{"title":"A note on quasisymmetric homeomorphisms","authors":"Shuan Tang, Pengcheng Wu","doi":"10.5186/aasfm.2020.4502","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4502","url":null,"abstract":"","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85379926","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}
is called the transfinite diameter (or logarithmic capacity) of E. It is known that a (closed or open) disc with radius R has transfinite diameter R, whereas an interval of lenght I has transfinite diameter I/4. In [7], Fekete has shown that every compact set E satisfying τ(E) < 1 contains only finitely many full sets of conjugate algebraic integers over Q. In particular, this result can be applied to every closed disc whose radius is smaller than 1 and to every real interval whose length is smaller than 4. In the opposite direction, Fekete and Szegö [8] proved that if E is a compact set which is stable under complex conjugation and satisfies τ(E) ≥ 1, then its every complex neighborhood F (so that E ⊂ F and F is an open set) contains infinitely many sets of conjugate algebraic integers. Furthermore, by the results of Robinson [15] and Ennola [4], every real interval of length strictly greater than 4 also contains infinitely many sets of conjugate algebraic integers. In [18], Zaïmi gave a lower bound for the length of a real interval containing an algebraic integer of degree d and its conjugates. His result asserts that the length I of such an interval should be at least 4 − φ(d), where φ(d) is some explicit positive function which tends to zero as d → ∞. (For instance, one can take φ(d) = (c log d)/d with some c > 0. Similar bound also follows from an earlier result of Schur [17].) On the other hand, the author has shown that, for infinitely many d ∈ N, every real interval of length 4+4(log log d)/ log d contains an algebraic integer of degree d and its conjugates (see [2] and [3]). It is not known whether there is an interval [t, t+ 4] with some t ∈ RZ containing infinitely many full sets of algebraic integers. For t ∈ Z, one can simply take infinitely many algebraic integers of the form t+2 cos(πr)+2, where r ∈ Q. By Kronecker’s theorem [13], these are the only such numbers in [t, t+4] if t ∈ Z.
{"title":"Small discs containing conjugate algebraic integers","authors":"A. Dubickas","doi":"10.5186/aasfm.2020.4524","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4524","url":null,"abstract":"is called the transfinite diameter (or logarithmic capacity) of E. It is known that a (closed or open) disc with radius R has transfinite diameter R, whereas an interval of lenght I has transfinite diameter I/4. In [7], Fekete has shown that every compact set E satisfying τ(E) < 1 contains only finitely many full sets of conjugate algebraic integers over Q. In particular, this result can be applied to every closed disc whose radius is smaller than 1 and to every real interval whose length is smaller than 4. In the opposite direction, Fekete and Szegö [8] proved that if E is a compact set which is stable under complex conjugation and satisfies τ(E) ≥ 1, then its every complex neighborhood F (so that E ⊂ F and F is an open set) contains infinitely many sets of conjugate algebraic integers. Furthermore, by the results of Robinson [15] and Ennola [4], every real interval of length strictly greater than 4 also contains infinitely many sets of conjugate algebraic integers. In [18], Zaïmi gave a lower bound for the length of a real interval containing an algebraic integer of degree d and its conjugates. His result asserts that the length I of such an interval should be at least 4 − φ(d), where φ(d) is some explicit positive function which tends to zero as d → ∞. (For instance, one can take φ(d) = (c log d)/d with some c > 0. Similar bound also follows from an earlier result of Schur [17].) On the other hand, the author has shown that, for infinitely many d ∈ N, every real interval of length 4+4(log log d)/ log d contains an algebraic integer of degree d and its conjugates (see [2] and [3]). It is not known whether there is an interval [t, t+ 4] with some t ∈ RZ containing infinitely many full sets of algebraic integers. For t ∈ Z, one can simply take infinitely many algebraic integers of the form t+2 cos(πr)+2, where r ∈ Q. By Kronecker’s theorem [13], these are the only such numbers in [t, t+4] if t ∈ Z.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77079898","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":"Absolute logics","authors":"Jyrki Akkanen","doi":"10.5186/aasfmd.1995.100","DOIUrl":"https://doi.org/10.5186/aasfmd.1995.100","url":null,"abstract":"","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74619467","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}
