{"title":"Spans of translates in weighted $ell^p$ spaces","authors":"K. Kellay, Florian Le Manach, M. Zarrabi","doi":"10.4171/rmi/1414","DOIUrl":"https://doi.org/10.4171/rmi/1414","url":null,"abstract":"","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49387458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On complete hypersurfaces with negative Ricci curvature in Euclidean spaces","authors":"A. P. Barreto, F. Fontenele","doi":"10.4171/rmi/1407","DOIUrl":"https://doi.org/10.4171/rmi/1407","url":null,"abstract":"","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46016995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Amenability and acyclicity in bounded cohomology","authors":"M. Moraschini, G. Raptis","doi":"10.4171/rmi/1406","DOIUrl":"https://doi.org/10.4171/rmi/1406","url":null,"abstract":"","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47824295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We discuss a system of third order PDEs for strictly convex smooth functions on domains of Euclidean space. We argue that it may be understood as a closure of sorts of the first order prolongation of a family of second order PDEs. We describe explicitly its real analytic solutions and all the solutions which satisfy a genericity condition; we also describe a family of non-generic solutions which has an application to Poisson geometry and Kahler structures on toric varieties. Our methods are geometric: we use the theory of Hessian metrics and symmetric spaces to link the analysis of the system of PDEs with properties of the manifold of matrices with orthogonal columns.
{"title":"Partial differential equations from matrices with orthogonal columns","authors":"D. Martínez Torres","doi":"10.4171/rmi/1405","DOIUrl":"https://doi.org/10.4171/rmi/1405","url":null,"abstract":"We discuss a system of third order PDEs for strictly convex smooth functions on domains of Euclidean space. We argue that it may be understood as a closure of sorts of the first order prolongation of a family of second order PDEs. We describe explicitly its real analytic solutions and all the solutions which satisfy a genericity condition; we also describe a family of non-generic solutions which has an application to Poisson geometry and Kahler structures on toric varieties. Our methods are geometric: we use the theory of Hessian metrics and symmetric spaces to link the analysis of the system of PDEs with properties of the manifold of matrices with orthogonal columns.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45052674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The large sieve with prime moduli","authors":"H. Iwaniec","doi":"10.4171/rmi/1381","DOIUrl":"https://doi.org/10.4171/rmi/1381","url":null,"abstract":"","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45472761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a set $Esubsetmathbb{R}^n$ that contains the origin we consider $I^m(E)$ -- the set of all $m^{text{th}}$ degree Taylor approximations (at the origin) of $C^m$ functions on $mathbb{R}^n$ that vanish on $E$. This set is an ideal in $mathcal{P}^m(mathbb{R}^n)$ -- the ring of all $m^{text{th}}$ degree Taylor approximations of $C^m$ functions on $mathbb{R}^n$. Which ideals in $mathcal{P}^m(mathbb{R}^n)$ arise as $I^m(E)$ for some $E$? In this paper we introduce the notion of a textit{closed} ideal in $mathcal{P}^m(mathbb{R}^n)$, and prove that any ideal of the form $I^m(E)$ is closed. We do not know whether in general any closed ideal is of the form $I^m(E)$ for some $E$, however we prove in [FS] that all closed ideals in $mathcal{P}^m(mathbb{R}^n)$ arise as $I^m(E)$ when $m+nleq5$.
{"title":"A property of ideals of jets of functions vanishing on a set","authors":"C. Fefferman, Ary Shaviv","doi":"10.4171/rmi/1423","DOIUrl":"https://doi.org/10.4171/rmi/1423","url":null,"abstract":"For a set $Esubsetmathbb{R}^n$ that contains the origin we consider $I^m(E)$ -- the set of all $m^{text{th}}$ degree Taylor approximations (at the origin) of $C^m$ functions on $mathbb{R}^n$ that vanish on $E$. This set is an ideal in $mathcal{P}^m(mathbb{R}^n)$ -- the ring of all $m^{text{th}}$ degree Taylor approximations of $C^m$ functions on $mathbb{R}^n$. Which ideals in $mathcal{P}^m(mathbb{R}^n)$ arise as $I^m(E)$ for some $E$? In this paper we introduce the notion of a textit{closed} ideal in $mathcal{P}^m(mathbb{R}^n)$, and prove that any ideal of the form $I^m(E)$ is closed. We do not know whether in general any closed ideal is of the form $I^m(E)$ for some $E$, however we prove in [FS] that all closed ideals in $mathcal{P}^m(mathbb{R}^n)$ arise as $I^m(E)$ when $m+nleq5$.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47484711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Dolbeault, D. Gontier, Fabio Pizzichillo, H. Bosch
We estimate the lowest eigenvalue in the gap of the essential spectrum of a Dirac operator with mass in terms of a Lebesgue norm of the potential. Such a bound is the counterpart for Dirac operators of the Keller estimates for the Schr"odinger operator, which are equivalent to Gagliardo-Nirenberg-Sobolev interpolation inequalities. Domain, self-adjointness, optimality and critical values of the norms are addressed, while the optimal potential is given by a Dirac equation with a Kerr nonlinearity. A new critical bound appears, which is the smallest value of the norm of the potential for which eigenvalues may reach the bottom of the gap in the essential spectrum. The Keller estimate is then extended to a Lieb-Thirring inequality for the eigenvalues in the gap. Most of our result are established in the Birman-Schwinger reformulation.
