A characterization is obtained for those pairs of weights $v$ and $w$ on $mathbb{R}^2_+$, for which the two--dimensional rectangular integration operator is bounded from a weighted Lebesgue space $L^p_v(mathbb{R}^2_+)$ to $L^q_w(mathbb{R}^2_+)$ for $1
得到了$mathbb{R}^2_+$上权重对$v$和$w$的一个刻画，对于$1
{"title":"On weighted Hardy inequality with two-dimensional rectangular operator -- extension of the E. Sawyer theorem","authors":"V. Stepanov, E. Ushakova","doi":"10.7153/mia-2021-24-3","DOIUrl":"https://doi.org/10.7153/mia-2021-24-3","url":null,"abstract":"A characterization is obtained for those pairs of weights $v$ and $w$ on $mathbb{R}^2_+$, for which the two--dimensional rectangular integration operator is bounded from a weighted Lebesgue space $L^p_v(mathbb{R}^2_+)$ to $L^q_w(mathbb{R}^2_+)$ for $1<pnot= q<infty$, which is an essential complement to E. Sawyer's result cite{Saw1} given for $1<pleq q<infty$. Besides, we declare that the E. Sawyer theorem is actual if $p=q$ only, for $p<q$ the criterion is less complicated. The case $q<p$ is new.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79312328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-10DOI: 10.7900/jot.2021nov26.2392
R. Curto, M. Ghasemi, M. Infusino, S. Kuhlmann
Let A be a unital commutative R-algebra, B a linear subspace of A and K a closed subset of the character space of A. For a linear functional L: B --> R, we investigate conditions under which L admits an integral representation with respect to a positive Radon measure supported in K. When A is equipped with a submultiplicative seminorm, we employ techniques from the theory of positive extensions of linear functionals to prove a criterion for the existence of such an integral representation for L. When no topology is prescribed on A, we identify suitable assumptions on B, A, L and K which allow us to construct a seminormed structure on A, so as to exploit our previous result to get an integral representation for L. We then use our main theorems to obtain, as applications, several well known results on the classical truncated moment problem, the moment problem for point processes, and the subnormal completion problem for 2-variable weighted shifts.
{"title":"The Truncated Moment Problem for Unital Commutative R-Algebras","authors":"R. Curto, M. Ghasemi, M. Infusino, S. Kuhlmann","doi":"10.7900/jot.2021nov26.2392","DOIUrl":"https://doi.org/10.7900/jot.2021nov26.2392","url":null,"abstract":"Let A be a unital commutative R-algebra, B a linear subspace of A and K a closed subset of the character space of A. For a linear functional L: B --> R, we investigate conditions under which L admits an integral representation with respect to a positive Radon measure supported in K. When A is equipped with a submultiplicative seminorm, we employ techniques from the theory of positive extensions of linear functionals to prove a criterion for the existence of such an integral representation for L. When no topology is prescribed on A, we identify suitable assumptions on B, A, L and K which allow us to construct a seminormed structure on A, so as to exploit our previous result to get an integral representation for L. We then use our main theorems to obtain, as applications, several well known results on the classical truncated moment problem, the moment problem for point processes, and the subnormal completion problem for 2-variable weighted shifts.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80059242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove two identities that connect some natural tensor products in the category $sf{LCS}$ of locally convex spaces with the tensor products in the category $sf{Ste}$ of stereotype spaces. In particular, we give sufficient conditions under which the identity $$ X^vartriangleodot Y^vartrianglecong (X^vartrianglecdot Y^vartriangle)^vartrianglecong (Xcdot Y)^vartriangle $$ holds, where $odot$ is the injective tensor product in the category $sf{Ste}$, $cdot$, the primary tensor product in the category $sf{LCS}$, and $vartriangle$, the pseudosaturation operation in the category $sf{LCS}$. Studying the relations of this type is justified by the fact that they turn out to be important instruments for constructing duality theory based on the notion of envelope. In particular, they are used in the construction of the duality theory for the class of (not necessarily, Abelian) countable discrete groups.
