We consider the compressible Navier–Stokes system in the critical Besov spaces. It is known that the system is (semi-)well-posed in the scaling semi-invariant spaces of the homogeneous Besov spaces Ḃ n p p,1 × Ḃ n p −1 p,1 for all 1 ≤ p < 2n. However, if the data is in a larger scaling invariant class such as p > 2n, then the system is not well-posed. In this paper, we demonstrate that for the critical case p = 2n the system is ill-posed by showing that a sequence of initial data is constructed to show discontinuity of the solution map in the critical space. Our result indicates that the well-posedness results due to Danchin [10] and Haspot [18] are indeed sharp in the framework of the homogeneous Besov spaces.
考虑临界Besov空间中的可压缩Navier-Stokes系统。已知系统在齐次Besov空间的缩放半不变空间中是(半)适定的Ḃ n p p,1 × Ḃ n p−1p,1对于所有1≤p < 2n。然而，如果数据是一个更大的尺度不变类，如p bbb20n，那么系统不是适定的。在本文中，我们通过构造一个初始数据序列来表示解映射在临界空间中的不连续，证明了对于临界情况p = 2n，系统是不适定的。我们的结果表明，在齐次Besov空间的框架下，由Danchin[10]和Haspot[18]引起的适定性结果确实是尖锐的。
{"title":"Ill-posedness for the compressible Navier–Stokes equations under barotropic condition in limiting Besov spaces","authors":"T. Iwabuchi, T. Ogawa","doi":"10.2969/JMSJ/81598159","DOIUrl":"https://doi.org/10.2969/JMSJ/81598159","url":null,"abstract":"We consider the compressible Navier–Stokes system in the critical Besov spaces. It is known that the system is (semi-)well-posed in the scaling semi-invariant spaces of the homogeneous Besov spaces Ḃ n p p,1 × Ḃ n p −1 p,1 for all 1 ≤ p < 2n. However, if the data is in a larger scaling invariant class such as p > 2n, then the system is not well-posed. In this paper, we demonstrate that for the critical case p = 2n the system is ill-posed by showing that a sequence of initial data is constructed to show discontinuity of the solution map in the critical space. Our result indicates that the well-posedness results due to Danchin [10] and Haspot [18] are indeed sharp in the framework of the homogeneous Besov spaces.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":"-1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69574279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let M be a non-doubling parabolic manifold with ends and L a non-negative self-adjoint operator on L2(M) which satisfies a suitable heat kernel upper bound named the upper bound of Gaussian type. These operators include the Schrödinger operators L = ∆ + V where ∆ is the Laplace-Beltrami operator and V is an arbitrary non-negative potential. This paper will investigate the behaviour of the Poisson semi-group kernels of L together with its time derivatives and then apply them to obtain the weak type (1, 1) estimate of the functional calculus of Laplace transform type of √ L which is defined by M( √ L)f(x) := ́∞ 0 [√ Le−t √ Lf(x) ] m(t)dt where m(t) is a bounded function on [0,∞). In the setting of our study, both doubling condition of the measure on M and the smoothness of the operators’ kernels are missing. The purely imaginary power Lis, s ∈ R, is a special case of our result and an example of weak type (1, 1) estimates of a singular integral with non-smooth kernels on non-doubling spaces.
