Pub Date : 2019-10-18DOI: 10.4310/arkiv.2021.v59.n1.a3
Y. Bonthonneau, N. Raymond, San Vũ Ngọc
We establish a magnetic Agmon estimate in the case of a purely magnetic single non-degenerate well, by means of the Fourier-Bros-Iagolnitzer transform and microlocal exponential estimates a la Martinez-Sjostrand.
{"title":"Exponential localization in 2D pure magnetic wells","authors":"Y. Bonthonneau, N. Raymond, San Vũ Ngọc","doi":"10.4310/arkiv.2021.v59.n1.a3","DOIUrl":"https://doi.org/10.4310/arkiv.2021.v59.n1.a3","url":null,"abstract":"We establish a magnetic Agmon estimate in the case of a purely magnetic single non-degenerate well, by means of the Fourier-Bros-Iagolnitzer transform and microlocal exponential estimates a la Martinez-Sjostrand.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393664","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}
Pub Date : 2019-10-11DOI: 10.4310/arkiv.2020.v58.n2.a6
A. Knauf, N. Martynchuk
Classical Morse theory proceeds by considering sublevel sets $f^{-1}(-infty, a]$ of a Morse function $f: M to R$, where $M$ is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets $f^{-1}(a)$ and give conditions under which the topology of $f^{-1}(a)$ changes when passing a critical value. We show that for a general class of functions, which includes all exhaustive Morse function, the topology of a regular level $f^{-1}(a)$ always changes when passing a single critical point, unless the index of the critical point is half the dimension of the manifold $M$. When $f$ is a natural Hamiltonian on a cotangent bundle, we obtain more precise results in terms of the topology of the configuration space. (Counter-)examples and applications to celestial mechanics are also discussed.
经典莫尔斯理论首先考虑莫尔斯函数$f: M to R$的子水平集$f^{-1}(-infty, a]$，其中$M$是光滑的有限维流形。本文研究了水平集$f^{-1}(a)$的拓扑结构，并给出了当通过一个临界值时$f^{-1}(a)$拓扑结构发生变化的条件。我们证明了对于包含所有穷举莫尔斯函数的一般函数类，正则能级$f^{-1}(a)$的拓扑结构在经过单个临界点时总是变化的，除非临界点的索引是流形的一半维数$M$。当$f$是余切束上的自然哈密顿量时，我们就构型空间的拓扑得到了更精确的结果。还讨论了反例及其在天体力学中的应用。
{"title":"Topology change of level sets in Morse theory","authors":"A. Knauf, N. Martynchuk","doi":"10.4310/arkiv.2020.v58.n2.a6","DOIUrl":"https://doi.org/10.4310/arkiv.2020.v58.n2.a6","url":null,"abstract":"Classical Morse theory proceeds by considering sublevel sets $f^{-1}(-infty, a]$ of a Morse function $f: M to R$, where $M$ is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets $f^{-1}(a)$ and give conditions under which the topology of $f^{-1}(a)$ changes when passing a critical value. We show that for a general class of functions, which includes all exhaustive Morse function, the topology of a regular level $f^{-1}(a)$ always changes when passing a single critical point, unless the index of the critical point is half the dimension of the manifold $M$. When $f$ is a natural Hamiltonian on a cotangent bundle, we obtain more precise results in terms of the topology of the configuration space. (Counter-)examples and applications to celestial mechanics are also discussed.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393543","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}
Pub Date : 2019-10-01DOI: 10.4310/arkiv.2019.v57.n2.a4
A. Freitas, H. F. Lima, E. Lima, Márcio S. Santos
We study weakly trapped submanifolds of codimension two in a standard static spacetime. In this setting, we apply some generalized maximum principles in order to investigate the geometry of these trapped submanifolds. For instance, we establish sufficient conditions to guarantee that such a spacelike submanifold must be a hypersurface of the Riemannian base of the ambient spacetime. As a consequence, we prove that there do not exist n-dimensional compact (without boundary) trapped submanifolds immersed in an (n+2)-dimensional standard static spacetime. Such a nonexistence result was originally obtained for stationary spacetimes by Mars and Senovilla [20]. Furthermore, we also investigate parabolic weakly trapped submanifolds immersed in a standard static spacetime.
