Ole Fredrik Brevig, Karl-Mikael Perfekt, Alexander Pushnitski
For $alpha > 0$ we consider the operator $K_alpha colon ell^2 to ell^2$ corresponding to the matrix [left(frac{(nm)^{-frac{1}{2}+alpha}}{[max(n,m)]^{2alpha}}right)_{n,m=1}^infty.] By interpreting $K_alpha$ as the inverse of an unbounded Jacobi matrix, we show that the absolutely continuous spectrum coincides with $[0, 2/alpha]$ (multiplicity one), and that there is no singular continuous spectrum. There are a finite number of eigenvalues above the continuous spectrum. We apply our results to demonstrate that the reproducing kernel thesis does not hold for composition operators on the Hardy space of Dirichlet series $mathscr{H}^2$.
对于$alpha > 0$，我们考虑对应于矩阵[left(frac{(nm)^{-frac{1}{2}+alpha}}{[max(n,m)]^{2alpha}}right)_{n,m=1}^infty.]的算子$K_alpha colon ell^2 to ell^2$，通过将$K_alpha$解释为无界Jacobi矩阵的逆，我们证明了绝对连续谱与$[0, 2/alpha]$重合(多重度为1)，并且不存在奇异连续谱。连续谱上有有限个特征值。我们应用我们的结果证明了在Dirichlet级数的Hardy空间$mathscr{H}^2$上复合算子的再现核命题不成立。
{"title":"The spectrum of some Hardy kernel matrices","authors":"Ole Fredrik Brevig, Karl-Mikael Perfekt, Alexander Pushnitski","doi":"10.5802/aif.3589","DOIUrl":"https://doi.org/10.5802/aif.3589","url":null,"abstract":"For $alpha > 0$ we consider the operator $K_alpha colon ell^2 to ell^2$ corresponding to the matrix [left(frac{(nm)^{-frac{1}{2}+alpha}}{[max(n,m)]^{2alpha}}right)_{n,m=1}^infty.] By interpreting $K_alpha$ as the inverse of an unbounded Jacobi matrix, we show that the absolutely continuous spectrum coincides with $[0, 2/alpha]$ (multiplicity one), and that there is no singular continuous spectrum. There are a finite number of eigenvalues above the continuous spectrum. We apply our results to demonstrate that the reproducing kernel thesis does not hold for composition operators on the Hardy space of Dirichlet series $mathscr{H}^2$.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47955589","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 $(X,omega)$ be a compact Hermitian manifold. We establish a stability result for solutions to complex Monge-Ampere equations with right-hand side in $L^p$, $p>1$. Using this we prove that the solutions are Holder continuous with the same exponent as in the Kahler case cite{DDGKPZ14}. Our techniques also apply to the setting of big cohomology classes on compact Kahler manifolds.
{"title":"Stability and Hölder regularity of solutions to complex Monge–Ampère equations on compact Hermitian manifolds","authors":"C. H. Lu, Trong Phung, T. Tô","doi":"10.5802/aif.3436","DOIUrl":"https://doi.org/10.5802/aif.3436","url":null,"abstract":"Let $(X,omega)$ be a compact Hermitian manifold. We establish a stability result for solutions to complex Monge-Ampere equations with right-hand side in $L^p$, $p>1$. Using this we prove that the solutions are Holder continuous with the same exponent as in the Kahler case cite{DDGKPZ14}. Our techniques also apply to the setting of big cohomology classes on compact Kahler manifolds.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41564622","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 notion of super weakly compact subsets of a Banach space, which can be described as a local version of super-reflexivity. Our first result is that the closed convex hull of a super weakly compact set is super weakly compact. This allows us to extend to the non convex setting the main properties of these sets. In particular, we give non linear characterizations of super weak compactness in terms of the (non) embeddability of special trees and graphs. We conclude with a few relevant examples of super weakly compact sets in non super-reflexive Banach spaces.
