Pub Date : 2021-01-25DOI: 10.4310/mrl.2022.v29.n6.a6
M. Lacey, H. Mousavi, Yaghoub Rahimi
Let $ Lambda $ denote von Mangoldt's function, and consider the averages begin{align*} A_N f (x)&=frac{1}{N}sum_{1leq n leq N}f(x-n)Lambda(n) . end{align*} We prove sharp $ ell ^{p}$-improving for these averages, and sparse bounds for the maximal function. The simplest inequality is that for sets $ F, Gsubset [0,N]$ there holds begin{equation*} N ^{-1} langle A_N mathbf 1_{F} , mathbf 1_{G} rangle ll frac{lvert Frvert cdot lvert Grvert} { N ^2 } Bigl( operatorname {Log} frac{lvert Frvert cdot lvert Grvert} { N ^2 } Bigr) ^{t}, end{equation*} where $ t=2$, or assuming the Generalized Riemann Hypothesis, $ t=1$. The corresponding sparse bound is proved for the maximal function $ sup_N A_N mathbf 1_{F}$. The inequalities for $ t=1$ are sharp. The proof depends upon the Circle Method, and an interpolation argument of Bourgain.
{"title":"Endpoint $ell^r$ improving estimates for prime averages","authors":"M. Lacey, H. Mousavi, Yaghoub Rahimi","doi":"10.4310/mrl.2022.v29.n6.a6","DOIUrl":"https://doi.org/10.4310/mrl.2022.v29.n6.a6","url":null,"abstract":"Let $ Lambda $ denote von Mangoldt's function, and consider the averages begin{align*} A_N f (x)&=frac{1}{N}sum_{1leq n leq N}f(x-n)Lambda(n) . end{align*} We prove sharp $ ell ^{p}$-improving for these averages, and sparse bounds for the maximal function. The simplest inequality is that for sets $ F, Gsubset [0,N]$ there holds begin{equation*} N ^{-1} langle A_N mathbf 1_{F} , mathbf 1_{G} rangle ll frac{lvert Frvert cdot lvert Grvert} { N ^2 } Bigl( operatorname {Log} frac{lvert Frvert cdot lvert Grvert} { N ^2 } Bigr) ^{t}, end{equation*} where $ t=2$, or assuming the Generalized Riemann Hypothesis, $ t=1$. The corresponding sparse bound is proved for the maximal function $ sup_N A_N mathbf 1_{F}$. The inequalities for $ t=1$ are sharp. The proof depends upon the Circle Method, and an interpolation argument of Bourgain.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43964086","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}
Pub Date : 2021-01-10DOI: 10.4310/MRL.2022.v29.n2.a9
Alvaro Liendo, G. Arteche
A BSTRACT . We give an explicit description of the automorphism group of a product of complete toric varieties over an arbitrary field in terms of the respective automorphism groups of its components. More precisely, we prove that, up to permutation of isomorphic components, an automorphism of a product corresponds to a product of automorphisms of its components. We also reprove, in modern language, the classic result by Demazure describing the group-scheme of automorphisms of a complete toric variety over an arbitrary field.