where ǫ > 0 is a parameter, V : R → R is a bounded continuous function. Under some mild conditions on V , Luo, Peng, Wang and the last named author of the present paper [22] proved the existence of multi-peak solutions to (1.1). As a continuation of the work [22], this paper is devoted to establish a local uniqueness result for the multi-peak solutions obtained there. For physical background for equation (1.1), the readers are referred to Luo et al. [22] and the references therein. To be precise, we first give the definition of k-peak solutions of Eq. (1.1) as usual. Definition 1.1. Let k ∈ N, bj ∈ R , 1 ≤ j ≤ k. We say that uǫ ∈ H (R) is a k-peak solution of (1.1) concentrated at {b1, b2, · · · , bk}, if (i) uǫ has k local maximum points x j ǫ ∈ R , j = 1, 2, . . . , k, satisfying xǫ → bj as ǫ→ 0 for each j; (ii) For any given τ > 0, there exists R ≫ 1, such that |uǫ(x)| ≤ τ for x ∈ R ∪j=1 BRǫ(x j ǫ);
{"title":"Local uniqueness of multi-peak solutions to a class of Kirchhoff equations","authors":"Gongbao Li, Yahui Niu, Chang-Lin Xiang","doi":"10.5186/aasfm.2020.4503","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4503","url":null,"abstract":"where ǫ > 0 is a parameter, V : R → R is a bounded continuous function. Under some mild conditions on V , Luo, Peng, Wang and the last named author of the present paper [22] proved the existence of multi-peak solutions to (1.1). As a continuation of the work [22], this paper is devoted to establish a local uniqueness result for the multi-peak solutions obtained there. For physical background for equation (1.1), the readers are referred to Luo et al. [22] and the references therein. To be precise, we first give the definition of k-peak solutions of Eq. (1.1) as usual. Definition 1.1. Let k ∈ N, bj ∈ R , 1 ≤ j ≤ k. We say that uǫ ∈ H (R) is a k-peak solution of (1.1) concentrated at {b1, b2, · · · , bk}, if (i) uǫ has k local maximum points x j ǫ ∈ R , j = 1, 2, . . . , k, satisfying xǫ → bj as ǫ→ 0 for each j; (ii) For any given τ > 0, there exists R ≫ 1, such that |uǫ(x)| ≤ τ for x ∈ R ∪j=1 BRǫ(x j ǫ);","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91176161","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}
Abstract. We discuss the boundedness of linear and sublinear operators in two types of weighted local Morrey spaces. One is defined by Natasha Samko in 2008. The other is defined by Yasuo Komori-Furuya and Satoru Shirai in 2009. We characterize the class of weights for which the Hardy–Littlewood maximal operator is bounded. Under a certain integral condition it turns out that the singular integral operators are bounded if and only if the Hardy–Littlewood maximal operator is bounded. As an application of the characterization, the power weight function | · | is considered. The condition on α for which the Hardy–Littlewood maximal operator is bounded can be described completely.
{"title":"Weighted local Morrey spaces","authors":"Shohei Nakamura, Y. Sawano, Hitoshi Tanaka","doi":"10.5186/aasfm.2020.4504","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4504","url":null,"abstract":"Abstract. We discuss the boundedness of linear and sublinear operators in two types of weighted local Morrey spaces. One is defined by Natasha Samko in 2008. The other is defined by Yasuo Komori-Furuya and Satoru Shirai in 2009. We characterize the class of weights for which the Hardy–Littlewood maximal operator is bounded. Under a certain integral condition it turns out that the singular integral operators are bounded if and only if the Hardy–Littlewood maximal operator is bounded. As an application of the characterization, the power weight function | · | is considered. The condition on α for which the Hardy–Littlewood maximal operator is bounded can be described completely.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90845404","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 , d, μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors establish a complete real-variable theory of Musielak–Orlicz Hardy spaces on (X , d, μ). To be precise, the authors first introduce the atomic Musielak–Orlicz Hardy space H at (X ) and then establish its various maximal function characterizations. The authors also investigate the Littlewood–Paley characterizations of H at (X ) via Lusin area functions, Littlewood– Paley g-functions and Littlewood–Paley g∗ λ-functions. The authors further obtain the finite atomic characterization of H at (X ) and its improved version in case q < ∞, and their applications to criteria of the boundedness of sublinear operators from H at (X ) to a quasi-Banach space, which are also applied to the boundedness of Calderón–Zygmund operators. Moreover, the authors find the dual space of H at (X ), namely, the Musielak–Orlicz BMO space BMO(X ), present its several equivalent characterizations, and apply it to establish a new characterization of the set of pointwise multipliers for the space BMO(X ). The main novelty of this article is that, throughout the article, except the last section, μ is not assumed to satisfy the reverse doubling condition.