{"title":"Nonlinear aspects of super weakly compact sets","authors":"G. Lancien, M. Raja","doi":"10.5802/aif.3488","DOIUrl":"https://doi.org/10.5802/aif.3488","url":null,"abstract":"We study the notion of super weakly compact subsets of a Banach space, which can be described as a local version of super-reflexivity. Our first result is that the closed convex hull of a super weakly compact set is super weakly compact. This allows us to extend to the non convex setting the main properties of these sets. In particular, we give non linear characterizations of super weak compactness in terms of the (non) embeddability of special trees and graphs. We conclude with a few relevant examples of super weakly compact sets in non super-reflexive Banach spaces.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71210054","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 $f$ be a univalent self-map of the unit disc. We introduce a technique, that we call {sl semigroup-fication}, which allows to construct a continuous semigroup $(phi_t)$ of holomorphic self-maps of the unit disc whose time one map $phi_1$ is, in a sense, very close to $f$. The semigrup-fication of $f$ is of the same type as $f$ (elliptic, hyperbolic, parabolic of positive step or parabolic of zero step) and there is a one-to-one correspondence between the set of boundary regular fixed points of $f$ with a given multiplier and the corresponding set for $phi_1$. Moreover, in case $f$ (and hence $phi_1$) has no interior fixed points, the slope of the orbits converging to the Denjoy-Wolff point is the same. The construction is based on holomorphic models, localization techniques and Gromov hyperbolicity. As an application of this construction, we prove that in the non-elliptic case, the orbits of $f$ converge non-tangentially to the Denjoy-Wolff point if and only if the Koenigs domain of $f$ is "almost symmetric" with respect to vertical lines.
{"title":"Semigroup-fication of univalent self-maps of the unit disc","authors":"F. Bracci, Oliver Roth","doi":"10.5802/aif.3517","DOIUrl":"https://doi.org/10.5802/aif.3517","url":null,"abstract":"Let $f$ be a univalent self-map of the unit disc. We introduce a technique, that we call {sl semigroup-fication}, which allows to construct a continuous semigroup $(phi_t)$ of holomorphic self-maps of the unit disc whose time one map $phi_1$ is, in a sense, very close to $f$. The semigrup-fication of $f$ is of the same type as $f$ (elliptic, hyperbolic, parabolic of positive step or parabolic of zero step) and there is a one-to-one correspondence between the set of boundary regular fixed points of $f$ with a given multiplier and the corresponding set for $phi_1$. Moreover, in case $f$ (and hence $phi_1$) has no interior fixed points, the slope of the orbits converging to the Denjoy-Wolff point is the same. The construction is based on holomorphic models, localization techniques and Gromov hyperbolicity. As an application of this construction, we prove that in the non-elliptic case, the orbits of $f$ converge non-tangentially to the Denjoy-Wolff point if and only if the Koenigs domain of $f$ is \"almost symmetric\" with respect to vertical lines.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48096100","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 $X$ be a complex manifold and $L$ be a holomorphic line bundle on $X$. Assume that $L$ is semi-positive, namely $L$ admits a smooth Hermitian metric with semi-positive Chern curvature. Let $Y$ be a compact Kahler submanifold of $X$ such that the restriction of $L$ to $Y$ is topologically trivial. We investigate the obstruction for $L$ to be unitary flat on a neighborhood of $Y$ in $X$. As an application, for example, we show the existence of nef, big, and non semi-positive line bundle on a non-singular projective surface.
{"title":"Linearization of transition functions of a semi-positive line bundle along a certain submanifold","authors":"T. Koike","doi":"10.5802/aif.3439","DOIUrl":"https://doi.org/10.5802/aif.3439","url":null,"abstract":"Let $X$ be a complex manifold and $L$ be a holomorphic line bundle on $X$. Assume that $L$ is semi-positive, namely $L$ admits a smooth Hermitian metric with semi-positive Chern curvature. Let $Y$ be a compact Kahler submanifold of $X$ such that the restriction of $L$ to $Y$ is topologically trivial. We investigate the obstruction for $L$ to be unitary flat on a neighborhood of $Y$ in $X$. As an application, for example, we show the existence of nef, big, and non semi-positive line bundle on a non-singular projective surface.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48404432","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 $W_0$ be a reflection subgroup of a finite complex reflection group $W$, and let $B_0$ and $B$ be their respective braid groups. In order to construct a Hecke algebra $widetilde{H}_0$ for the normalizer $N_W(W_0)$, one first considers a natural subquotient $widetilde{B}_0$ of $B$ which is an extension of $N_W(W_0)/W_0$ by $B_0$. We prove that this extension is split when $W$ is a Coxeter group, and deduce a standard basis for the Hecke algebra $widetilde{H}_0$. We also give classes of both split and non-split examples in the non-Coxeter case.
