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{"title":"Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic","authors":"Bin Shu, Lisun Zheng, Ye Ren","doi":"10.1515/forum-2023-0326","DOIUrl":"https://doi.org/10.1515/forum-2023-0326","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>𝔤</m:mi> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>𝔤</m:mi> <m:mover accent=\"true\"> <m:mn>0</m:mn> <m:mo stretchy=\"false\">¯</m:mo> </m:mover> </m:msub> <m:mo>⊕</m:mo> <m:msub> <m:mi>𝔤</m:mi> <m:mover accent=\"true\"> <m:mn>1</m:mn> <m:mo stretchy=\"false\">¯</m:mo> </m:mover> </m:msub> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0326_eq_0687.png\"/> <jats:tex-math>{{mathfrak{g}}={mathfrak{g}}_{bar{0}}oplus{mathfrak{g}}_{bar{1}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a basic classical Lie superalgebra over an algebraically closed field <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝐤</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0326_eq_0676.png\"/> <jats:tex-math>{{mathbf{k}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> of characteristic <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0326_eq_0586.png\"/> <jats:tex-math>{p>2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Denote by <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">𝒵</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0326_eq_0376.png\"/> <jats:tex-math>{mathcal{Z}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> the center of the universal enveloping algebra <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>U</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>𝔤</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0326_eq_0135.png\"/> <jats:tex-math>{U({mathfrak{g}})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Then <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">𝒵</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0326_eq_0376.png\"/> <jats:tex-math>{mathcal{Z}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> turns out to be finitely-generated purely-even commutative algebra without nonzero divisors. In this paper, we demonstrate that the fraction <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Frac</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo st","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"8 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503337","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":"Representations of non-finitely graded Lie algebras related to Virasoro algebra","authors":"Chunguang Xia, Tianyu Ma, Xiao Dong, Mingjing Zhang","doi":"10.1515/forum-2023-0320","DOIUrl":"https://doi.org/10.1515/forum-2023-0320","url":null,"abstract":"In this paper, we study representations of non-finitely graded Lie algebras <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"script\">𝒲</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ϵ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0320_eq_0340.png\"/> <jats:tex-math>{mathcal{W}(epsilon)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> related to Virasoro algebra, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ϵ</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo>±</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0320_eq_0321.png\"/> <jats:tex-math>{epsilon=pm 1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Precisely speaking, we completely classify the free <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"script\">𝒰</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>𝔥</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0320_eq_0333.png\"/> <jats:tex-math>{mathcal{U}(mathfrak{h})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-modules of rank one over <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"script\">𝒲</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ϵ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0320_eq_0340.png\"/> <jats:tex-math>{mathcal{W}(epsilon)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and find that these module structures are rather different from those of other graded Lie algebras. We also determine the simplicity and isomorphism classes of these modules.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"45 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528930","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":"Geometric approach to the Moore–Penrose inverse and the polar decomposition of perturbations by operator ideals","authors":"Eduardo Chiumiento, Pedro Massey","doi":"10.1515/forum-2024-0010","DOIUrl":"https://doi.org/10.1515/forum-2024-0010","url":null,"abstract":"We study the Moore–Penrose inverse of perturbations by a proper symmetrically-normed ideal of a closed range operator on a Hilbert space. We show that the notion of essential codimension of projections gives a characterization of subsets of such perturbations in which the Moore–Penrose inverse is continuous with respect to the metric induced by the operator ideal. These subsets are maximal satisfying the continuity property, and they carry the structure of real analytic Banach manifolds, which are acted upon transitively by the Banach–Lie group consisting of invertible operators associated with the ideal. This geometric construction allows us to prove that the Moore–Penrose inverse is indeed a real bianalytic map between infinite-dimensional manifolds. We use these results to study the polar decomposition of closed range operators from a similar geometric perspective. At this point we prove that operator monotone functions are real analytic in the norm of any proper symmetrically-normed ideal. Finally, we show that the maps defined by the operator modulus and the polar factor in the polar decomposition of closed range operators are real analytic fiber bundles.