{"title":"Functional calculus of Laplace transform type on non-doubling parabolic manifolds with ends","authors":"Hong Chuong Doan","doi":"10.2969/JMSJ/83348334","DOIUrl":"https://doi.org/10.2969/JMSJ/83348334","url":null,"abstract":"Let M be a non-doubling parabolic manifold with ends and L a non-negative self-adjoint operator on L2(M) which satisfies a suitable heat kernel upper bound named the upper bound of Gaussian type. These operators include the Schrödinger operators L = ∆ + V where ∆ is the Laplace-Beltrami operator and V is an arbitrary non-negative potential. This paper will investigate the behaviour of the Poisson semi-group kernels of L together with its time derivatives and then apply them to obtain the weak type (1, 1) estimate of the functional calculus of Laplace transform type of √ L which is defined by M( √ L)f(x) := ́∞ 0 [√ Le−t √ Lf(x) ] m(t)dt where m(t) is a bounded function on [0,∞). In the setting of our study, both doubling condition of the measure on M and the smoothness of the operators’ kernels are missing. The purely imaginary power Lis, s ∈ R, is a special case of our result and an example of weak type (1, 1) estimates of a singular integral with non-smooth kernels on non-doubling spaces.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":"-1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69574353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The notion of coupled Kähler-Einstein metrics was introduced recently by Hultgren-WittNyström. In this paper we discuss deformation of a coupled KählerEinstein metric on a Fano manifold. We obtain a necessary and sufficient condition for a coupled Kähler-Einstein metric to be deformed to another coupled Kähler-Einstein metric for a Fano manifold admitting non-trivial holomorphic vector fields. In addition we also discuss deformation for a coupled Käher-Einstein metric on a Fano manifold when the complex structure varies.
{"title":"Deformation for coupled Kähler–Einstein metrics","authors":"Satoshi X. Nakamura","doi":"10.2969/JMSJ/84408440","DOIUrl":"https://doi.org/10.2969/JMSJ/84408440","url":null,"abstract":"The notion of coupled Kähler-Einstein metrics was introduced recently by Hultgren-WittNyström. In this paper we discuss deformation of a coupled KählerEinstein metric on a Fano manifold. We obtain a necessary and sufficient condition for a coupled Kähler-Einstein metric to be deformed to another coupled Kähler-Einstein metric for a Fano manifold admitting non-trivial holomorphic vector fields. In addition we also discuss deformation for a coupled Käher-Einstein metric on a Fano manifold when the complex structure varies.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47300822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show finiteness results on torsion points of commutative algebraic groups over a p-adic field K with values in various algebraic extensions L/K of infinite degree. We mainly study the following cases: (1) L is an abelian extension which is a splitting field of a crystalline character (such as a Lubin-Tate extension). (2) L is a certain iterate extension of K associated with Lubin-Tate formal groups, which is familiar with Kummer theory.
{"title":"Torsion of algebraic groups and iterate extensions associated with Lubin–Tate formal groups","authors":"Yoshiyasu Ozeki","doi":"10.2969/jmsj/87238723","DOIUrl":"https://doi.org/10.2969/jmsj/87238723","url":null,"abstract":"We show finiteness results on torsion points of commutative algebraic groups over a p-adic field K with values in various algebraic extensions L/K of infinite degree. We mainly study the following cases: (1) L is an abelian extension which is a splitting field of a crystalline character (such as a Lubin-Tate extension). (2) L is a certain iterate extension of K associated with Lubin-Tate formal groups, which is familiar with Kummer theory.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43446645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Ohno relation for multiple zeta values can be formulated as saying that a certain operator, defined for indices, is invariant under taking duals. In this paper, we generalize the Ohno relation to regularized multiple zeta values by showing that, although the suitably generalized operator is not invariant under taking duals, the relation between its values at an index and at its dual index can be written explicitly in terms of the gamma function.
{"title":"Ohno relation for regularized multiple zeta values","authors":"M. Hirose, H. Murahara, Shingo Saito","doi":"10.2969/jmsj/89088908","DOIUrl":"https://doi.org/10.2969/jmsj/89088908","url":null,"abstract":"The Ohno relation for multiple zeta values can be formulated as saying that a certain operator, defined for indices, is invariant under taking duals. In this paper, we generalize the Ohno relation to regularized multiple zeta values by showing that, although the suitably generalized operator is not invariant under taking duals, the relation between its values at an index and at its dual index can be written explicitly in terms of the gamma function.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47935591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a homogenization problem for symmetric jumpdiffusion processes by using the Mosco convergence and the two-scale convergence of the corresponding Dirichlet forms. Moreover, we show the weak convergence of the processes.
{"title":"Homogenization of symmetric Dirichlet forms","authors":"M. Tomisaki, T. Uemura","doi":"10.2969/JMSJ/85268526","DOIUrl":"https://doi.org/10.2969/JMSJ/85268526","url":null,"abstract":"We consider a homogenization problem for symmetric jumpdiffusion processes by using the Mosco convergence and the two-scale convergence of the corresponding Dirichlet forms. Moreover, we show the weak convergence of the processes.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":"-1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41317674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In the present article, we obtain an optimal support function of weighted L 2 integrations on superlevel sets of psh weights, which implies the strong openness property of multiplier ideal sheaves.
