In this paper we present a measurable version of the Smale’s spectral decomposition theorem for homeomorphisms on compact metric spaces. More precisely, we prove that if a homeomorphism f on a compact metric space X is invariantly measure expanding on its chain recurrent set CR(f) and has the eventually shadowing property on CR(f), then f has the spectral decomposition. Moreover we show that f is invariantly measure expanding on X if and only if its restriction on CR(f) is invariantly measure expanding. Using this, we characterize the measure expanding diffeomorphisms on compact smooth manifolds via the notion of Ω-stability.
{"title":"EXPANDING MEASURES FOR HOMEOMORPHISMS WITH EVENTUALLY SHADOWING PROPERTY","authors":"M. Dong, Keonhee Lee, Ngocthach Nguyen","doi":"10.4134/JKMS.J190453","DOIUrl":"https://doi.org/10.4134/JKMS.J190453","url":null,"abstract":"In this paper we present a measurable version of the Smale’s spectral decomposition theorem for homeomorphisms on compact metric spaces. More precisely, we prove that if a homeomorphism f on a compact metric space X is invariantly measure expanding on its chain recurrent set CR(f) and has the eventually shadowing property on CR(f), then f has the spectral decomposition. Moreover we show that f is invariantly measure expanding on X if and only if its restriction on CR(f) is invariantly measure expanding. Using this, we characterize the measure expanding diffeomorphisms on compact smooth manifolds via the notion of Ω-stability.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"935-955"},"PeriodicalIF":0.6,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45667563","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 this paper, we compute the Hankel matrix representation of the Hankel operator on the Hardy space of a general bounded domain with respect to special orthonormal bases for the Hardy space and its orthogonal complement. Moreover we obtain the compact form of the Hankel matrix for the unit disc case with respect to these bases. One can see that the Hankel matrix generated by this computation turns out to be a generalization of the case of the unit disc from the single simply connected domain to multiply connected domains with much diversities of bases.
{"title":"Computation of Hankel matrices in terms of classical kernel functions in potential theory","authors":"Young-Bok Chung","doi":"10.4134/JKMS.J190487","DOIUrl":"https://doi.org/10.4134/JKMS.J190487","url":null,"abstract":"In this paper, we compute the Hankel matrix representation of the Hankel operator on the Hardy space of a general bounded domain with respect to special orthonormal bases for the Hardy space and its orthogonal complement. Moreover we obtain the compact form of the Hankel matrix for the unit disc case with respect to these bases. One can see that the Hankel matrix generated by this computation turns out to be a generalization of the case of the unit disc from the single simply connected domain to multiply connected domains with much diversities of bases.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"973-986"},"PeriodicalIF":0.6,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44973840","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}
To extend the well-known extremal characterization of the geometric mean of two n x n positive definite matrices A and B, we solve the following problem:
为了推广众所周知的两个n × n正定矩阵A和B的几何平均值的极值性质，我们解决了以下问题:
{"title":"EXTENSION OF BLOCK MATRIX REPRESENTATION OF THE GEOMETRIC MEAN","authors":"Hana Choi, Hayoung Choi, Sejong Kim, Hosoo Lee","doi":"10.4134/JKMS.J190272","DOIUrl":"https://doi.org/10.4134/JKMS.J190272","url":null,"abstract":"To extend the well-known extremal characterization of the geometric mean of two n x n positive definite matrices A and B, we solve the following problem:","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"641-653"},"PeriodicalIF":0.6,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47319567","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}
{"title":"GEOMETRY OF ISOPARAMETRIC NULL HYPERSURFACES OF LORENTZIAN MANIFOLDS","authors":"S. Ssekajja","doi":"10.4134/JKMS.J190001","DOIUrl":"https://doi.org/10.4134/JKMS.J190001","url":null,"abstract":"","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"195-213"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70511174","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}
Following Matsumoto’s definition of continuous orbit equivalence for one-sided subshifts of finite type, we introduce the notion of orbit equivalence to canonically associated dynamical systems, called the limit dynamical systems, of self-similar groups. We show that the limit dynamical systems of two self-similar groups are orbit equivalent if and only if their associated Deaconu groupoids are isomorphic as topological groupoids. We also show that the equivalence class of Cuntz-Pimsner groupoids and the stably isomorphism class of Cuntz-Pimsner algebras of self-similar groups are invariants for orbit equivalence of limit dynamical systems.