{"title":"Automorphisms of products of toric varieties","authors":"Alvaro Liendo, G. Arteche","doi":"10.4310/MRL.2022.v29.n2.a9","DOIUrl":"https://doi.org/10.4310/MRL.2022.v29.n2.a9","url":null,"abstract":"A BSTRACT . We give an explicit description of the automorphism group of a product of complete toric varieties over an arbitrary field in terms of the respective automorphism groups of its components. More precisely, we prove that, up to permutation of isomorphic components, an automorphism of a product corresponds to a product of automorphisms of its components. We also reprove, in modern language, the classic result by Demazure describing the group-scheme of automorphisms of a complete toric variety over an arbitrary field.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43031033","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}
Pub Date : 2021-01-01DOI: 10.4310/MRL.2021.V28.N3.A12
O. Yakimova
{"title":"Commutative subalgebras of $mathcal{U}(mathfrak{q})$ of maximal transcendence degree","authors":"O. Yakimova","doi":"10.4310/MRL.2021.V28.N3.A12","DOIUrl":"https://doi.org/10.4310/MRL.2021.V28.N3.A12","url":null,"abstract":"","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"28 1","pages":"907-924"},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70516687","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}
Pub Date : 2021-01-01DOI: 10.4310/MRL.2021.V28.N1.A1
Bingyi Chen, S. Yau, S. Yau, Huaiqing Zuo
{"title":"$4d$ $N=2$ SCFT and singularity theory Part IV: Isolated rational Gorenstein non-complete intersection singularities with at least one-dimensional deformation and nontrivial $T^2$","authors":"Bingyi Chen, S. Yau, S. Yau, Huaiqing Zuo","doi":"10.4310/MRL.2021.V28.N1.A1","DOIUrl":"https://doi.org/10.4310/MRL.2021.V28.N1.A1","url":null,"abstract":"","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70516435","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}
Pub Date : 2021-01-01DOI: 10.4310/MRL.2021.V28.N1.A5
Joe Kramer-Miller
{"title":"Slope filtrations of $F$-isocrystals and logarithmic decay","authors":"Joe Kramer-Miller","doi":"10.4310/MRL.2021.V28.N1.A5","DOIUrl":"https://doi.org/10.4310/MRL.2021.V28.N1.A5","url":null,"abstract":"","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"28 1","pages":"107-125"},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70516498","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}
Pub Date : 2021-01-01DOI: 10.4310/mrl.2021.v28.n6.a11
Yu Yang
Let p be a prime number, and let Mg,n be the moduli stack over an algebraic closure Fp of the finite field Fp parameterizing pointed stable curves of type (g, n), Mg,n the open substack of Mg,n parameterizing smooth pointed stable curves, Mg,n the coarse moduli space of Mg,n, Mg,n the coarse moduli space of Mg,n, q ∈ Mg,n an arbitrary point, and Πq the admissible fundamental group of a pointed stable curve corresponding to a geometric point over q. In the present paper, we prove that, there exists q1, q2 ∈ Mg,n Mg,n such that q1 is a specialization of q2, that q1 ̸= q2, and that a specialization homomorphism sp : Πq2 ↠ Πq1 is an isomorphism.
{"title":"On the existence of specialization isomorphisms of admissible fundamental groups in positive characteristic","authors":"Yu Yang","doi":"10.4310/mrl.2021.v28.n6.a11","DOIUrl":"https://doi.org/10.4310/mrl.2021.v28.n6.a11","url":null,"abstract":"Let p be a prime number, and let Mg,n be the moduli stack over an algebraic closure Fp of the finite field Fp parameterizing pointed stable curves of type (g, n), Mg,n the open substack of Mg,n parameterizing smooth pointed stable curves, Mg,n the coarse moduli space of Mg,n, Mg,n the coarse moduli space of Mg,n, q ∈ Mg,n an arbitrary point, and Πq the admissible fundamental group of a pointed stable curve corresponding to a geometric point over q. In the present paper, we prove that, there exists q1, q2 ∈ Mg,n Mg,n such that q1 is a specialization of q2, that q1 ̸= q2, and that a specialization homomorphism sp : Πq2 ↠ Πq1 is an isomorphism.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"15 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70516919","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}
Pub Date : 2021-01-01DOI: 10.4310/MRL.2021.V28.N1.A12
Xifeng Su, P. Thieullen
{"title":"Gottschalk–Hedlund theorem revisited","authors":"Xifeng Su, P. Thieullen","doi":"10.4310/MRL.2021.V28.N1.A12","DOIUrl":"https://doi.org/10.4310/MRL.2021.V28.N1.A12","url":null,"abstract":"","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"28 1","pages":"285-300"},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70516449","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}
Pub Date : 2021-01-01DOI: 10.4310/mrl.2021.v28.n4.a4
Jiao Chen, Liang Huang, Guozhen Lu
The main aim of this paper to establish a bi-parameter version of a theorem of Baernstein and Sawyer [1] on boundedness of Fourier multipliers on one-parameter Hardy spaces H p ( R n ) which improves an earlier result of Calder´on and Torchinsky [2]. More pre-cisely, we prove the boundedness of the bi-parameter Fourier multiplier operators on the Lebesgue spaces L p ( R n 1 × R n 2 ) (1 < p < ∞ ) and bi-parameter Hardy spaces H p ( R n 1 × R n 2 ) (0 < p ≤ 1) with optimal regularity for the multiplier being in the bi-parameter Besov spaces B ( n 12 , n 22 ) 2 , 1 ( R n 1 × R n 2 ) and B ( s 1 ,s 2 ) 2 ,q ( R n 1 × R n 2 ). The Besov regularity assumption is clearly weaker than the assumption of the Sobolev regularity. Thus our results sharpen the known H¨ormander multiplier theorem for the bi-parameter Fourier multipliers using the Sobolev regularity in the same spirit as Baernstein and Sawyer improved the result of Calder´on and Torchinsky. Our method is differential from the one used by Baernstein and Sawyer in the one-parameter setting. We employ the bi-parameter Littlewood-Paley-Stein theory and atomic decomposition for the bi-parameter Hardy spaces H p ( R n 1 × R n 2 ) (0 < p ≤ 1) to establish our main result (Theorem 1.6). Moreover, the bi-parameter nature involves much more subtlety in our situation where atoms are supported on arbitrary open sets instead of rectangles.
