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The Scattered Range Problem of Elementary Operators on ({cal B}({cal H})) 初等算子在({cal B}({cal H}))上的散射范围问题
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-11-15 DOI: 10.1007/s10114-024-3346-0
Peng Cao, Cun Wang
{"title":"The Scattered Range Problem of Elementary Operators on ({cal B}({cal H}))","authors":"Peng Cao,&nbsp;Cun Wang","doi":"10.1007/s10114-024-3346-0","DOIUrl":"10.1007/s10114-024-3346-0","url":null,"abstract":"<div><p>A scattered operator is a bounded linear operator with at most countable spectrum. In this paper, we prove that for any elementary operator on <span>({cal B}({cal H}))</span>, not only for finite length but also for infinite length, if the range of the elementary operator is contained in scattered operators, then the corresponding sum of multipliers is a compact operator. We also prove that for some special classes of elementary operators, such as the elementary operators of length 2, higher order inner derivations and generalized inner derivation, if the range of the elementary operator is contained in the set of scattered operators, then the range is contained in the set of power compact operators. At the same time, the multipliers of the corresponding elementary operators are characterized.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 11","pages":"2684 - 2692"},"PeriodicalIF":0.8,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142694748","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}
引用次数: 0
A Weak ∞-Functor in Morse Theory 莫尔斯理论中的弱∞矢量
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-11-15 DOI: 10.1007/s10114-024-2523-5
Shan Zhong Sun, Chen Xi Wang
{"title":"A Weak ∞-Functor in Morse Theory","authors":"Shan Zhong Sun,&nbsp;Chen Xi Wang","doi":"10.1007/s10114-024-2523-5","DOIUrl":"10.1007/s10114-024-2523-5","url":null,"abstract":"<div><p>In the spirit of Morse homology initiated by Witten and Floer, we construct two ∞-categories <span>({cal A})</span> and <span>({cal B})</span>. The weak one <span>({cal A})</span> comes out of the Morse–Smale pairs and their higher homotopies, and the strict one <span>({cal B})</span> concerns the chain complexes of the Morse functions. Based on the boundary structures of the compactified moduli space of gradient flow lines of Morse functions with parameters, we build up a weak ∞-functor <span>({cal F}:{cal A} rightarrow {cal B})</span>. Higher algebraic structures behind Morse homology are revealed with the perspective of defects in topological quantum field theory.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 11","pages":"2571 - 2614"},"PeriodicalIF":0.8,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142694749","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}
引用次数: 0
Characterizations of Weighted Besov Spaces with Variable Exponents 具有可变指数的加权贝索夫空间的特征
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-11-15 DOI: 10.1007/s10114-024-2623-2
Sheng Rong Wang, Peng Fei Guo, Jing Shi Xu
{"title":"Characterizations of Weighted Besov Spaces with Variable Exponents","authors":"Sheng Rong Wang,&nbsp;Peng Fei Guo,&nbsp;Jing Shi Xu","doi":"10.1007/s10114-024-2623-2","DOIUrl":"10.1007/s10114-024-2623-2","url":null,"abstract":"<div><p>In this paper, we first give characterizations of weighted Besov spaces with variable exponents via Peetre’s maximal functions. Then we obtain decomposition characterizations of these spaces by atom, molecule and wavelet. As an application, we obtain the boundedness of the pseudo-differential operators on these spaces.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 11","pages":"2855 - 2878"},"PeriodicalIF":0.8,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142694747","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}
引用次数: 0
Compactness of Extremals for Trudinger-Moser Functionals on the Unit Ball in ℝ2 ℝ2中单位球上特鲁丁格-莫泽函数极值的紧凑性
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-11-15 DOI: 10.1007/s10114-024-3046-9
Wei Wei Shan, Xiao Meng Li
