In this paper, we study representations of non-finitely graded Lie algebras 𝒲(ϵ){mathcal{W}(epsilon)} related to Virasoro algebra, where ϵ=±1{epsilon=pm 1}. Precisely speaking, we completely classify the free 𝒰(𝔥){mathcal{U}(mathfrak{h})}-modules of rank one over 𝒲(ϵ){mathcal{W}(epsilon)}, 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.
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.
{"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}
Let <jats:italic>p</jats:italic> be an odd prime and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ρ</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>→</m:mo> <m:mrow> <m:msub> <m:mi>GL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ℤ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2022-0279_eq_0655.png"/> <jats:tex-math>{rho:mathbb{Z}/prightarrowoperatorname{{GL}}_{n}(mathbb{Z})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be an action of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2022-0279_eq_0555.png"/> <jats:tex-math>{mathbb{Z}/p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> on a lattice and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>:=</m:mo> <m:mrow> <m:mrow> <m:msup> <m:mi>ℤ</m:mi> <m:mi>n</m:mi> </m:msup> <m:msub> <m:mo>⋊</m:mo> <m:mi>ρ</m:mi> </m:msub> <m:mi>ℤ</m:mi> </m:mrow> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2022-0279_eq_0490.png"/> <jats:tex-math>{Gamma:=mathbb{Z}^{n}rtimes_{rho}mathbb{Z}/p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the corresponding semidirect product. The torus bundle <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>M</m:mi> <m:mo>:=</m:mo> <m:mrow> <m:msubsup> <m:mi>T</m:mi> <m:mi>ρ</m:mi> <m:mi>n</m:mi> </m:msubsup> <m:msub> <m:mo>×</m:mo> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:msup> <m:mi>S</m:mi> <m:mi mathvariant="normal">ℓ</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2022-0279_eq_0440.png"/> <jats:tex-math>{M:=T^{n}_{rho}times_{mathbb{Z}/p}S^{ell}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> over the lens space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>S</m:mi> <m:mi mathvariant="normal">ℓ</m:mi> </m:msup> <m:mo>/</m:mo> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2022-0279_eq_0463.png"/> <jats:tex-math>{S^{ell}/mathbb{Z}/p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> has fundamental group Γ. When <jats:inline-formula>
让 p 是奇素数，让 ρ : ℤ / p → GL n ( ℤ ) {rho:mathbb{Z}/prightarrowoperatorname{{GL}}_{n}(mathbb{Z})} } 是 ℤ / p {mathbb{Z}/p} 在网格上的作用，让 Γ ：= ℤ n ⋊ ρ ℤ / p {Gamma:=mathbb{Z}^{n}rtimes_{rho}mathbb{Z}/p} 是相应的半间接积。透镜空间 S ℓ / ℤ / p {S^{ell}/mathbb{Z}/p} 上的环束 M := T ρ n × ℤ / p S ℓ {M:=T^{n}_{rho}times_{mathbb{Z}/p}S^{ell}} 具有基群 Γ。当ℤ / p {mathbb{Z}/p} 只固定了ℤ n {mathbb{Z}^{n} 的原点时｝ Davis 和 Lück (2021) 计算了 L 群 L m 〈 j 〉 ( ℤ [ Γ ] ) {L^{langle jrangle}_{m}(mathbb{Z}[Gamma])} 和结构集 𝒮 geo , s ( M ) {mathcal{S}}^{rm geo},s}(M)} 。在本文中，我们将这些计算扩展到ℤ / p {mathbb{Z}/p} 对ℤ n {mathbb{Z}^{n} 的所有作用。｝ .具体而言，我们计算 L m 〈 j 〉 ( ℤ [ Γ ] ) {L^{langle jrangle}_{m}(mathbb{Z}[Gamma])} 和 𝒮 geo 、s ( M ) {mathcal{S}}^{rm geo},s}(M)} 在 E ¯ Γ {underline{E}Gamma} 有一个非离散奇异集的情况下。
{"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 <jats:italic>p</jats:italic> be an odd prime and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ρ</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>→</m:mo> <m:mrow> <m:msub> <m:mi>GL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ℤ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0655.png\"/> <jats:tex-math>{rho:mathbb{Z}/prightarrowoperatorname{{GL}}_{n}(mathbb{Z})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be an action of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0555.png\"/> <jats:tex-math>{mathbb{Z}/p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> on a lattice and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"normal\">Γ</m:mi> <m:mo>:=</m:mo> <m:mrow> <m:mrow> <m:msup> <m:mi>ℤ</m:mi> <m:mi>n</m:mi> </m:msup> <m:msub> <m:mo>⋊</m:mo> <m:mi>ρ</m:mi> </m:msub> <m:mi>ℤ</m:mi> </m:mrow> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0490.png\"/> <jats:tex-math>{Gamma:=mathbb{Z}^{n}rtimes_{rho}mathbb{Z}/p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the corresponding semidirect product. The torus bundle <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>M</m:mi> <m:mo>:=</m:mo> <m:mrow> <m:msubsup> <m:mi>T</m:mi> <m:mi>ρ</m:mi> <m:mi>n</m:mi> </m:msubsup> <m:msub> <m:mo>×</m:mo> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:msup> <m:mi>S</m:mi> <m:mi mathvariant=\"normal\">ℓ</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0440.png\"/> <jats:tex-math>{M:=T^{n}_{rho}times_{mathbb{Z}/p}S^{ell}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> over the lens space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>S</m:mi> <m:mi mathvariant=\"normal\">ℓ</m:mi> </m:msup> <m:mo>/</m:mo> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0463.png\"/> <jats:tex-math>{S^{ell}/mathbb{Z}/p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> has fundamental group Γ. When <jats:inline-formula>","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}
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].