{"title":"On tensor fractions and tensor products in the category of stereotype spaces","authors":"S. Akbarov","doi":"10.1070/SM9508","DOIUrl":"https://doi.org/10.1070/SM9508","url":null,"abstract":"We prove two identities that connect some natural tensor products in the category $sf{LCS}$ of locally convex spaces with the tensor products in the category $sf{Ste}$ of stereotype spaces. In particular, we give sufficient conditions under which the identity $$ X^vartriangleodot Y^vartrianglecong (X^vartrianglecdot Y^vartriangle)^vartrianglecong (Xcdot Y)^vartriangle $$ holds, where $odot$ is the injective tensor product in the category $sf{Ste}$, $cdot$, the primary tensor product in the category $sf{LCS}$, and $vartriangle$, the pseudosaturation operation in the category $sf{LCS}$. Studying the relations of this type is justified by the fact that they turn out to be important instruments for constructing duality theory based on the notion of envelope. In particular, they are used in the construction of the duality theory for the class of (not necessarily, Abelian) countable discrete groups.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80021649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-08-18DOI: 10.22075/ijnaa.2019.16797.1894
M. Aibinu, O. Mewomo
In this paper, we consider the class of generalized {Phi}-strongly monotone mappings and the methods of approximating a solution of equations of Hammerstein type. Auxiliary mapping is defined for nonlinear integral equations of Hammerstein type. The auxiliary mapping is the composition of bounded generalized {Phi}-strongly monotone mappings which satisfy the range condition. Suitable conditions are imposed to obtain the boundedness and to show that the auxiliary mapping is a generalized {Phi}-strongly which satisfies the range condition. A sequence is constructed and it is shown that it converges strongly to a solution of equations of Hammerstein type. The results in this paper improve and extend some recent corresponding results on the approximation of a solution of equations of Hammerstein type.
{"title":"On generalized {Phi}-strongly monotone mappings and algorithms for the solution of equations of Hammerstein type","authors":"M. Aibinu, O. Mewomo","doi":"10.22075/ijnaa.2019.16797.1894","DOIUrl":"https://doi.org/10.22075/ijnaa.2019.16797.1894","url":null,"abstract":"In this paper, we consider the class of generalized {Phi}-strongly monotone mappings and the methods of approximating a solution of equations of Hammerstein type. Auxiliary mapping is defined for nonlinear integral equations of Hammerstein type. The auxiliary mapping is the composition of bounded generalized {Phi}-strongly monotone mappings which satisfy the range condition. Suitable conditions are imposed to obtain the boundedness and to show that the auxiliary mapping is a generalized {Phi}-strongly which satisfies the range condition. A sequence is constructed and it is shown that it converges strongly to a solution of equations of Hammerstein type. The results in this paper improve and extend some recent corresponding results on the approximation of a solution of equations of Hammerstein type.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86229915","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $E$ be a uniformly smooth and uniformly convex real Banach space and $E^*$ be its dual space. Suppose $A : Erightarrow E^*$ is bounded, strongly monotone and satisfies the range condition such that $A^{-1}(0)neq emptyset$. Inspired by Alber [2], we introduce Lyapunov functions and use the new geometric properties of Banach spaces to show the strong convergence of an iterative algorithm to the solution of $Ax=0$.
{"title":"Strong convergence theorems for strongly monotone mappings in Banach spaces","authors":"M. Aibinu, O. Mewomo","doi":"10.5269/bspm.37655","DOIUrl":"https://doi.org/10.5269/bspm.37655","url":null,"abstract":"Let $E$ be a uniformly smooth and uniformly convex real Banach space and $E^*$ be its dual space. Suppose $A : Erightarrow E^*$ is bounded, strongly monotone and satisfies the range condition such that $A^{-1}(0)neq emptyset$. Inspired by Alber [2], we introduce Lyapunov functions and use the new geometric properties of Banach spaces to show the strong convergence of an iterative algorithm to the solution of $Ax=0$.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74631731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We demonstrate the result stated in the title, thus answering an open question asked by Julio Becerra Guerrero, Gines Lopez-Perez and Abraham Rueda Zoca in J. Conv. Anal. textbf{25}, no. 3 (2018).
{"title":"The diametral strong diameter 2 property of Banach spaces is the same as the Daugavet property","authors":"V. Kadets","doi":"10.1090/proc/15448","DOIUrl":"https://doi.org/10.1090/proc/15448","url":null,"abstract":"We demonstrate the result stated in the title, thus answering an open question asked by Julio Becerra Guerrero, Gines Lopez-Perez and Abraham Rueda Zoca in J. Conv. Anal. textbf{25}, no. 3 (2018).","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78997816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we prove the proximinality of closed unit ball of $M$-ideals of compact operators. We also prove the ball proximinality of $M$-embedded spaces in their biduals. Moreover, we show that $mathcal{K}(ell_1)$, the space of compact operators on $ell_1$, is ball proximinal in $mathcal{B}(ell_1)$, the space of bounded operators on $ell_1$, even though $mathcal{K}(ell_1)$ is not an $M$-ideal in $mathcal{B}(ell_1)$.