本文的主要目的是建立Baernstein和Sawyer[1]关于单参数Hardy空间H p (R n)上傅里叶乘子有界性定理的双参数版本，它改进了Calder´on和Torchinsky[1]先前的结果。更pre-cisely,我们证明了有界性bi-parameter傅里叶乘数运营商的勒贝格空间L p (R n 1×R n 2) (1 < p <∞)和bi-parameter哈代空间H p (R n 1×R n 2) (0 < p≤1)优化规律的乘数是bi-parameter Besov空间B (n 12日22)2、1 (R n 1×R n 2)和B(1,年代2)2,问(R n 1×R n 2)。Besov正则性假设明显弱于Sobolev正则性假设。因此，我们的结果锐化了已知的双参数傅里叶乘子的H¨ormander乘子定理，使用Sobolev正则性，就像Baernstein和Sawyer改进了Calder ' on和Torchinsky的结果一样。我们的方法不同于Baernstein和Sawyer在单参数设置中使用的方法。我们利用双参数littlewood - paly - stein理论和双参数Hardy空间hp (rn1 × rn2) (0 < p≤1)的原子分解来建立我们的主要结果(定理1.6)。此外，在我们的情况下，双参数的性质涉及到更多的微妙之处，其中原子被支持在任意开集而不是矩形上。
{"title":"Hörmander Fourier multiplier theorems with optimal regularity in bi-parameter Besov spaces","authors":"Jiao Chen, Liang Huang, Guozhen Lu","doi":"10.4310/mrl.2021.v28.n4.a4","DOIUrl":"https://doi.org/10.4310/mrl.2021.v28.n4.a4","url":null,"abstract":"The main aim of this paper to establish a bi-parameter version of a theorem of Baernstein and Sawyer [1] on boundedness of Fourier multipliers on one-parameter Hardy spaces H p ( R n ) which improves an earlier result of Calder´on and Torchinsky [2]. More pre-cisely, we prove the boundedness of the bi-parameter Fourier multiplier operators on the Lebesgue spaces L p ( R n 1 × R n 2 ) (1 < p < ∞ ) and bi-parameter Hardy spaces H p ( R n 1 × R n 2 ) (0 < p ≤ 1) with optimal regularity for the multiplier being in the bi-parameter Besov spaces B ( n 12 , n 22 ) 2 , 1 ( R n 1 × R n 2 ) and B ( s 1 ,s 2 ) 2 ,q ( R n 1 × R n 2 ). The Besov regularity assumption is clearly weaker than the assumption of the Sobolev regularity. Thus our results sharpen the known H¨ormander multiplier theorem for the bi-parameter Fourier multipliers using the Sobolev regularity in the same spirit as Baernstein and Sawyer improved the result of Calder´on and Torchinsky. Our method is differential from the one used by Baernstein and Sawyer in the one-parameter setting. We employ the bi-parameter Littlewood-Paley-Stein theory and atomic decomposition for the bi-parameter Hardy spaces H p ( R n 1 × R n 2 ) (0 < p ≤ 1) to establish our main result (Theorem 1.6). Moreover, the bi-parameter nature involves much more subtlety in our situation where atoms are supported on arbitrary open sets instead of rectangles.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70516813","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}
Pub Date : 2021-01-01DOI: 10.4310/MRL.2021.V28.N3.A4
J. Fornæss, F. Rong
{"title":"The boundary rigidity for holomorphic self-maps of some fibered domains","authors":"J. Fornæss, F. Rong","doi":"10.4310/MRL.2021.V28.N3.A4","DOIUrl":"https://doi.org/10.4310/MRL.2021.V28.N3.A4","url":null,"abstract":"","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"28 1","pages":"697-706"},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70516734","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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