{"title":"Weakly trapped submanifolds in standard static spacetimes","authors":"A. Freitas, H. F. Lima, E. Lima, Márcio S. Santos","doi":"10.4310/arkiv.2019.v57.n2.a4","DOIUrl":"https://doi.org/10.4310/arkiv.2019.v57.n2.a4","url":null,"abstract":"We study weakly trapped submanifolds of codimension two in a standard static spacetime. In this setting, we apply some generalized maximum principles in order to investigate the geometry of these trapped submanifolds. For instance, we establish sufficient conditions to guarantee that such a spacelike submanifold must be a hypersurface of the Riemannian base of the ambient spacetime. As a consequence, we prove that there do not exist n-dimensional compact (without boundary) trapped submanifolds immersed in an (n+2)-dimensional standard static spacetime. Such a nonexistence result was originally obtained for stationary spacetimes by Mars and Senovilla [20]. Furthermore, we also investigate parabolic weakly trapped submanifolds immersed in a standard static spacetime.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393680","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}
Pub Date : 2019-10-01DOI: 10.4310/arkiv.2019.v57.n2.a11
J. Xiao, Yuan Zhou
{"title":"A reverse quasiconformal composition problem for $Q_alpha(mathbb{R}^n)$","authors":"J. Xiao, Yuan Zhou","doi":"10.4310/arkiv.2019.v57.n2.a11","DOIUrl":"https://doi.org/10.4310/arkiv.2019.v57.n2.a11","url":null,"abstract":"","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393638","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}
Pub Date : 2019-08-30DOI: 10.4310/arkiv.2020.v58.n2.a5
Christina Karafyllia
Let $psi $ be a conformal map on $mathbb{D}$ with $ psi left( 0 right)=0$ and let ${F_alpha }=left{ {z in mathbb{D}:left| {psi left( z right)} right| = alpha } right}$ for $alpha >0$. Denote by ${H^p}left( mathbb{D} right)$ the classical Hardy space with exponent $p>0$ and by ${tt h}left( psi right)$ the Hardy number of $psi$. Consider the limits $$ L:= lim_{alphato+infty}left( log omega_{mathbb D}(0,F_{alpha})^{-1}/log alpha right), ,, mu:= lim_{alphato+infty}left( d_{mathbb D}(0,F_{alpha})/logalpha right),$$ where $omega _mathbb{D}left( {0,{F_alpha }} right)$ denotes the harmonic measure at $0$ of $F_alpha $ and $d_mathbb{D} {left( {0,{F_alpha }} right)}$ denotes the hyperbolic distance between $0$ and $F_alpha$ in $mathbb{D}$. We study a problem posed by P. Poggi-Corradini. What is the relation between $L$, $mu$ and ${tt h}left( psi right)$? We also provide conditions for the existence of $L$ and $mu$ and for the equalities $L=mu={tt h}left( psi right)$. Poggi-Corradini proved that $psi notin {H^{mu}}left( mathbb{D} right)$ for a wide class of conformal maps $psi$. We present an example of $psi$ such that $psi in {H^mu {left( mathbb{D} right)} }$.