本更正纠正了 [A. Dasgupta, L. Mohan and S. S. Mondal, Multilinear Fourier Integral operators on modulation spaces, Forum Math.Dasgupta、L. Mohan 和 S. S. Mondal，调制空间上的多线性傅里叶积分算子，Forum Math.2024, 10.1515/forum-2023-0158].
{"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}
Marco Cantarini, Alessandro Gambini, Alessandro Zaccagnini
Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>G</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>g</m:mi> <m:mo>;</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>:=</m:mo> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∑</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mi>g</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2023-0259_eq_0246.png" /> <jats:tex-math>{G(g;x):=sum_{nleq x}g(n)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the summatory function of an arithmetical function <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:mo stretchy="false">(</m:mo> <m:mi>n</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-2023-0259_eq_0403.png" /> <jats:tex-math>{g(n)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we prove that we can write weighted averages of an arbitrary fixed number <jats:italic>N</jats:italic> of arithmetical functions <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>g</m:mi> <m:mi>j</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace="4.2pt">,</m:mo> <m:mi>j</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:mi>N</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-2023-0259_eq_0414.png" /> <jats:tex-math>{g_{j}(n),,jin{1,dots,N}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> as an integral involving the convolution (in the sense of Laplace) of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mi>j</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</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-2023-0259_eq_0257.png" /> <jats:tex-math>{G_{j}(x)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>j</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false
设 G ( g ; x ) := ∑ n ≤ x g ( n ) {G(g;x):=sum_{nleq x}g(n)} 为算术函数 g ( n ) {g(n)} 的求和函数。本文将证明，我们可以写出任意固定数量 N 的算术函数 g j ( n ) , j ∈ { 1 , ... , N } 的加权平均数 {g_{j}(n),,jin{1,dots,N}} 是一个涉及 G j ( x ) {G_{j}(x)} 的卷积（拉普拉斯意义上）的积分，j∈ { 1 , ... , N }。 {jin{1,dots,N}} . .此外，我们还证明了一个特性，它使我们能够以非常简单自然的方式获得关于算术函数平均数的已知结果，并克服了一些著名问题的技术限制。
{"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 <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>G</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>g</m:mi> <m:mo>;</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>:=</m:mo> <m:mrow> <m:msub> <m:mo largeop=\"true\" symmetric=\"true\">∑</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mi>g</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0259_eq_0246.png\" /> <jats:tex-math>{G(g;x):=sum_{nleq x}g(n)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the summatory function of an arithmetical function <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:mo stretchy=\"false\">(</m:mo> <m:mi>n</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-2023-0259_eq_0403.png\" /> <jats:tex-math>{g(n)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we prove that we can write weighted averages of an arbitrary fixed number <jats:italic>N</jats:italic> of arithmetical functions <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mi>g</m:mi> <m:mi>j</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>n</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace=\"4.2pt\">,</m:mo> <m:mi>j</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">…</m:mi> <m:mo>,</m:mo> <m:mi>N</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-2023-0259_eq_0414.png\" /> <jats:tex-math>{g_{j}(n),,jin{1,dots,N}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> as an integral involving the convolution (in the sense of Laplace) of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mi>j</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</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-2023-0259_eq_0257.png\" /> <jats:tex-math>{G_{j}(x)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>j</m:mi> <m:mo>∈</m:mo> <m:mrow> <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}
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.
{"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}
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.
我们根据共轭作用的广义轨道来描述（连通的、简单连通的）可解李群的可方整表示。我们证明，通过普卡恩斯基对应关系，与开放共轭轨道相对应的正则表达式是 I 型，而不一定是平方可整合表达式。我们证明了 I 型方整表示的准等价类与简单连接的开放共轭轨道是双射的，而开放共轭轨道的存在保证了群的原始理想空间的紧凑开放子集的存在。当无根性的标度为 1 时，我们证明了基元理想空间的孤立点总是 I 型的。当标度大于 2 时，情况并非总是如此，这一点可以通过具有致密但非局部封闭的共轭轨道的可解李群的具体例子来证明。
{"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}
We show that for each abelian number field K of sufficiently large degree d there exists an element α∈K{alphain K} with K=ℚ(α){K=mathbb{Q}(alpha)} and absolute Weil height H(α)≪d|ΔK|12d{H(alpha)ll_{d}|Delta_{K}|^{frac{1}{2d}}}, where ΔK{Delta_{K}} denotes the discriminant of K. This answers a question of Ruppert from 1998 in the case of abelian extensions of sufficiently large degree. We also show that the exponent 12d{frac{1}{2d}} is best-possible when d is even.
我们证明，对于每个阶数为 d 的无性数域 K，都存在一个元素 α∈K {alphain K} ，其中 K = ℚ ( α ) {K=mathbb{Q}(alpha)} 且绝对韦尔高 H ( α ) ≪ d | Δ K | 1 2 d {H(alpha)ll_{d}|Delta_{K}|^{frac{1}{2d}} ，其中 Δ K {Delta_{K}} 表示 K 的判别式。 其中 Δ K {Delta_{K}} 表示 K 的判别式。这回答了鲁珀特在 1998 年提出的一个问题，即在阶数足够大的无性扩展的情况下。我们还证明了当 d 为偶数时，指数 1 2 d {frac{1}{2d}} 是最可能的。
{"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}
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.
{"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}
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