{"title":"Compactness of Extremals for Trudinger-Moser Functionals on the Unit Ball in ℝ2","authors":"Wei Wei Shan,&nbsp;Xiao Meng Li","doi":"10.1007/s10114-024-3046-9","DOIUrl":"10.1007/s10114-024-3046-9","url":null,"abstract":"<div><p>Let <span>(mathbb{B})</span> be a unit ball in ℝ<sup>2</sup>, <span>(W_{0}^{1,2}(mathbb{B}))</span> be the standard Sobolev space. For any <i>ϵ</i> &gt; 0, de Figueiredo, do Ó, dos Santons, Yang and Zhu proved the existence of extremals of a Trudinger-Moser inequality in the unit ball. Precisely, </p><div><div><span>$$mathop {sup }limits_{u in W_0^{1,2}left( mathbb{B} right),int_mathbb{B} {|nabla u{|^2}dx le 1} } int_mathbb{B} {{{left| x right|}^{2epsilon }}{rm{e}^{4pi left( {1 + epsilon } right){u^2}}}} dx$$</span></div></div><p> can be attained by some radially symmetric function <span>(u_{epsilon}in W_{0}^{1,2}(mathbb{B}))</span> with <span>(int_{mathbb{B}}vertnabla u_{epsilon}vert^{2}dx=1)</span>. In this note, we concern the compactness of the function family {<i>u</i><sub><i>ϵ</i></sub>}<sub><i>ϵ</i>&gt;0</sub> and prove that up to a subsequence <i>u</i><sub><i>ϵ</i></sub> converges to some function <i>u</i>\u0000<sub>0</sub> in <span>(C^{1}(overline{mathbb{B}}))</span> as <i>ϵ</i> → 0. Furthermore, <i>u</i><sub>0</sub> is an extremal function of the supremum </p><div><div><span>$$mathop {sup }limits_{u in W_0^{1,2}left( mathbb{B} right),int_mathbb{B} {|nabla u{|^2}dx le 1} } int_{mathbb{B}}rm{e}^{4pi u^{2}}dx.$$</span></div></div><p> Let us explain the result in geometry. Denote <span>(omega_{0}=dx_{1}^{2}+dx_{2}^{2})</span> be the standard Euclidean metric. Define a conical metric <span>(omega_{epsilon}=vert xvert^{2epsilon}omega_{0})</span> for <span>(xinmathbb{B})</span>. Then the extremal family {<i>u</i>\u0000<sub><i>ϵ</i></sub>}<sub><i>ϵ</i>&gt;0</sub> of the following Trudinger-Moser functionals </p><div><div><span>$$int_{mathbb{B}}rm{e}^{4pi(1+epsilon)u^{2}}dv_{w_{epsilon}}$$</span></div></div><p> under the constraint <span>(uin W_{0}^{1,2}(mathbb{B}))</span> and <span>(int_{mathbb{B}}vertnabla_{omega_{epsilon}u}vert^{2}dv_{omega_{epsilon}}leq 1)</span> is compact as <i>ϵ</i> → 0.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 11","pages":"2840 - 2854"},"PeriodicalIF":0.8,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142694742","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}
引用次数: 0
On pj-rank of Even K-groups of Rings of Integers 论整数环偶数 K 群的 pj 级
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-10-15 DOI: 10.1007/s10114-024-1312-5
Meng Fai Lim
{"title":"On pj-rank of Even K-groups of Rings of Integers","authors":"Meng Fai Lim","doi":"10.1007/s10114-024-1312-5","DOIUrl":"10.1007/s10114-024-1312-5","url":null,"abstract":"<div><p>Let <i>L/F</i> be a finite Galois extension of number fields of degree <i>n</i> and let <i>p</i> be a prime which does not divide <i>n</i>. We shall study the <i>p</i><sup><i>j</i></sup>-rank of <span>(K_{2i}(mathcal{O}_{L}))</span> via its Galois module structure following the approaches of Iwasawa and Komatsu–Nakano. Along the way, we generalize previous observations of Browkin, Wu and Zhou on <i>K</i><sub>2</sub>-groups to higher even <i>K</i>-groups. We also give examples to illustrate our results. Finally, we apply our discussion to refine a result of Kitajima pertaining to the <i>p</i>-rank of even <i>K</i>-groups in the cyclotomic ℤ<sub><i>l</i></sub>-extension, where <i>l</i> ≠ <i>p</i>.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 10","pages":"2481 - 2496"},"PeriodicalIF":0.8,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142595981","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}
引用次数: 0
Completeness of Induced Cotorsion Pairs in Representation Categories of Rooted Quivers 有根四元组表示范畴中诱导共旋对的完备性
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-07-18 DOI: 10.1007/s10114-024-3041-1
Zhen Xing Di, Li Ping Li, Li Liang, Fei Xu
{"title":"Completeness of Induced Cotorsion Pairs in Representation Categories of Rooted Quivers","authors":"Zhen Xing Di,&nbsp;Li Ping Li,&nbsp;Li Liang,&nbsp;Fei Xu","doi":"10.1007/s10114-024-3041-1","DOIUrl":"10.1007/s10114-024-3041-1","url":null,"abstract":"<div><p>This paper focuses on a question raised by Holm and Jørgensen, who asked if the induced cotorsion pairs (Φ(X), Φ(X)<sup>⊥</sup>) and (<sup>⊥</sup>Ψ(Y), Ψ(Y)) in Rep(<i>Q</i>, A)—the category of all A-valued representations of a quiver <i>Q</i>—are complete whenever (X, Y) is a complete cotorsion pair in an abelian category A satisfying some mild conditions. We give an affirmative answer if the quiver <i>Q</i> is rooted. As an application, we show under certain mild conditions that if a subcategory L, which is not necessarily closed under direct summands, of A is special precovering (resp., preenveloping), then Φ(L)(resp., Ψ(L)) is special precovering (resp., preenveloping) in Rep(<i>Q</i>, A).</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 10","pages":"2436 - 2452"},"PeriodicalIF":0.8,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141824347","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}