{"title":"On the Hardy number of a domain in terms of harmonic measure and hyperbolic distance","authors":"Christina Karafyllia","doi":"10.4310/arkiv.2020.v58.n2.a5","DOIUrl":"https://doi.org/10.4310/arkiv.2020.v58.n2.a5","url":null,"abstract":"Let $psi $ be a conformal map on $mathbb{D}$ with $ psi left( 0 right)=0$ and let ${F_alpha }=left{ {z in mathbb{D}:left| {psi left( z right)} right| = alpha } right}$ for $alpha >0$. Denote by ${H^p}left( mathbb{D} right)$ the classical Hardy space with exponent $p>0$ and by ${tt h}left( psi right)$ the Hardy number of $psi$. Consider the limits $$ L:= lim_{alphato+infty}left( log omega_{mathbb D}(0,F_{alpha})^{-1}/log alpha right), ,, mu:= lim_{alphato+infty}left( d_{mathbb D}(0,F_{alpha})/logalpha right),$$ where $omega _mathbb{D}left( {0,{F_alpha }} right)$ denotes the harmonic measure at $0$ of $F_alpha $ and $d_mathbb{D} {left( {0,{F_alpha }} right)}$ denotes the hyperbolic distance between $0$ and $F_alpha$ in $mathbb{D}$. We study a problem posed by P. Poggi-Corradini. What is the relation between $L$, $mu$ and ${tt h}left( psi right)$? We also provide conditions for the existence of $L$ and $mu$ and for the equalities $L=mu={tt h}left( psi right)$. Poggi-Corradini proved that $psi notin {H^{mu}}left( mathbb{D} right)$ for a wide class of conformal maps $psi$. We present an example of $psi$ such that $psi in {H^mu {left( mathbb{D} right)} }$.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393506","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}
Pub Date : 2019-08-22DOI: 10.4310/arkiv.2021.v59.n1.a2
P. Freitas, J. Lagac'e, Jordan Payette
Given a bounded Euclidean domain $Omega$, we consider the sequence of optimisers of the $k^{rm th}$ Laplacian eigenvalue within the family consisting of all possible disjoint unions of scaled copies of $Omega$ with fixed total volume. We show that this sequence encodes information yielding conditions for $Omega$ to satisfy Polya's conjecture with either Dirichlet or Neumann boundary conditions. This is an extension of a result by Colbois and El Soufi which applies only to the case where the family of domains consists of all bounded domains. Furthermore, we fully classify the different possible behaviours for such sequences, depending on whether Polya's conjecture holds for a given specific domain or not. This approach allows us to recover a stronger version of Polya's original results for tiling domains satisfying some dynamical billiard conditions, and a strenghtening of Urakawa's bound in terms of packing density.
{"title":"Optimal unions of scaled copies of domains and Pólya's conjecture","authors":"P. Freitas, J. Lagac'e, Jordan Payette","doi":"10.4310/arkiv.2021.v59.n1.a2","DOIUrl":"https://doi.org/10.4310/arkiv.2021.v59.n1.a2","url":null,"abstract":"Given a bounded Euclidean domain $Omega$, we consider the sequence of optimisers of the $k^{rm th}$ Laplacian eigenvalue within the family consisting of all possible disjoint unions of scaled copies of $Omega$ with fixed total volume. We show that this sequence encodes information yielding conditions for $Omega$ to satisfy Polya's conjecture with either Dirichlet or Neumann boundary conditions. This is an extension of a result by Colbois and El Soufi which applies only to the case where the family of domains consists of all bounded domains. Furthermore, we fully classify the different possible behaviours for such sequences, depending on whether Polya's conjecture holds for a given specific domain or not. This approach allows us to recover a stronger version of Polya's original results for tiling domains satisfying some dynamical billiard conditions, and a strenghtening of Urakawa's bound in terms of packing density.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393617","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}
Pub Date : 2019-08-20DOI: 10.4310/arkiv.2020.v58.n2.a2
J. Flesch, A. Predtetchinski, Ville Suomala
We introduce the so--called doubling metric on the collection of non--empty bounded open subsets of a metric space. Given a subset $U$ of a metric space $X$, the predecessor $U_{*}$ of $U$ is defined by doubling the radii of all open balls contained inside $U$, and taking their union. If $U$ is open, the predecessor of $U$ is an open set containing $U$. The directed doubling distance between $U$ and another subset $V$ is the number of times that the predecessor operation needs to be applied to $U$ to obtain a set that contains $V$. Finally, the doubling distance between $U$ and $V$ is the maximum of the directed distance between $U$ and $V$ and the directed distance between $V$ and $U$.
