We give a complete classification, up to isometric isomorphism and scaling, of $4$-dimensional metric Lie algebras $(mathfrak{g},langle cdot,cdot rangle)$ that admit a non-zero parallel skew-symmetric endomorphism. In particular, we distinguish those metric Lie algebras that admit such an endomorphism which is not a multiple of a complex structure, and for each of them we obtain the de Rham decomposition of the associated simply connected Lie group with the corresponding left invariant metric. On the other hand, we find that the associated simply connected Lie group is irreducible as a Riemannian manifold for those metric Lie algebras where each parallel skew-symmetric endomorphism is a multiple of a complex structure.
{"title":"Parallel skew-symmetric tensors on 4-dimensional metric Lie algebras","authors":"Herrera, A. C.","doi":"10.33044/revuma.2451","DOIUrl":"https://doi.org/10.33044/revuma.2451","url":null,"abstract":"We give a complete classification, up to isometric isomorphism and scaling, of $4$-dimensional metric Lie algebras $(mathfrak{g},langle cdot,cdot rangle)$ that admit a non-zero parallel skew-symmetric endomorphism. In particular, we distinguish those metric Lie algebras that admit such an endomorphism which is not a multiple of a complex structure, and for each of them we obtain the de Rham decomposition of the associated simply connected Lie group with the corresponding left invariant metric. On the other hand, we find that the associated simply connected Lie group is irreducible as a Riemannian manifold for those metric Lie algebras where each parallel skew-symmetric endomorphism is a multiple of a complex structure.","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":"38 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135430946","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 aim of this paper is to obtain mixed weak-type inequalities for multilinear fractional operators, extending results by F. Berra, M. Carena and G. Pradolini cite{BCP}. We prove that, under certain conditions on the weights, there exists a constant $C$ such that $$Bigg| frac{mathcal G_{alpha}(vec f ,)}{v}Bigg|_{L^{q, infty}(nu v^q)} leq C prod_{i=1}^m{|f_i|_{L^1(u_i)}},$$ where $mathcal G_{alpha}(vec f ,)$ is the multilinear maximal function $mathcal M_{alpha}(vec f,)$ that was introduced by K. Moen in cite{M} or the multilineal fractional integral $mathcal I_{alpha}(vec f ,)$. As an application a vector-valued weighted mixed inequality for $mathcal I_{alpha}(vec f ,)$ will be provided as well.
{"title":"Weighted mixed weak-type inequalities for multilinear fractional operators","authors":"M. Belén Picardi","doi":"10.33044/revuma.3017","DOIUrl":"https://doi.org/10.33044/revuma.3017","url":null,"abstract":"The aim of this paper is to obtain mixed weak-type inequalities for multilinear fractional operators, extending results by F. Berra, M. Carena and G. Pradolini cite{BCP}. We prove that, under certain conditions on the weights, there exists a constant $C$ such that $$Bigg| frac{mathcal G_{alpha}(vec f ,)}{v}Bigg|_{L^{q, infty}(nu v^q)} leq C prod_{i=1}^m{|f_i|_{L^1(u_i)}},$$ where $mathcal G_{alpha}(vec f ,)$ is the multilinear maximal function $mathcal M_{alpha}(vec f,)$ that was introduced by K. Moen in cite{M} or the multilineal fractional integral $mathcal I_{alpha}(vec f ,)$. As an application a vector-valued weighted mixed inequality for $mathcal I_{alpha}(vec f ,)$ will be provided as well.","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":"186 2‐3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135679538","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 $A$ be a right linear operator on a two-sided quaternionic Banach space $X$. The paper studies the Drazin inverse for right linear operators on a quaternionic Banach space. It is shown that if $A$ is Drazin invertible then the Drazin inverse of $A$ is given by $f(A)$ where $f$ is $0$ in an axially symmetric neighborhood of $0$ and $f(q) = q^{-1}$ in an axially symmetric neighborhood of the nonzero spherical spectrum of $A$. Some results analogous to the ones concerning the Drazin inverse of operators on complex Banach spaces are proved in the quaternionic context.
