Pub Date : 2020-06-30DOI: 10.2422/2036-2145.201710_001
Jeffrey S. Case, A. Gover
We establish an algorithm which computes formulae for the CR GJMS operators, the P0-operator, and the Q0-curvature in terms of CR tractors. When applied to torsion-free pseudo-Einstein contact forms, this algorithm both gives an explicit factorisation of the CR GJMS operators and the P0-operator, and shows that the Q0-curvature is constant, with the constant explicitly given in terms of the Webster scalar curvature. We also use our algorithm to derive local formulae for the P0-operator and Q0-curvature of a five-dimensional pseudo-Einstein manifold. Comparison with Marugame’s formulation of the Burns-Epstein invariant as the integral of a pseudohermitian invariant yields new insights into the class of local pseudohermitian invariants for which the total integral is independent of the choice of pseudo-Einstein contact form.
{"title":"The P'-operator, the Q'-curvature, and the CR tractor calculus","authors":"Jeffrey S. Case, A. Gover","doi":"10.2422/2036-2145.201710_001","DOIUrl":"https://doi.org/10.2422/2036-2145.201710_001","url":null,"abstract":"We establish an algorithm which computes formulae for the CR GJMS \u0000operators, the P0-operator, and the Q0-curvature in terms of CR tractors. When \u0000applied to torsion-free pseudo-Einstein contact forms, this algorithm both gives an \u0000explicit factorisation of the CR GJMS operators and the P0-operator, and shows \u0000that the Q0-curvature is constant, with the constant explicitly given in terms of \u0000the Webster scalar curvature. We also use our algorithm to derive local formulae \u0000for the P0-operator and Q0-curvature of a five-dimensional pseudo-Einstein manifold. Comparison with Marugame’s formulation of the Burns-Epstein invariant \u0000as the integral of a pseudohermitian invariant yields new insights into the class of \u0000local pseudohermitian invariants for which the total integral is independent of the \u0000choice of pseudo-Einstein contact form.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"28 1","pages":"565-618"},"PeriodicalIF":1.4,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85905659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-06-30DOI: 10.2422/2036-2145.201706_027
Riccardo Scala
We consider the relaxed area functional for vector valued maps and its exact value on the triple junction function u : B1(O) → R, a specific function which represents the first example of map whose graph area shows nonlocal effects. This is a map taking only three different values α, β, γ ∈ R in three equal circular sectors of the unit radius ball B1(O). We prove a conjecture due to G. Bellettini and M. Paolini asserting that the recovery sequence provided in [5] (and the corresponding upper bound for the relaxed area functional of the map u) is optimal. At the same time, we show by means of a counterexample that such construction is not optimal if we consider different domains than B1(O), which still contain the same discontinuity set of u in B1(O). Such domains are obtained from B1(O) erasing part of interior of the sectors where u is constant.
{"title":"Optimal estimates for the triple junction function and other surprising aspects of the area functional","authors":"Riccardo Scala","doi":"10.2422/2036-2145.201706_027","DOIUrl":"https://doi.org/10.2422/2036-2145.201706_027","url":null,"abstract":"We consider the relaxed area functional for vector valued maps and its exact value on the triple junction function u : B1(O) → R, a specific function which represents the first example of map whose graph area shows nonlocal effects. This is a map taking only three different values α, β, γ ∈ R in three equal circular sectors of the unit radius ball B1(O). We prove a conjecture due to G. Bellettini and M. Paolini asserting that the recovery sequence provided in [5] (and the corresponding upper bound for the relaxed area functional of the map u) is optimal. At the same time, we show by means of a counterexample that such construction is not optimal if we consider different domains than B1(O), which still contain the same discontinuity set of u in B1(O). Such domains are obtained from B1(O) erasing part of interior of the sectors where u is constant.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"93 1","pages":"491-564"},"PeriodicalIF":1.4,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83865847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-03-23DOI: 10.2422/2036-2145.202001_ERRATA
G. Metafune, M. Sobajima, C. Spina
The proof of Proposition 3.18 in [1] contains a mistake since an r2 appeared erroneously, instead of r, in the equation for w. The statement is however correct and below is an amended proof.