{"title":"The doubling metric and doubling measures","authors":"J. Flesch, A. Predtetchinski, Ville Suomala","doi":"10.4310/arkiv.2020.v58.n2.a2","DOIUrl":"https://doi.org/10.4310/arkiv.2020.v58.n2.a2","url":null,"abstract":"We introduce the so--called doubling metric on the collection of non--empty bounded open subsets of a metric space. Given a subset $U$ of a metric space $X$, the predecessor $U_{*}$ of $U$ is defined by doubling the radii of all open balls contained inside $U$, and taking their union. If $U$ is open, the predecessor of $U$ is an open set containing $U$. The directed doubling distance between $U$ and another subset $V$ is the number of times that the predecessor operation needs to be applied to $U$ to obtain a set that contains $V$. Finally, the doubling distance between $U$ and $V$ is the maximum of the directed distance between $U$ and $V$ and the directed distance between $V$ and $U$.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47580501","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}
Pub Date : 2019-07-26DOI: 10.4310/arkiv.2020.v58.n2.a10
E. Rydhe
We investigate so-called Laplace--Carleson embeddings for large exponents. In particular, we extend some results by Jacob, Partington, and Pott. We also discuss some related results for Sobolev- and Besov spaces, and mapping properties of the Fourier transform. These variants of the Hausdorff--Young theorem appear difficult to find in the literature. We conclude the paper with an example related to an open problem.
{"title":"On Laplace–Carleson embeddings, and $L^p$-mapping properties of the Fourier transform","authors":"E. Rydhe","doi":"10.4310/arkiv.2020.v58.n2.a10","DOIUrl":"https://doi.org/10.4310/arkiv.2020.v58.n2.a10","url":null,"abstract":"We investigate so-called Laplace--Carleson embeddings for large exponents. In particular, we extend some results by Jacob, Partington, and Pott. We also discuss some related results for Sobolev- and Besov spaces, and mapping properties of the Fourier transform. These variants of the Hausdorff--Young theorem appear difficult to find in the literature. We conclude the paper with an example related to an open problem.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393442","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}
Pub Date : 2019-06-06DOI: 10.4310/ARKIV.2020.v58.n2.a1
G. Sala, B. Lamel, M. Reiter
In this paper we continue our study of local rigidity for maps of CR submanifolds of the complex space. We provide a linear sufficient condition for local rigidity of finitely nondegenerate maps between minimal CR manifolds. Furthermore, we show higher order infinitesimal conditions can be used to give a characterization of local rigidity.
{"title":"Sufficient and necessary conditions for local rigidity of CR mappings and higher order infinitesimal deformations","authors":"G. Sala, B. Lamel, M. Reiter","doi":"10.4310/ARKIV.2020.v58.n2.a1","DOIUrl":"https://doi.org/10.4310/ARKIV.2020.v58.n2.a1","url":null,"abstract":"In this paper we continue our study of local rigidity for maps of CR submanifolds of the complex space. We provide a linear sufficient condition for local rigidity of finitely nondegenerate maps between minimal CR manifolds. Furthermore, we show higher order infinitesimal conditions can be used to give a characterization of local rigidity.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47544624","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}
Pub Date : 2019-05-25DOI: 10.4310/arkiv.2020.v58.n2.a8
Long Li
The aim of this paper is to study the Lelong number, the integrability index and the Monge-Ampere mass at the origin of an $S^1$-invariant plurisubharmonic function on a balanced domain in $mathbb{C}^n$ under the Schwarz symmetrization. We prove that $n$ times the integrability index is exactly the Lelong number of the symmetrization, and if the function is further toric with a single pole at the origin, then the Monge-Ampere mass is always decreasing under the symmetrization
{"title":"The Lelong number, the Monge–Ampère mass, and the Schwarz symmetrization of plurisubharmonic functions","authors":"Long Li","doi":"10.4310/arkiv.2020.v58.n2.a8","DOIUrl":"https://doi.org/10.4310/arkiv.2020.v58.n2.a8","url":null,"abstract":"The aim of this paper is to study the Lelong number, the integrability index and the Monge-Ampere mass at the origin of an $S^1$-invariant plurisubharmonic function on a balanced domain in $mathbb{C}^n$ under the Schwarz symmetrization. We prove that $n$ times the integrability index is exactly the Lelong number of the symmetrization, and if the function is further toric with a single pole at the origin, then the Monge-Ampere mass is always decreasing under the symmetrization","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2019-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393599","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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