{"title":"Drazin invertibility of linear operators on quaternionic Banach spaces","authors":"El Hassan Benabdi, Mohamed Barraa","doi":"10.33044/revuma.2700","DOIUrl":"https://doi.org/10.33044/revuma.2700","url":null,"abstract":"Let $A$ be a right linear operator on a two-sided quaternionic Banach space $X$. The paper studies the Drazin inverse for right linear operators on a quaternionic Banach space. It is shown that if $A$ is Drazin invertible then the Drazin inverse of $A$ is given by $f(A)$ where $f$ is $0$ in an axially symmetric neighborhood of $0$ and $f(q) = q^{-1}$ in an axially symmetric neighborhood of the nonzero spherical spectrum of $A$. Some results analogous to the ones concerning the Drazin inverse of operators on complex Banach spaces are proved in the quaternionic context.","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":"88 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135765833","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 maps preserving the Jordan product of $C$-symmetric operators","authors":"Zouheir Amara, Mourad Oudghiri","doi":"10.33044/revuma.2950","DOIUrl":"https://doi.org/10.33044/revuma.2950","url":null,"abstract":"","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136318182","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 describe the full group of isometries of absolutely simple, compact, connected real Lie groups, of SO(4) and of U(n), endowed with suitable bi-invariant Riemannian metrics.
我们描述了具有适当双不变黎曼度量的绝对简单、紧致、连通的实李群SO(4)和U(n)的完整等距群。
{"title":"The full group of isometries of some compact Lie groups endowed with a bi-invariant metric","authors":"Alberto Dolcetti, Donato Pertici","doi":"10.33044/revuma.2737","DOIUrl":"https://doi.org/10.33044/revuma.2737","url":null,"abstract":"We describe the full group of isometries of absolutely simple, compact, connected real Lie groups, of SO(4) and of U(n), endowed with suitable bi-invariant Riemannian metrics.","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135322754","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":"Characterization of essential spectra by quasi-compact perturbations","authors":"Faiçal Abdmouleh, Hamadi Chaâben, Ines Walha","doi":"10.33044/revuma.2771","DOIUrl":"https://doi.org/10.33044/revuma.2771","url":null,"abstract":"","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135367232","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 representation obtained as a tensor product of a rational elliptic curve with a weight 1 modular form is in general an irreducible four-dimensional representation. However, there are some instances where such representation is reducible. In this short article we give an exotic way to obtain such a reducible representation.
{"title":"An exotic example of a tensor product of a CM elliptic curve and a weight 1 form","authors":"Ariel Pacetti","doi":"10.33044/revuma.2284","DOIUrl":"https://doi.org/10.33044/revuma.2284","url":null,"abstract":". The representation obtained as a tensor product of a rational elliptic curve with a weight 1 modular form is in general an irreducible four-dimensional representation. However, there are some instances where such representation is reducible. In this short article we give an exotic way to obtain such a reducible representation.","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136079947","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}
Víctor Almeida, Jorge J. Betancor, Juan C. Fariña, Lourdes Rodríguez-Mesa
Let $mathcal{L}$ be the Schr"odinger operator with potential $V$, that is, $mathcal L=-Delta+V$, where it is assumed that $V$ satisfies a reverse H"older inequality. We consider weighted Morrey-Campanato spaces $BMO_{mathcal L,w}^alpha (mathbb R^d)$ and $BLO_{L,w}^alpha (mathbb R^d)$ in the Schr"odinger setting. We prove that the variation operator $V_sigma ({T_t}_{t>0})$, $sigma>2$, and the oscillation operator $O({T_t}_{t>0}, {t_j}_{jin mathbb Z})$, where $t_j0$, with $kin mathbb N$, are bounded operators from $BMO_{mathcal L,w}^alpha (mathbb R^d)$ into $BLO_{mathcal L,w}^alpha (mathbb R^d)$. We also establish the same property for the maximal operators defined by ${t^kpartial_t^k e^{-tmathcal L}}_{t>0}$, $kin mathbb N$.