{"title":"Braid groups of normalizers of reflection subgroups","authors":"T. Gobet, A. Henderson, Ivan Marin","doi":"10.5802/aif.3440","DOIUrl":"https://doi.org/10.5802/aif.3440","url":null,"abstract":"Let $W_0$ be a reflection subgroup of a finite complex reflection group $W$, and let $B_0$ and $B$ be their respective braid groups. In order to construct a Hecke algebra $widetilde{H}_0$ for the normalizer $N_W(W_0)$, one first considers a natural subquotient $widetilde{B}_0$ of $B$ which is an extension of $N_W(W_0)/W_0$ by $B_0$. We prove that this extension is split when $W$ is a Coxeter group, and deduce a standard basis for the Hecke algebra $widetilde{H}_0$. We also give classes of both split and non-split examples in the non-Coxeter case.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43830235","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 essential minimum and equidistribution of small points are two well-established interrelated subjects in arithmetic geometry. However, there is lack of an analogue of essential minimum dealing with higher dimensional subvarieties, and the equidistribution of these is a far less explored topic. In this paper, we introduce a new notion of higher dimensional essential minimum and use it to prove equidistribution of generic and small effective cycles. The latter generalizes the previous higher dimensional equidistribution theorems by considering cycles and by allowing more fexibility on the arithmetic datum.
{"title":"Higher dimensional essential minima and equidistribution of cycles","authors":"R. Gualdi, C. Mart'inez","doi":"10.5802/aif.3500","DOIUrl":"https://doi.org/10.5802/aif.3500","url":null,"abstract":"The essential minimum and equidistribution of small points are two well-established interrelated subjects in arithmetic geometry. However, there is lack of an analogue of essential minimum dealing with higher dimensional subvarieties, and the equidistribution of these is a far less explored topic. \u0000In this paper, we introduce a new notion of higher dimensional essential minimum and use it to prove equidistribution of generic and small effective cycles. The latter generalizes the previous higher dimensional equidistribution theorems by considering cycles and by allowing more fexibility on the arithmetic datum.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48781261","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 extend Yves Andre's theory of solution algebras in differential Galois theory to a general Tannakian context. As applications, we establish analogues of his correspondence between solution fields and observable subgroups of the Galois group for iterated differential equations in positive characteristic and for difference equations. The use of solution algebras in the difference algebraic context also allows a new approach to recent results of Philippon and Adamczewski--Faverjon in transcendence theory.
{"title":"A general theory of André’s solution algebras","authors":"L. Nagy, Tam'as Szamuely","doi":"10.5802/AIF.3383","DOIUrl":"https://doi.org/10.5802/AIF.3383","url":null,"abstract":"We extend Yves Andre's theory of solution algebras in differential Galois theory to a general Tannakian context. As applications, we establish analogues of his correspondence between solution fields and observable subgroups of the Galois group for iterated differential equations in positive characteristic and for difference equations. The use of solution algebras in the difference algebraic context also allows a new approach to recent results of Philippon and Adamczewski--Faverjon in transcendence theory.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44847254","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 Zakharov-Kuznetsov equation in space dimension $dgeq 3$ is considered. It is proved that the Cauchy problem is locally well-posed in $H^s(mathbb{R}^d)$ in the full subcritical range $s>(d-4)/2$, which is optimal up to the endpoint. As a corollary, global well-posedness in $L^2(mathbb{R}^3)$ and, under a smallness condition, in $H^1(mathbb{R}^4)$, follow.