． 在psh权的超水平集上，我们得到了加权l2积分的最优支持函数，它表明了乘法器理想轮的强开性。
{"title":"An optimal support function related to the strong openness conjecture","authors":"Q. Guan, Zheng Yuan","doi":"10.2969/jmsj/87048704","DOIUrl":"https://doi.org/10.2969/jmsj/87048704","url":null,"abstract":". In the present article, we obtain an optimal support function of weighted L 2 integrations on superlevel sets of psh weights, which implies the strong openness property of multiplier ideal sheaves.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45225667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the distribution of values of automorphic L-functions in a family of holomorphic cusp forms with prime level. We prove an asymptotic formula for a certain density function closely related to this value-distribution. The formula is applied to estimate large values of L-functions.
{"title":"Large deviations for values of $L$-functions attached to cusp forms in the level aspect","authors":"Masahiro Mine","doi":"10.2969/jmsj/88888888","DOIUrl":"https://doi.org/10.2969/jmsj/88888888","url":null,"abstract":"We study the distribution of values of automorphic L-functions in a family of holomorphic cusp forms with prime level. We prove an asymptotic formula for a certain density function closely related to this value-distribution. The formula is applied to estimate large values of L-functions.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42146861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We revisit the localization formulas of cohomology intersection numbers associated to a logarithmic connection. The main contribution of this paper is threefold: we prove the localization formula of the cohomology intersection number of logarithmic forms in terms of residue of a connection; we prove that the leading term of the Laurent expansion of the cohomology intersection number is Grothendieck residue when the connection is hypergeometric; and we prove that the leading term of stringy integral discussed by Arkani-Hamed, He and Lam is nothing but the self-cohomology intersection number of the canonical form.
{"title":"Localization formulas of cohomology intersection numbers","authors":"Saiei-Jaeyeong Matsubara-Heo","doi":"10.2969/jmsj/87738773","DOIUrl":"https://doi.org/10.2969/jmsj/87738773","url":null,"abstract":"We revisit the localization formulas of cohomology intersection numbers associated to a logarithmic connection. The main contribution of this paper is threefold: we prove the localization formula of the cohomology intersection number of logarithmic forms in terms of residue of a connection; we prove that the leading term of the Laurent expansion of the cohomology intersection number is Grothendieck residue when the connection is hypergeometric; and we prove that the leading term of stringy integral discussed by Arkani-Hamed, He and Lam is nothing but the self-cohomology intersection number of the canonical form.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49506370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let (X, L) denote a quasi-polarized manifold of dimension n ≥ 5 defined over the field of complex numbers such that the canonical line bundle KX of X is numerically equivalent to zero. In this paper, we consider the dimension of the global sections of KX + mL in this case, and we prove that h(KX + mL) > 0 for every positive integer m with m ≥ n − 3. In particular, a Beltrametti-Sommese conjecture is true for quasi-polarized manifolds with numerically trivial canonical divisors.
{"title":"On the positivity of the dimension of the global sections of\u0000 adjoint bundle for quasi-polarized manifold with numerically trivial canonical bundle","authors":"Y. Fukuma","doi":"10.2969/JMSJ/84588458","DOIUrl":"https://doi.org/10.2969/JMSJ/84588458","url":null,"abstract":"Let (X, L) denote a quasi-polarized manifold of dimension n ≥ 5 defined over the field of complex numbers such that the canonical line bundle KX of X is numerically equivalent to zero. In this paper, we consider the dimension of the global sections of KX + mL in this case, and we prove that h(KX + mL) > 0 for every positive integer m with m ≥ n − 3. In particular, a Beltrametti-Sommese conjecture is true for quasi-polarized manifolds with numerically trivial canonical divisors.","PeriodicalId":49988,"journal":{"name":"Journal of the Mathematical Society of Japan","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43339968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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