引用次数: 0
On Some Sums Involving Small Arithmetic Functions 论涉及小算术函数的一些和
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-07-18 DOI: 10.1007/s10114-024-2129-y
Wen Guang Zhai
{"title":"On Some Sums Involving Small Arithmetic Functions","authors":"Wen Guang Zhai","doi":"10.1007/s10114-024-2129-y","DOIUrl":"10.1007/s10114-024-2129-y","url":null,"abstract":"<div><p>Let <i>f</i> be any arithmetic function and define <span>({S_{f}}(x):=sumnolimits_{{n le x}}f([x/n]))</span>. If the function <i>f</i> is small, namely, <i>f</i>(<i>n</i>) ≪ <i>n</i><sup><i>ε</i></sup>, then the error term <i>E</i><sub><i>f</i></sub>(<i>x</i>) in the asymptotic formula of <i>S</i><sub><i>f</i></sub>(<i>x</i>) has the form <i>O</i>(<i>x</i><sup>1/2+<i>ε</i></sup>). In this paper, we shall study the mean square of <i>E</i><sub><i>f</i></sub>(<i>x</i>) and establish some new results of <i>E</i><sub><i>f</i></sub>(<i>x</i>) for some special functions.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 10","pages":"2497 - 2518"},"PeriodicalIF":0.8,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141825074","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}
引用次数: 0
Supnorm Estimates for ({bar partial}) on Product Domains in ℂn ℂn中积域上$${bar partial}$ 的超矩形估计值
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-07-18 DOI: 10.1007/s10114-024-2463-0
Martino Fassina, Yi Fei Pan
{"title":"Supnorm Estimates for ({bar partial}) on Product Domains in ℂn","authors":"Martino Fassina,&nbsp;Yi Fei Pan","doi":"10.1007/s10114-024-2463-0","DOIUrl":"10.1007/s10114-024-2463-0","url":null,"abstract":"<div><p>Using methods from complex analysis in one variable, we define an integral operator that solves <span>({bar partial})</span> with supnorm estimates on product domains in ℂ<sup><i>n</i></sup>.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 10","pages":"2307 - 2323"},"PeriodicalIF":0.8,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141825922","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}
引用次数: 0
On Variational Principles of Metric Mean Dimension on Subsets in Feldman–Katok Metric 论费尔德曼-卡托克公设子集上公设平均维度的变分原理
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-07-10 DOI: 10.1007/s10114-024-2517-3
Kun Mei Gao, Rui Feng Zhang
{"title":"On Variational Principles of Metric Mean Dimension on Subsets in Feldman–Katok Metric","authors":"Kun Mei Gao,&nbsp;Rui Feng Zhang","doi":"10.1007/s10114-024-2517-3","DOIUrl":"10.1007/s10114-024-2517-3","url":null,"abstract":"<div><p>In this paper, we studied the metric mean dimension in Feldman–Katok (FK for short) metric. We introduced the notions of FK-Bowen metric mean dimension and FK-Packing metric mean dimension on subsets. And we established two variational principles.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 10","pages":"2519 - 2536"},"PeriodicalIF":0.8,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572163","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}
引用次数: 0
Essentially Commuting Truncated Toeplitz Operators 本质归一的截断托普利兹算子
IF 0.8 3区 数学
Acta Mathematica Sinica-English Series Pub Date : 2024-07-10 DOI: 10.1007/s10114-024-2696-y
Xi Zhao, Tao Yu
{"title":"Essentially Commuting Truncated Toeplitz Operators","authors":"Xi Zhao,&nbsp;Tao Yu","doi":"10.1007/s10114-024-2696-y","DOIUrl":"10.1007/s10114-024-2696-y","url":null,"abstract":"<div><p>A model space is a subspace of the Hardy space which is invariant under the backward shift, and a truncated Toeplitz operator is the compression of a Toeplitz operator on some model space. In this paper we prove a necessary and sufficient condition for the commutator of two truncated Toeplitz operators on a model space to be compact.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 10","pages":"2453 - 2480"},"PeriodicalIF":0.8,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572167","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}
引用次数: 0
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