{"title":"Subcritical well-posedness results for the Zakharov–Kuznetsov equation in dimension three and higher","authors":"S. Herr, S. Kinoshita","doi":"10.5802/aif.3547","DOIUrl":"https://doi.org/10.5802/aif.3547","url":null,"abstract":"The Zakharov-Kuznetsov equation in space dimension $dgeq 3$ is considered. It is proved that the Cauchy problem is locally well-posed in $H^s(mathbb{R}^d)$ in the full subcritical range $s>(d-4)/2$, which is optimal up to the endpoint. As a corollary, global well-posedness in $L^2(mathbb{R}^3)$ and, under a smallness condition, in $H^1(mathbb{R}^4)$, follow.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48122282","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 goal of this article is to provide several sharp results on the minimal time required for observability of several Grushin-type equations. Namely, it is by now well-known that Grushin-type equations are degenerate parabolic equations for which some geometric conditions are needed to get observability properties, contrarily to the usual parabolic equations. Our results concern the Grushin operator $partial_t - Delta_{x} - |x|^2 Delta_{y}$ observed from the whole boundary in the multi-dimensional setting (meaning that $x in Omega_x$, where $Omega_x$ is a subset of $mathbb{R}^{d_x}$ with $d_x geq 1$, $y in Omega_y$, where $Omega_y$ is a subset of $mathbb{R}^{d_y}$ with $d_y geq 1$, and the observation is done on $Gamma = partial Omega_x times Omega_y$), from one lateral boundary in the one-dimensional setting (i.e. $d_x = 1$), including some generalized version of the form $partial_t - partial_{x}^2 - (q(x))^2 partial_{y}^2$ for suitable functions $q$, and the Heisenberg operator $partial_t - partial_{x}^2 -(x partial_z + partial_y)^2$ observed from one lateral boundary. In all these cases, our approach strongly relies on the analysis of the family of equations obtained by using the Fourier expansion of the equations in the $y$ (or $(y,z)$) variables, and in particular the asymptotic of the cost of observability in the Fourier parameters. Combining these estimates with results on the rate of dissipation of each of these equations, we obtain observability estimates in suitably large times. We then show that the times we obtain to get observability are optimal in several cases using Agmon type estimates.
{"title":"Minimal time issues for the observability of Grushin-type equations","authors":"K. Beauchard, J. Dardé, S. Ervedoza","doi":"10.5802/AIF.3313","DOIUrl":"https://doi.org/10.5802/AIF.3313","url":null,"abstract":"The goal of this article is to provide several sharp results on the minimal time required for observability of several Grushin-type equations. Namely, it is by now well-known that Grushin-type equations are degenerate parabolic equations for which some geometric conditions are needed to get observability properties, contrarily to the usual parabolic equations. Our results concern the Grushin operator $partial_t - Delta_{x} - |x|^2 Delta_{y}$ observed from the whole boundary in the multi-dimensional setting (meaning that $x in Omega_x$, where $Omega_x$ is a subset of $mathbb{R}^{d_x}$ with $d_x geq 1$, $y in Omega_y$, where $Omega_y$ is a subset of $mathbb{R}^{d_y}$ with $d_y geq 1$, and the observation is done on $Gamma = partial Omega_x times Omega_y$), from one lateral boundary in the one-dimensional setting (i.e. $d_x = 1$), including some generalized version of the form $partial_t - partial_{x}^2 - (q(x))^2 partial_{y}^2$ for suitable functions $q$, and the Heisenberg operator $partial_t - partial_{x}^2 -(x partial_z + partial_y)^2$ observed from one lateral boundary. In all these cases, our approach strongly relies on the analysis of the family of equations obtained by using the Fourier expansion of the equations in the $y$ (or $(y,z)$) variables, and in particular the asymptotic of the cost of observability in the Fourier parameters. Combining these estimates with results on the rate of dissipation of each of these equations, we obtain observability estimates in suitably large times. We then show that the times we obtain to get observability are optimal in several cases using Agmon type estimates.","PeriodicalId":50781,"journal":{"name":"Annales De L Institut Fourier","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71208544","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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