{"title":"Keller and Lieb–Thirring estimates of the eigenvalues in the gap of Dirac operators","authors":"J. Dolbeault, D. Gontier, Fabio Pizzichillo, H. Bosch","doi":"10.4171/rmi/1443","DOIUrl":"https://doi.org/10.4171/rmi/1443","url":null,"abstract":"We estimate the lowest eigenvalue in the gap of the essential spectrum of a Dirac operator with mass in terms of a Lebesgue norm of the potential. Such a bound is the counterpart for Dirac operators of the Keller estimates for the Schr\"odinger operator, which are equivalent to Gagliardo-Nirenberg-Sobolev interpolation inequalities. Domain, self-adjointness, optimality and critical values of the norms are addressed, while the optimal potential is given by a Dirac equation with a Kerr nonlinearity. A new critical bound appears, which is the smallest value of the norm of the potential for which eigenvalues may reach the bottom of the gap in the essential spectrum. The Keller estimate is then extended to a Lieb-Thirring inequality for the eigenvalues in the gap. Most of our result are established in the Birman-Schwinger reformulation.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46182551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a conjecture of Brevig, Ortega-Cerd`a, Seip and Zhao about contractive inequalities between Dirichlet and Hardy spaces and discuss its consequent connection with the Riesz projection.
{"title":"Contractive inequalities between Dirichlet and Hardy spaces","authors":"A. Llinares","doi":"10.4171/RMI/1418","DOIUrl":"https://doi.org/10.4171/RMI/1418","url":null,"abstract":"We prove a conjecture of Brevig, Ortega-Cerd`a, Seip and Zhao about contractive inequalities between Dirichlet and Hardy spaces and discuss its consequent connection with the Riesz projection.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46864008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Eriksson-Bique, T. Rajala, Elefterios Soultanis
The tensorization problem for Sobolev spaces asks for a characterization of how the Sobolev space on a product metric measure space $Xtimes Y$ can be determined from its factors. We show that two natural descriptions of the Sobolev space from the literature coincide, $W^{1,2}(Xtimes Y)=J^{1,2}(X,Y)$, thus settling the tensorization problem for Sobolev spaces in the case $p=2$, when $X$ and $Y$ are infinitesimally quasi-Hilbertian, i.e. the Sobolev space $W^{1,2}$ admits an equivalent renorming by a Dirichlet form. This class includes in particular metric measure spaces $X,Y$ of finite Hausdorff dimension as well as infinitesimally Hilbertian spaces. More generally for $pin (1,infty)$ we obtain the norm-one inclusion $|f|_{J^{1,p}(X,Y)}le |f|_{W^{1,p}(Xtimes Y)}$ and show that the norms agree on the algebraic tensor product $W^{1,p}(X)otimes W^{1,p}(Y)subset W^{1,p}(Xtimes Y)$. When $p=2$ and $X$ and $Y$ are infinitesimally quasi-Hilbertian, standard Dirichlet form theory yields the density of $W^{1,2}(X)otimes W^{1,2}(Y)$ in $J^{1,2}(X,Y)$ thus implying the equality of the spaces. Our approach raises the question of the density of $W^{1,p}(X)otimes W^{1,p}(Y)$ in $J^{1,p}(X,Y)$ in the general case.
{"title":"Tensorization of quasi-Hilbertian Sobolev spaces","authors":"S. Eriksson-Bique, T. Rajala, Elefterios Soultanis","doi":"10.4171/rmi/1433","DOIUrl":"https://doi.org/10.4171/rmi/1433","url":null,"abstract":"The tensorization problem for Sobolev spaces asks for a characterization of how the Sobolev space on a product metric measure space $Xtimes Y$ can be determined from its factors. We show that two natural descriptions of the Sobolev space from the literature coincide, $W^{1,2}(Xtimes Y)=J^{1,2}(X,Y)$, thus settling the tensorization problem for Sobolev spaces in the case $p=2$, when $X$ and $Y$ are infinitesimally quasi-Hilbertian, i.e. the Sobolev space $W^{1,2}$ admits an equivalent renorming by a Dirichlet form. This class includes in particular metric measure spaces $X,Y$ of finite Hausdorff dimension as well as infinitesimally Hilbertian spaces. More generally for $pin (1,infty)$ we obtain the norm-one inclusion $|f|_{J^{1,p}(X,Y)}le |f|_{W^{1,p}(Xtimes Y)}$ and show that the norms agree on the algebraic tensor product $W^{1,p}(X)otimes W^{1,p}(Y)subset W^{1,p}(Xtimes Y)$. When $p=2$ and $X$ and $Y$ are infinitesimally quasi-Hilbertian, standard Dirichlet form theory yields the density of $W^{1,2}(X)otimes W^{1,2}(Y)$ in $J^{1,2}(X,Y)$ thus implying the equality of the spaces. Our approach raises the question of the density of $W^{1,p}(X)otimes W^{1,p}(Y)$ in $J^{1,p}(X,Y)$ in the general case.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2022-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48190937","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：