{"title":"Ball proximinality of $M$-ideals of compact operators","authors":"C. R. Jayanarayanan, Sreejith Siju","doi":"10.1090/PROC/15446","DOIUrl":"https://doi.org/10.1090/PROC/15446","url":null,"abstract":"In this article, we prove the proximinality of closed unit ball of $M$-ideals of compact operators. We also prove the ball proximinality of $M$-embedded spaces in their biduals. Moreover, we show that $mathcal{K}(ell_1)$, the space of compact operators on $ell_1$, is ball proximinal in $mathcal{B}(ell_1)$, the space of bounded operators on $ell_1$, even though $mathcal{K}(ell_1)$ is not an $M$-ideal in $mathcal{B}(ell_1)$.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85172824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that well-established methods from the theory of Banach modules and time-frequency analysis allow to derive completeness results for the collection of shifted and dilated version of a given (test) function in a quite general setting. While the basic ideas show strong similarity to the arguments used in a recent paper by V.~Katsnelson we extend his results in several directions, both relaxing the assumptions and widening the range of applications. There is no need for the Banach spaces considered to be embedded into $(L^2(mathbb{R}), ||cdot{}||_2)$, nor is the Hilbert space structure relevant. We choose to present the results in the setting of the Euclidean spaces, because then the Schwartz space $mathcal{S}^{prime}(mathbb{R}^d)$ ($d geq 1$) of tempered distributions provides a well-established environment for mathematical analysis. We also establish connections to modulation spaces and Shubin classes $({Q}_{s}(mathbb{R}^d), ||cdot{}||_{Q_s})$, showing that they are special cases of Katsnelson's setting (only) for $s geq 0$.
{"title":"Completeness of shifted dilates in invariant Banach spaces of tempered distributions","authors":"H. Feichtinger, Anupam Gumber","doi":"10.1090/PROC/15564","DOIUrl":"https://doi.org/10.1090/PROC/15564","url":null,"abstract":"We show that well-established methods from the theory of Banach modules and time-frequency analysis allow to derive completeness results for the collection of shifted and dilated version of a given (test) function in a quite general setting. While the basic ideas show strong similarity to the arguments used in a recent paper by V.~Katsnelson we extend his results in several directions, both relaxing the assumptions and widening the range of applications. There is no need for the Banach spaces considered to be embedded into $(L^2(mathbb{R}), ||cdot{}||_2)$, nor is the Hilbert space structure relevant. We choose to present the results in the setting of the Euclidean spaces, because then the Schwartz space $mathcal{S}^{prime}(mathbb{R}^d)$ ($d geq 1$) of tempered distributions provides a well-established environment for mathematical analysis. We also establish connections to modulation spaces and Shubin classes $({Q}_{s}(mathbb{R}^d), ||cdot{}||_{Q_s})$, showing that they are special cases of Katsnelson's setting (only) for $s geq 0$.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88858299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a Sobolev homeomorphism $fin W^{2,1}$ in the plane we find a piecewise quadratic homeomorphism that approximates it up to a set of $epsilon$ measure. We show that this piecewise quadratic map can be approximated by diffeomorphisms in the $W^{2,1}$ norm on this set.
{"title":"Approximation of planar Sobolev W2,1 homeomorphisms by piecewise quadratic homeomorphisms and diffeomorphisms","authors":"D. Campbell, S. Hencl","doi":"10.1051/COCV/2021019","DOIUrl":"https://doi.org/10.1051/COCV/2021019","url":null,"abstract":"Given a Sobolev homeomorphism $fin W^{2,1}$ in the plane we find a piecewise quadratic homeomorphism that approximates it up to a set of $epsilon$ measure. We show that this piecewise quadratic map can be approximated by diffeomorphisms in the $W^{2,1}$ norm on this set.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91454729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
F. Albiac, J. L. Ansorena, S. Dilworth, D. Kutzarova
Our aim in this article is to contribute to the study of the structure of subsymmetric basic sequences in Banach spaces (even, more generally, in quasi-Banach spaces). For that we introduce the notion of positioning and develop new tools which lead to a dichotomy theorem that holds for general spaces with subsymmetric bases. As an illustration of how to use this dichotomy theorem we obtain the classification of all subsymmetric sequences in certain types of spaces. To be more specific, we show that Garling sequence spaces have a unique symmetric basic sequence but no symmetric basis and that these spaces have a continuum of subsymmetric basic sequences.
{"title":"A dichotomy for subsymmetric basic sequences with applications to Garling spaces","authors":"F. Albiac, J. L. Ansorena, S. Dilworth, D. Kutzarova","doi":"10.1090/tran/8278","DOIUrl":"https://doi.org/10.1090/tran/8278","url":null,"abstract":"Our aim in this article is to contribute to the study of the structure of subsymmetric basic sequences in Banach spaces (even, more generally, in quasi-Banach spaces). For that we introduce the notion of positioning and develop new tools which lead to a dichotomy theorem that holds for general spaces with subsymmetric bases. As an illustration of how to use this dichotomy theorem we obtain the classification of all subsymmetric sequences in certain types of spaces. To be more specific, we show that Garling sequence spaces have a unique symmetric basic sequence but no symmetric basis and that these spaces have a continuum of subsymmetric basic sequences.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75176374","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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