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"26 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141059890","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":"Torus bundles over lens spaces","authors":"Oliver H. Wang","doi":"10.1515/forum-2022-0279","DOIUrl":"https://doi.org/10.1515/forum-2022-0279","url":null,"abstract":"Let &lt;jats:italic&gt;p&lt;/jats:italic&gt; be an odd prime and let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;ρ&lt;/m:mi&gt; &lt;m:mo&gt;:&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;ℤ&lt;/m:mi&gt; &lt;m:mo&gt;/&lt;/m:mo&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;→&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:msub&gt; &lt;m:mi&gt;GL&lt;/m:mi&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;m:mo&gt;⁡&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;ℤ&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0655.png\"/&gt; &lt;jats:tex-math&gt;{rho:mathbb{Z}/prightarrowoperatorname{{GL}}_{n}(mathbb{Z})}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be an action of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;ℤ&lt;/m:mi&gt; &lt;m:mo&gt;/&lt;/m:mo&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0555.png\"/&gt; &lt;jats:tex-math&gt;{mathbb{Z}/p}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; on a lattice and let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"normal\"&gt;Γ&lt;/m:mi&gt; &lt;m:mo&gt;:=&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mi&gt;ℤ&lt;/m:mi&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;m:msub&gt; &lt;m:mo&gt;⋊&lt;/m:mo&gt; &lt;m:mi&gt;ρ&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;m:mi&gt;ℤ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;/&lt;/m:mo&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0490.png\"/&gt; &lt;jats:tex-math&gt;{Gamma:=mathbb{Z}^{n}rtimes_{rho}mathbb{Z}/p}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be the corresponding semidirect product. The torus bundle &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;M&lt;/m:mi&gt; &lt;m:mo&gt;:=&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:msubsup&gt; &lt;m:mi&gt;T&lt;/m:mi&gt; &lt;m:mi&gt;ρ&lt;/m:mi&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:msubsup&gt; &lt;m:msub&gt; &lt;m:mo&gt;×&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;ℤ&lt;/m:mi&gt; &lt;m:mo&gt;/&lt;/m:mo&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msub&gt; &lt;m:msup&gt; &lt;m:mi&gt;S&lt;/m:mi&gt; &lt;m:mi mathvariant=\"normal\"&gt;ℓ&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0440.png\"/&gt; &lt;jats:tex-math&gt;{M:=T^{n}_{rho}times_{mathbb{Z}/p}S^{ell}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; over the lens space &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mi&gt;S&lt;/m:mi&gt; &lt;m:mi mathvariant=\"normal\"&gt;ℓ&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;m:mo&gt;/&lt;/m:mo&gt; &lt;m:mi&gt;ℤ&lt;/m:mi&gt; &lt;m:mo&gt;/&lt;/m:mo&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0463.png\"/&gt; &lt;jats:tex-math&gt;{S^{ell}/mathbb{Z}/p}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; has fundamental group Γ. When &lt;jats:inline-formula&gt;","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"66 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141059902","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":"Multilinear fourier integral operators on modulation spaces","authors":"Aparajita Dasgupta, Lalit Mohan, Shyam Swarup Mondal","doi":"10.1515/forum-2024-0088","DOIUrl":"https://doi.org/10.1515/forum-2024-0088","url":null,"abstract":"This corrigendum corrects Proposition 5.2 in [A. Dasgupta, L. Mohan and S. S. Mondal, Multilinear Fourier Integral operators on modulation spaces, Forum Math. 2024, 10.1515/forum-2023-0158].","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"50 8 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801915","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":"Laplace convolutions of weighted averages of arithmetical functions","authors":"Marco Cantarini, Alessandro Gambini, Alessandro Zaccagnini","doi":"10.1515/forum-2023-0259","DOIUrl":"https://doi.org/10.1515/forum-2023-0259","url":null,"abstract":"Let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;G&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;g&lt;/m:mi&gt; &lt;m:mo&gt;;&lt;/m:mo&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;:=&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:msub&gt; &lt;m:mo largeop=\"true\" symmetric=\"true\"&gt;∑&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;m:mo&gt;≤&lt;/m:mo&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msub&gt; &lt;m:mrow&gt; &lt;m:mi&gt;g&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0259_eq_0246.png\" /&gt; &lt;jats:tex-math&gt;{G(g;x):=sum_{nleq x}g(n)}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be the summatory function of an arithmetical function &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;g&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0259_eq_0403.png\" /&gt; &lt;jats:tex-math&gt;{g(n)}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. In this paper, we prove that we can write weighted averages of an arbitrary fixed number &lt;jats:italic&gt;N&lt;/jats:italic&gt; of arithmetical functions &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:msub&gt; &lt;m:mi&gt;g&lt;/m:mi&gt; &lt;m:mi&gt;j&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo rspace=\"4.2pt\"&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;j&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;∈&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;{&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi mathvariant=\"normal\"&gt;…&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;N&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;}&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0259_eq_0414.png\" /&gt; &lt;jats:tex-math&gt;{g_{j}(n),,jin{1,dots,N}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; as an integral involving the convolution (in the sense of Laplace) of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:msub&gt; &lt;m:mi&gt;G&lt;/m:mi&gt; &lt;m:mi&gt;j&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0259_eq_0257.png\" /&gt; &lt;jats:tex-math&gt;{G_{j}(x)}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;j&lt;/m:mi&gt; &lt;m:mo&gt;∈&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"105 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801659","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":"Weighted bilinear multiplier theorems in Dunkl setting via singular integrals","authors":"Suman Mukherjee, Sanjay Parui","doi":"10.1515/forum-2023-0398","DOIUrl":"https://doi.org/10.1515/forum-2023-0398","url":null,"abstract":"The purpose of this article is to present one and two-weight inequalities for bilinear multiplier operators in Dunkl setting with multiple Muckenhoupt weights. In order to do so, new results regarding Littlewood–Paley type theorems and weighted inequalities for multilinear Calderón–Zygmund operators in Dunkl setting are also proved.