400204","url":null,"abstract":"<p>In this paper, we introduce five two-parameter <span></span><math>\u0000 <semantics>\u0000 <mi>B</mi>\u0000 <annotation>$mathbf {B}$</annotation>\u0000 </semantics></math>-valued martingale Hardy–Lorentz–Karamata spaces and establish some atomic decomposition theorems via atomic martingales as well as atomic functions. With the aid of atomic decompositions, we obtain some martingale inequalities and characterize the duals of these spaces. Our conclusions strongly depend on the geometric properties of the underlying Banach spaces and remove the restrictive condition that the slowly varying function <span></span><math>\u0000 <semantics>\u0000 <mi>b</mi>\u0000 <annotation>$b$</annotation>\u0000 </semantics></math> is nondecreasing as in the previous literature.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 2","pages":"677-708"},"PeriodicalIF":0.8,"publicationDate":"2024-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143397209","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}