设(X, d， μ)是Coifman和Weiss意义上的齐次型空间。本文建立了(X, d， μ)上的Musielak-Orlicz Hardy空间的完全实变量理论。准确地说，作者首先引入原子的Musielak-Orlicz Hardy空间H at (X)，然后建立了它的各种极大函数表征。作者还利用Lusin面积函数、Littlewood - Paley g-函数和Littlewood - Paley g * λ-函数研究了H (X)的Littlewood - Paley表征。进一步得到了H at (X)的有限原子刻划及其在q <∞情况下的改进刻划，并将其应用于从H at (X)到拟banach空间的次线性算子的有界性判据，同时也应用于Calderón-Zygmund算子的有界性判据。此外，作者找到了H at (X)的对偶空间，即Musielak-Orlicz BMO空间BMO(X)，给出了它的几个等价表征，并应用它建立了空间BMO(X)的点向乘子集的一个新的表征。本文的主要新颖之处在于，在整篇文章中，除了最后一节之外，都没有假设μ满足反向加倍条件。
{"title":"Real-variable characterizations of Musielak–Orlicz Hardy spaces on spaces of homogeneous type","authors":"Xing Fu, T. Ma, Dachun Yang","doi":"10.5186/aasfm.2020.4519","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4519","url":null,"abstract":"Let (X , d, μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the authors establish a complete real-variable theory of Musielak–Orlicz Hardy spaces on (X , d, μ). To be precise, the authors first introduce the atomic Musielak–Orlicz Hardy space H at (X ) and then establish its various maximal function characterizations. The authors also investigate the Littlewood–Paley characterizations of H at (X ) via Lusin area functions, Littlewood– Paley g-functions and Littlewood–Paley g∗ λ-functions. The authors further obtain the finite atomic characterization of H at (X ) and its improved version in case q < ∞, and their applications to criteria of the boundedness of sublinear operators from H at (X ) to a quasi-Banach space, which are also applied to the boundedness of Calderón–Zygmund operators. Moreover, the authors find the dual space of H at (X ), namely, the Musielak–Orlicz BMO space BMO(X ), present its several equivalent characterizations, and apply it to establish a new characterization of the set of pointwise multipliers for the space BMO(X ). The main novelty of this article is that, throughout the article, except the last section, μ is not assumed to satisfy the reverse doubling condition.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81307098","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":"On mappings whose inverses satisfy the Poletsky inequality","authors":"E. Sevost’yanov, S. Skvortsov","doi":"10.5186/aasfm.2020.4520","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4520","url":null,"abstract":"","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72865186","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":"Boundary growth of Sobolev functions for double phase functionals","authors":"Y. Mizuta, T. Shimomura","doi":"10.5186/aasfm.2020.4510","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4510","url":null,"abstract":"","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84431805","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 continue our study of Sobolev spaces on locally compact abelian groups. In this paper we mainly focus on the case of metrizable groups. We show the density of the Bruhat–Schwartz space in Sobolev space. We prove the trace theorem on the cartesian product of topological groups. The comparison of Sobolev and fractional Sobolev spaces are given. In particular, it is proved that in the case of any abelian connected Lie group Sobolev and fractional Sobolev spaces coincide. Most of the theorems are illustrated by p-adic groups.
{"title":"A second look of Sobolev spaces on metrizable groups","authors":"P. Górka, Tomasz Kostrzewa","doi":"10.5186/aasfm.2020.4507","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4507","url":null,"abstract":"We continue our study of Sobolev spaces on locally compact abelian groups. In this paper we mainly focus on the case of metrizable groups. We show the density of the Bruhat–Schwartz space in Sobolev space. We prove the trace theorem on the cartesian product of topological groups. The comparison of Sobolev and fractional Sobolev spaces are given. In particular, it is proved that in the case of any abelian connected Lie group Sobolev and fractional Sobolev spaces coincide. Most of the theorems are illustrated by p-adic groups.","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74208790","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":"Higher integrability of minimizers of degenerate functionals in Carnot–Carathéodory spaces","authors":"Patrizia Di Gironimo, F. Giannetti","doi":"10.5186/aasfm.2020.4509","DOIUrl":"https://doi.org/10.5186/aasfm.2020.4509","url":null,"abstract":"","PeriodicalId":50787,"journal":{"name":"Annales Academiae Scientiarum Fennicae-Mathematica","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75638645","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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