{"title":"Variation and oscillation operators on weighted Morrey–Campanato spaces in the Schrödinger setting","authors":"Víctor Almeida, Jorge J. Betancor, Juan C. Fariña, Lourdes Rodríguez-Mesa","doi":"10.33044/revuma.4327","DOIUrl":"https://doi.org/10.33044/revuma.4327","url":null,"abstract":"Let $mathcal{L}$ be the Schr\"odinger operator with potential $V$, that is, $mathcal L=-Delta+V$, where it is assumed that $V$ satisfies a reverse H\"older inequality. We consider weighted Morrey-Campanato spaces $BMO_{mathcal L,w}^alpha (mathbb R^d)$ and $BLO_{L,w}^alpha (mathbb R^d)$ in the Schr\"odinger setting. We prove that the variation operator $V_sigma ({T_t}_{t>0})$, $sigma>2$, and the oscillation operator $O({T_t}_{t>0}, {t_j}_{jin mathbb Z})$, where $t_j<t_{j+1}$, $jin mathbb Z$, $lim_{jrightarrow +infty}t_j=+infty$ and $lim_{jrightarrow -infty} t_j=0$, being $T_t=t^kpartial_t^k e^{-tmathcal L}$, $t>0$, with $kin mathbb N$, are bounded operators from $BMO_{mathcal L,w}^alpha (mathbb R^d)$ into $BLO_{mathcal L,w}^alpha (mathbb R^d)$. We also establish the same property for the maximal operators defined by ${t^kpartial_t^k e^{-tmathcal L}}_{t>0}$, $kin mathbb N$.","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136235959","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 short note, we discuss a boundary value problem for a matrix valued ∂ equation.
{"title":"Remarks on a boundary value problem for a matrix valued $overline{partial}$ equation","authors":"Carlos E. Kenig","doi":"10.33044/revuma.4326","DOIUrl":"https://doi.org/10.33044/revuma.4326","url":null,"abstract":". In this short note, we discuss a boundary value problem for a matrix valued ∂ equation.","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136136415","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}
Given an $m$-tuple of weights $vec{v}=(v_1,dots,v_m)$, we characterize the classes of pairs $(w,vec{v})$ involved with the boundedness properties of the multilinear fractional integral operator from $prod_{i=1}^mL^{p_i}left(v_i^{p_i}right)$ into suitable Lipschitz spaces associated to a parameter $delta$, $mathcal{L}_w(delta)$. Our results generalize some previous estimates not only for the linear case but also for the unweighted problem in the multilinear context. We emphasize the study related to the range of the parameters involved with the problem described above, which is optimal in the sense that they become trivial outside of the region obtained. We also exhibit nontrivial examples of pairs of weights in this region.
{"title":"Two-weighted estimates of the multilinear fractional integral operator between weighted Lebesgue and Lipschitz spaces with optimal parameters","authors":"Fabio Berra, Gladis Pradolini, Wilfredo Ramos","doi":"10.33044/revuma.4346","DOIUrl":"https://doi.org/10.33044/revuma.4346","url":null,"abstract":"Given an $m$-tuple of weights $vec{v}=(v_1,dots,v_m)$, we characterize the classes of pairs $(w,vec{v})$ involved with the boundedness properties of the multilinear fractional integral operator from $prod_{i=1}^mL^{p_i}left(v_i^{p_i}right)$ into suitable Lipschitz spaces associated to a parameter $delta$, $mathcal{L}_w(delta)$. Our results generalize some previous estimates not only for the linear case but also for the unweighted problem in the multilinear context. We emphasize the study related to the range of the parameters involved with the problem described above, which is optimal in the sense that they become trivial outside of the region obtained. We also exhibit nontrivial examples of pairs of weights in this region.","PeriodicalId":54469,"journal":{"name":"Revista De La Union Matematica Argentina","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136236101","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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