{"title":"Errata corrige. Elliptic and parabolic problems for a class of operators with discontinuous coefficients","authors":"G. Metafune, M. Sobajima, C. Spina","doi":"10.2422/2036-2145.202001_ERRATA","DOIUrl":"https://doi.org/10.2422/2036-2145.202001_ERRATA","url":null,"abstract":"The proof of Proposition 3.18 in [1] contains a mistake since an r2 appeared erroneously, instead of r, in the equation for w. The statement is however \u0000correct and below is an amended proof.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"18 1","pages":"413-414"},"PeriodicalIF":1.4,"publicationDate":"2020-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81036628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.2422/2036-2145.201901_011
A. Bendikov, Eryan Hu, Jiaxin Hu
We consider a class of jump measures on ultrametric spaces and the associated non-local regular Dirichlet forms. We obtain equivalent conditions for certain heat kernel upper and lower estimates in terms of the properties of the jump measure. In particular, heat kernel estimates are given for quite degenerate/singular jump measures as shown in a number of examples.
{"title":"Heat kernels and non-local Dirichlet forms on ultrametric spaces","authors":"A. Bendikov, Eryan Hu, Jiaxin Hu","doi":"10.2422/2036-2145.201901_011","DOIUrl":"https://doi.org/10.2422/2036-2145.201901_011","url":null,"abstract":"We consider a class of jump measures on ultrametric spaces and the associated non-local regular Dirichlet forms. We obtain equivalent conditions for certain heat kernel upper and lower estimates in terms of the properties of the jump measure. In particular, heat kernel estimates are given for quite degenerate/singular jump measures as shown in a number of examples.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"15 1","pages":"1"},"PeriodicalIF":1.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74238077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.2422/2036-2145.201811_016
J. M. Masqué, L. M. P. Coronado, M. E. Rosado
{"title":"Differential p-forms and q-vector fields with constant coefficients","authors":"J. M. Masqué, L. M. P. Coronado, M. E. Rosado","doi":"10.2422/2036-2145.201811_016","DOIUrl":"https://doi.org/10.2422/2036-2145.201811_016","url":null,"abstract":"","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"156 1","pages":"1"},"PeriodicalIF":1.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73448061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.2422/2036-2145.201811_017
C. Dupont, Johan Taflin
{"title":"Dynamics of fibered endomorphisms of CP(k)","authors":"C. Dupont, Johan Taflin","doi":"10.2422/2036-2145.201811_017","DOIUrl":"https://doi.org/10.2422/2036-2145.201811_017","url":null,"abstract":"","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"109 1","pages":"1"},"PeriodicalIF":1.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76345813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.2422/2036-2145.201801_001
R. Camporesi
Let S = N A be a Damek-Ricci space, identified with the unit ball B in s via the Cayley transform. Let S p+q = @B be the unit sphere in s, p = dim v, q = dim z. The metric in the ball model was computed in [1] both in Euclidean (or geodesic) polar coordinates and in Cartesian coordinates on B. The induced metric on the Euclidean sphere S(R) of radius R is the sum of a constant curvature term, plus a correction term proportional to h1, where h1 is a suitable differential expression which is smooth on S(R) for R < 1, but becomes (possibly) singular on the unit sphere at the pole (0, 0, 1). It has a simple geometric interpretation, namely h1 = |2|2, where 2 is, up to a conformal factor, the pull-back of the canonical 1-form on the group N (defining the horizontal distribution on N ) by the generalized stereographic projection. In the symmetric case h1, as well as the transported distribution on S p+q {(0, 0, 1)}, have a smooth extension to the whole sphere. This can be interpreted by the Hopf fibration of S p+q . In the general case no such structure is allowed on the unit sphere, and the question was left open in [1] whether or not h1 extends smoothly at the pole. In this paper we prove that h1 does not extend, except in the symmetric case. More precisely, writing h1 in the coordinates (V, Z) on S p+q as h1 = P h i j dzi dz j + P h i j dvi dv j + P h i j dzi dv j , we prove that, in the non-symmetric case, the coefficients h i j do not have a limit at the pole, but remain bounded there, whereas the coefficients h i j and h (zv) i j extend smoothly at the pole. In order to do this, we obtain an explicit formula for the 1-form 2 valid for any Damek-Ricci space. From this formula we deduce that2 does not extend to the pole, except for q = 1 (Hermitian symmetric case). The square of 2 and the distribution ker2 do not extend, unless S is symmetric. Indeed, we prove that the singular part of h1 vanishes identically if and only if the J2-condition holds. Mathematics Subject Classification (2010): 22E25 (primary); 43A85, 53C30 (secondary).