{"title":"ORBIT EQUIVALENCE ON SELF-SIMILAR GROUPS AND THEIR C*-ALGEBRAS","authors":"Inhyeop Yi","doi":"10.4134/JKMS.J190090","DOIUrl":"https://doi.org/10.4134/JKMS.J190090","url":null,"abstract":"Following Matsumoto’s definition of continuous orbit equivalence for one-sided subshifts of finite type, we introduce the notion of orbit equivalence to canonically associated dynamical systems, called the limit dynamical systems, of self-similar groups. We show that the limit dynamical systems of two self-similar groups are orbit equivalent if and only if their associated Deaconu groupoids are isomorphic as topological groupoids. We also show that the equivalence class of Cuntz-Pimsner groupoids and the stably isomorphism class of Cuntz-Pimsner algebras of self-similar groups are invariants for orbit equivalence of limit dynamical systems.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"27 1","pages":"383-399"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70511479","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 Λ be an Artin algebra and mod Λ the category of finitely generated right Λ-modules. We prove that the radical layer length of Λ is an upper bound for the radical layer length of mod Λ. We give an upper bound for the extension dimension of mod Λ in terms of the injective dimension of a certain class of simple right Λ-modules and the radical layer length of DΛ.
{"title":"On the extension dimension of module categories","authors":"Yeyang Peng, Tiwei Zhao","doi":"10.4134/JKMS.J190681","DOIUrl":"https://doi.org/10.4134/JKMS.J190681","url":null,"abstract":"Let Λ be an Artin algebra and mod Λ the category of finitely generated right Λ-modules. We prove that the radical layer length of Λ is an upper bound for the radical layer length of mod Λ. We give an upper bound for the extension dimension of mod Λ in terms of the injective dimension of a certain class of simple right Λ-modules and the radical layer length of DΛ.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"1389-1406"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70512136","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 L K denote the hypothetical automorphic Langlands gr- oup of a number field K . In our recent study, we briefly introduced a certain unconditional non-commutative topological group W A ϕ K , called the Weil-Arthur id`ele group of K , which, assuming the existence of L K , comes equipped with a natural topological group homomorphism NR ϕ Langlands K : W A ϕ K → L K that we called the “Langlands form” of the global nonabelian norm-residue symbol of K . In this work, we present a detailed construction of W A ϕ K and NR ϕ Langlands K : W A ϕ K → L K , and discuss their basic properties.
． 设lk表示数域K的假设自同构朗兰群。在我们最近的研究中，我们简要地介绍了一个无条件的非交换拓扑群W a φ K，称为K的Weil-Arthur id 'ele群，它在假设L K存在的情况下，具有一个自然的拓扑群同态NR φ Langlands K: W a φ K→L K，我们称之为K的全局非阿贝尔范数-残差符号的“Langlands形式”。在这项工作中，我们给出了W a φ K和NR φ朗兰兹K的详细构造:W a φ K→L K，并讨论了它们的基本性质。
{"title":"ON A GROUP CLOSELY RELATED WITH THE AUTOMORPHIC LANGLANDS GROUP","authors":"K. I. Ikeda","doi":"10.4134/JKMS.J180475","DOIUrl":"https://doi.org/10.4134/JKMS.J180475","url":null,"abstract":". Let L K denote the hypothetical automorphic Langlands gr- oup of a number field K . In our recent study, we briefly introduced a certain unconditional non-commutative topological group W A ϕ K , called the Weil-Arthur id`ele group of K , which, assuming the existence of L K , comes equipped with a natural topological group homomorphism NR ϕ Langlands K : W A ϕ K → L K that we called the “Langlands form” of the global nonabelian norm-residue symbol of K . In this work, we present a detailed construction of W A ϕ K and NR ϕ Langlands K : W A ϕ K → L K , and discuss their basic properties.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"21-59"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70510461","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}
. For a nonautonomous dynamics defined by a sequence of bounded linear operators on a Banach space, we give a characterization of the existence of an exponential dichotomy with respect to a sequence of norms in terms of the invertibility of a certain linear operator be- tween general admissible spaces. This notion of an exponential dichotomy contains as very special cases the notions of uniform, nonuniform and tempered exponential dichotomies. As applications, we detail the conse-quences of our results for the class of tempered exponential dichotomies, which are ubiquitous in the context of ergodic theory, and we show that the notion of an exponential dichotomy under sufficiently small parame- terized perturbations persists and that their stable and unstable spaces are as regular as the perturbation.