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"41 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801707","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":"Square-integrable representations and the coadjoint action of solvable Lie groups","authors":"Ingrid Beltiţă, Daniel Beltiţă","doi":"10.1515/forum-2024-0025","DOIUrl":"https://doi.org/10.1515/forum-2024-0025","url":null,"abstract":"We characterize the square-integrable representations of (connected, simply connected) solvable Lie groups in terms of the generalized orbits of the coadjoint action. We prove that the normal representations corresponding, via the Pukánszky correspondence, to open coadjoint orbits are type I, not necessarily square-integrable representations. We show that the quasi-equivalence classes of type I square-integrable representations are in bijection with the simply connected open coadjoint orbits, and the existence of an open coadjoint orbit guarantees the existence of a compact open subset of the space of primitive ideals of the group. When the nilradical has codimension 1, we prove that the isolated points of the primitive ideal space are always of type I. This is not always true for codimension greater than 2, as shown by specific examples of solvable Lie groups that have dense, but not locally closed, coadjoint orbits.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"32 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801660","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":"Small generators of abelian number fields","authors":"Martin Widmer","doi":"10.1515/forum-2023-0467","DOIUrl":"https://doi.org/10.1515/forum-2023-0467","url":null,"abstract":"We show that for each abelian number field <jats:italic>K</jats:italic> of sufficiently large degree <jats:italic>d</jats:italic> there exists an element <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mi>K</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0467_eq_0135.png\" /> <jats:tex-math>{alphain K}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>K</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>ℚ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>α</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0467_eq_0100.png\" /> <jats:tex-math>{K=mathbb{Q}(alpha)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and absolute Weil height <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>H</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>α</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:msub> <m:mo>≪</m:mo> <m:mi>d</m:mi> </m:msub> <m:msup> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:msub> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mi>K</m:mi> </m:msub> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>d</m:mi> </m:mrow> </m:mfrac> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0467_eq_0091.png\" /> <jats:tex-math>{H(alpha)ll_{d}|Delta_{K}|^{frac{1}{2d}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mi>K</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0467_eq_0128.png\" /> <jats:tex-math>{Delta_{K}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> denotes the discriminant of <jats:italic>K</jats:italic>. This answers a question of Ruppert from 1998 in the case of abelian extensions of sufficiently large degree. We also show that the exponent <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mfrac> <m:mn>1</m:mn> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>d</m:mi> </m:mrow> </m:mfrac> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0467_eq_0152.png\" /> <jats:tex-math>{frac{1}{2d}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is best-possible when <jats:italic>d</jats:italic> is even.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"104 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140801916","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":"Weighted estimates for product singular integral operators in Journé’s class on RD-spaces","authors":"Taotao Zheng, Yanmei Xiao, Xiangxing Tao","doi":"10.1515/forum-2023-0273","DOIUrl":"https://doi.org/10.1515/forum-2023-0273","url":null,"abstract":"An RD-space 𝑀 is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in 𝑀. In this paper, firstly, the authors give the Plancherel–Pôlya characterization of product weighted Triebel–Lizorkin spaces and product weighted Besov spaces on RD-spaces and make some estimates for the product singular integral operators in Journé’s class on these function spaces. As a result of these conclusions, they present some sufficient conditions for the boundedness of product singular integral operators on the product Lipschitz spaces and product weighted Hardy spaces. Secondly, by the boundedness of lifting and projection operators, they also obtain that the dual spaces of the product weighted Hardy spaces are product weighted Carleson measure spaces. Using the idea of dual, the authors obtain the weighted boundedness of singular integral operators on the product weighted Carleson measure spaces.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"219 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140611388","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}