设S = A为Damek-Ricci空间，通过Cayley变换与S中的单位球B识别。让S p + q = @B单位球的年代,p =昏暗的v, q =昏暗的z。球模型计算的指标[1]在欧几里得(或测地线)极坐标和在笛卡尔坐标b上的感应度规欧几里得球体半径为R的年代(R)是一个常曲率项的总和,加上一个修正项比例h1, h1是一个合适的微分表达式顺利在S (R) R < 1,但是就会(可能)单数单位球上的杆(0,0,1).它有一个简单的几何解释，即h1 = |2|2，其中2在保形因子范围内，是正则1-形式在群N(定义N上的水平分布)上的广义立体投影的回拉。在对称情况下h1，以及S p+q {(0,0,1)}上的输运分布，对整个球面有光滑的扩展。这可以用S p+q的霍普夫振动来解释。在一般情况下，在单位球上不允许有这样的结构，在[1]中，h1在极点处是否平滑延伸的问题是开放的。本文证明了除对称情况外，h1不扩展。更准确地说，将(V, Z)坐标下的h1在S p+q上写成h1 = p h i j dzi dz j + p h i j dvi dv j + p h i j dzi dv j，我们证明了，在非对称情况下，系数h i j在极点处没有极限，在极点处保持有界，而系数h i j和h (zv) i j在极点处平滑扩展。为了做到这一点，我们得到了对任何Damek-Ricci空间有效的1-form - 2的显式公式。从这个公式我们推断出，除了q = 1(厄米对称情况)，2不能扩展到极点。2的平方和分布ker2不会扩展，除非S是对称的。事实上，我们证明了h1的奇异部分当且仅当j2条件成立时完全消失。数学学科分类(2010):22E25(初级);43A85, 53C30(二级)。
{"title":"The metric at infinity on Damek-Ricci spaces","authors":"R. Camporesi","doi":"10.2422/2036-2145.201801_001","DOIUrl":"https://doi.org/10.2422/2036-2145.201801_001","url":null,"abstract":"Let S = N A be a Damek-Ricci space, identified with the unit ball B in s via the Cayley transform. Let S p+q = @B be the unit sphere in s, p = dim v, q = dim z. The metric in the ball model was computed in [1] both in Euclidean (or geodesic) polar coordinates and in Cartesian coordinates on B. The induced metric on the Euclidean sphere S(R) of radius R is the sum of a constant curvature term, plus a correction term proportional to h1, where h1 is a suitable differential expression which is smooth on S(R) for R < 1, but becomes (possibly) singular on the unit sphere at the pole (0, 0, 1). It has a simple geometric interpretation, namely h1 = |2|2, where 2 is, up to a conformal factor, the pull-back of the canonical 1-form on the group N (defining the horizontal distribution on N ) by the generalized stereographic projection. In the symmetric case h1, as well as the transported distribution on S p+q {(0, 0, 1)}, have a smooth extension to the whole sphere. This can be interpreted by the Hopf fibration of S p+q . In the general case no such structure is allowed on the unit sphere, and the question was left open in [1] whether or not h1 extends smoothly at the pole. In this paper we prove that h1 does not extend, except in the symmetric case. More precisely, writing h1 in the coordinates (V, Z) on S p+q as h1 = P h i j dzi dz j + P h i j dvi dv j + P h i j dzi dv j , we prove that, in the non-symmetric case, the coefficients h i j do not have a limit at the pole, but remain bounded there, whereas the coefficients h i j and h (zv) i j extend smoothly at the pole. In order to do this, we obtain an explicit formula for the 1-form 2 valid for any Damek-Ricci space. From this formula we deduce that2 does not extend to the pole, except for q = 1 (Hermitian symmetric case). The square of 2 and the distribution ker2 do not extend, unless S is symmetric. Indeed, we prove that the singular part of h1 vanishes identically if and only if the J2-condition holds. Mathematics Subject Classification (2010): 22E25 (primary); 43A85, 53C30 (secondary).","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"24 1","pages":"1"},"PeriodicalIF":1.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86963447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.2422/2036-2145.201710_010
Ching-Jui Lai
{"title":"On anticanonical volumes of weak Q-Fano terminal threefolds of Picard rank two","authors":"Ching-Jui Lai","doi":"10.2422/2036-2145.201710_010","DOIUrl":"https://doi.org/10.2422/2036-2145.201710_010","url":null,"abstract":"","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"107 1","pages":"1"},"PeriodicalIF":1.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79721107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.2422/2036-2145.201703_004