{"title":"Characterization of tempered exponential dichotomies","authors":"L. Barreira, J. Rijo, C. Valls","doi":"10.4134/JKMS.J180880","DOIUrl":"https://doi.org/10.4134/JKMS.J180880","url":null,"abstract":". For a nonautonomous dynamics defined by a sequence of bounded linear operators on a Banach space, we give a characterization of the existence of an exponential dichotomy with respect to a sequence of norms in terms of the invertibility of a certain linear operator be- tween general admissible spaces. This notion of an exponential dichotomy contains as very special cases the notions of uniform, nonuniform and tempered exponential dichotomies. As applications, we detail the conse-quences of our results for the class of tempered exponential dichotomies, which are ubiquitous in the context of ergodic theory, and we show that the notion of an exponential dichotomy under sufficiently small parame- terized perturbations persists and that their stable and unstable spaces are as regular as the perturbation.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"171-194"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70511112","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 obtain infinitely many solutions for a class of fractional Schr¨odinger equation, where the nonlinearity is superquadratic or involves a combination of superquadratic and subquadratic terms at infinity. By using some weaker conditions, our results extend and improve some ex- isting results in the literature.
{"title":"INFINITELY MANY SOLUTIONS FOR FRACTIONAL SCHRÖDINGER EQUATION WITH SUPERQUADRATIC CONDITIONS OR COMBINED NONLINEARITIES","authors":"M. Timoumi","doi":"10.4134/JKMS.J190368","DOIUrl":"https://doi.org/10.4134/JKMS.J190368","url":null,"abstract":". We obtain infinitely many solutions for a class of fractional Schr¨odinger equation, where the nonlinearity is superquadratic or involves a combination of superquadratic and subquadratic terms at infinity. By using some weaker conditions, our results extend and improve some ex- isting results in the literature.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"825-844"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70511733","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 are concerned with the following elliptic equations: { (−∆)pu = λf(x, u) in Ω, u = 0 on RNΩ, where λ are real parameters, (−∆)p is the fractional p-Laplacian operator, 0 < s < 1 < p < +∞, sp < N , and f : Ω × R → R satisfies a Carathéodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters λ for which our problem admits at least one or two nontrivial weak solutions when the nonlinearity f has the subcritical growth condition. In addition, under adequate conditions, we establish an apriori estimate in L∞(Ω) of any possible weak solution by applying the bootstrap argument.
我们研究了以下椭圆方程:{(−∆)pu = λf(x, u)在Ω上，u = 0在RNΩ上，其中λ为实参数，(−∆)p为分数阶p-拉普拉斯算子，0 < s < 1 < p < +∞，sp < N, f: Ω × R→R满足carathacriodory条件。利用抽象临界点结果，建立了当非线性函数f具有亚临界生长条件时，至少存在一个或两个非平凡弱解的参数λ正区间的估计。此外，在适当的条件下，我们利用自举参数在L∞(Ω)上建立了任意可能弱解的先验估计。
{"title":"Existence, multiplicity and regularity of solutions for the fractional $p$-Laplacian equation","authors":"Yun-Ho Kim","doi":"10.4134/JKMS.J190693","DOIUrl":"https://doi.org/10.4134/JKMS.J190693","url":null,"abstract":"We are concerned with the following elliptic equations: { (−∆)pu = λf(x, u) in Ω, u = 0 on RNΩ, where λ are real parameters, (−∆)p is the fractional p-Laplacian operator, 0 < s < 1 < p < +∞, sp < N , and f : Ω × R → R satisfies a Carathéodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters λ for which our problem admits at least one or two nontrivial weak solutions when the nonlinearity f has the subcritical growth condition. In addition, under adequate conditions, we establish an apriori estimate in L∞(Ω) of any possible weak solution by applying the bootstrap argument.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"57 1","pages":"1451-1470"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70513218","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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