N. Q. Thang
We introduce a new equivalence relation on k-points of connected reductive groups over an arbitrary field, which coincides with the usual Brauer equivalence when the characteristic is 0, and study its relation with R-equivalence relation and other basic arithmetic-geometric invariants of the given group over local and global fields of any characteristic via some local-global exact sequences. Mathematics Subject Classification (2010): 11E72 (primary); 20G10, 20G30, 20G35 (secondary).
{"title":"Tate-Shafarevich kernel, weak Brauer and R-equivalence on connected reductive groups over local and global fields","authors":"N. Q. Thang","doi":"10.2422/2036-2145.201703_004","DOIUrl":"https://doi.org/10.2422/2036-2145.201703_004","url":null,"abstract":"We introduce a new equivalence relation on k-points of connected reductive groups over an arbitrary field, which coincides with the usual Brauer equivalence when the characteristic is 0, and study its relation with R-equivalence relation and other basic arithmetic-geometric invariants of the given group over local and global fields of any characteristic via some local-global exact sequences. Mathematics Subject Classification (2010): 11E72 (primary); 20G10, 20G30, 20G35 (secondary).","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"22 1","pages":"1"},"PeriodicalIF":1.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89444017","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.2422/2036-2145.201707_009
Abed Bounemoura, J. Féjoz
We prove a new invariant torus theorem, for α-Gevrey smooth Hamiltonian systems , under an arithmetic assumption which we call the α-Bruno-Russmann condition , and which reduces to the classical Bruno-Russmann condition in the analytic category. Our proof is direct in the sense that, for analytic Hamiltonians, we avoid the use of complex extensions and, for non-analytic Hamiltonians, we do not use analytic approximation nor smoothing operators. Following Bessi, we also show that if a slightly weaker arithmetic condition is not satisfied, the invariant torus may be destroyed. Crucial to this work are new functional estimates in the Gevrey class.
{"title":"KAM, α-Gevrey regularity and the α-Bruno-Rüssmann condition","authors":"Abed Bounemoura, J. Féjoz","doi":"10.2422/2036-2145.201707_009","DOIUrl":"https://doi.org/10.2422/2036-2145.201707_009","url":null,"abstract":"We prove a new invariant torus theorem, for α-Gevrey smooth Hamiltonian systems , under an arithmetic assumption which we call the α-Bruno-Russmann condition , and which reduces to the classical Bruno-Russmann condition in the analytic category. Our proof is direct in the sense that, for analytic Hamiltonians, we avoid the use of complex extensions and, for non-analytic Hamiltonians, we do not use analytic approximation nor smoothing operators. Following Bessi, we also show that if a slightly weaker arithmetic condition is not satisfied, the invariant torus may be destroyed. Crucial to this work are new functional estimates in the Gevrey class.","PeriodicalId":50966,"journal":{"name":"Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze","volume":"46 1","pages":"1225-1279"},"PeriodicalIF":1.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80684831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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