Let $L$ be the distinguished Laplacian on the Iwasawa $AN$ group associated with a semisimple Lie group $G$. Assume $F$ is a Borel function on $mathbb{R}^+$. We give a condition on $F$ such that the kernels of the functions $F(L)$ are uniformly bounded. This condition involves the decay of $F$ only and not its derivatives. By a known correspondence, this implies pointwise estimates for a wide range of functions of the Laplace-Beltrami operator on symmetric spaces. In particular, when $G$ is of real rank one and $F(x)={rm e}^{itsqrt x}psi(sqrt x)$, our bounds are sharp.
{"title":"Pointwise and uniform bounds for functions of the Laplacian on non-compact symmetric spaces","authors":"Yulia Kuznetsova, Zhipeng Song","doi":"arxiv-2409.02688","DOIUrl":"https://doi.org/arxiv-2409.02688","url":null,"abstract":"Let $L$ be the distinguished Laplacian on the Iwasawa $AN$ group associated\u0000with a semisimple Lie group $G$. Assume $F$ is a Borel function on\u0000$mathbb{R}^+$. We give a condition on $F$ such that the kernels of the\u0000functions $F(L)$ are uniformly bounded. This condition involves the decay of\u0000$F$ only and not its derivatives. By a known correspondence, this implies\u0000pointwise estimates for a wide range of functions of the Laplace-Beltrami\u0000operator on symmetric spaces. In particular, when $G$ is of real rank one and\u0000$F(x)={rm e}^{itsqrt x}psi(sqrt x)$, our bounds are sharp.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208175","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 introduce the notion of a generalized complex (GC) Stein manifold and provide complete characterizations in three fundamental aspects. First, we extend Cartan's Theorem A and B within the framework of GC geometry. Next, we define $L$-plurisubharmonic functions and develop an associated $L^2$ theory. This leads to a characterization of GC Stein manifolds using $L$-plurisubharmonic exhaustion functions. Finally, we establish the existence of a proper GH embedding from any GC Stein manifold into $mathbb{R}^{2n-2k} times mathbb{C}^{2k+1}$, where $2n$ and $k$ denote the dimension and type of the GC Stein manifold, respectively. This provides a characterization of GC Stein manifolds via GH embeddings. Several examples of GC Stein manifolds are given.
{"title":"Generalized complex Stein manifold","authors":"Debjit Pal","doi":"arxiv-2409.01912","DOIUrl":"https://doi.org/arxiv-2409.01912","url":null,"abstract":"We introduce the notion of a generalized complex (GC) Stein manifold and\u0000provide complete characterizations in three fundamental aspects. First, we\u0000extend Cartan's Theorem A and B within the framework of GC geometry. Next, we\u0000define $L$-plurisubharmonic functions and develop an associated $L^2$ theory.\u0000This leads to a characterization of GC Stein manifolds using\u0000$L$-plurisubharmonic exhaustion functions. Finally, we establish the existence\u0000of a proper GH embedding from any GC Stein manifold into $mathbb{R}^{2n-2k}\u0000times mathbb{C}^{2k+1}$, where $2n$ and $k$ denote the dimension and type of\u0000the GC Stein manifold, respectively. This provides a characterization of GC\u0000Stein manifolds via GH embeddings. Several examples of GC Stein manifolds are\u0000given.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226621","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 use Grothendieck's dessins d'enfant to show that if $P$ and $Q$ are two real polynomials, any real function of the form $x^alpha(1-x)^{beta} P - Q$, has at most $deg P +deg Q + 2$ roots in the interval $]0,~1[$. As a consequence, we obtain an upper bound on the number of positive solutions to a real polynomial system $f=g=0$ in two variables where $f$ has three monomials terms, and $g$ has $t$ terms. The approach we adopt for tackling this Fewnomial bound relies on the theory of Wronskians, which was used in Koiran et. al. (J. Symb. Comput., 2015) for producing the first upper bound which is polynomial in $t$.
{"title":"Improved fewnomial upper bounds from Wronskians and dessins d'enfant","authors":"Boulos El Hilany, Sébastien Tavenas","doi":"arxiv-2409.01651","DOIUrl":"https://doi.org/arxiv-2409.01651","url":null,"abstract":"We use Grothendieck's dessins d'enfant to show that if $P$ and $Q$ are two\u0000real polynomials, any real function of the form $x^alpha(1-x)^{beta} P - Q$,\u0000has at most $deg P +deg Q + 2$ roots in the interval $]0,~1[$. As a\u0000consequence, we obtain an upper bound on the number of positive solutions to a\u0000real polynomial system $f=g=0$ in two variables where $f$ has three monomials\u0000terms, and $g$ has $t$ terms. The approach we adopt for tackling this Fewnomial\u0000bound relies on the theory of Wronskians, which was used in Koiran et. al.\u0000(J. Symb. Comput., 2015) for producing the first upper bound which is\u0000polynomial in $t$.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"62 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208190","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 provide a characterization of two expansive dilation matrices yielding equal discrete anisotropic Triebel-Lizorkin spaces. For two such matrices $A$ and $B$, it is shown that $dot{mathbf{f}}^{alpha}_{p,q}(A) = dot{mathbf{f}}^{alpha}_{p,q}(B)$ for all $alpha in mathbb{R}$ and $p, q in (0, infty]$ if and only if the set ${A^j B^{-j} : j in mathbb{Z}}$ is finite, or in the trivial case when $p = q$ and $|det(A)|^{alpha + 1/2 - 1/p} = |det(B)|^{alpha + 1/2 - 1/p}$. This provides an extension of a result by Triebel for diagonal dilations to arbitrary expansive matrices. The obtained classification of dilations is different from corresponding results for anisotropic Triebel-Lizorkin function spaces.
{"title":"Discrete Triebel-Lizorkin spaces and expansive matrices","authors":"Jordy Timo van Velthoven, Felix Voigtlaender","doi":"arxiv-2409.01849","DOIUrl":"https://doi.org/arxiv-2409.01849","url":null,"abstract":"We provide a characterization of two expansive dilation matrices yielding\u0000equal discrete anisotropic Triebel-Lizorkin spaces. For two such matrices $A$\u0000and $B$, it is shown that $dot{mathbf{f}}^{alpha}_{p,q}(A) =\u0000dot{mathbf{f}}^{alpha}_{p,q}(B)$ for all $alpha in mathbb{R}$ and $p, q\u0000in (0, infty]$ if and only if the set ${A^j B^{-j} : j in mathbb{Z}}$ is\u0000finite, or in the trivial case when $p = q$ and $|det(A)|^{alpha + 1/2 - 1/p}\u0000= |det(B)|^{alpha + 1/2 - 1/p}$. This provides an extension of a result by\u0000Triebel for diagonal dilations to arbitrary expansive matrices. The obtained\u0000classification of dilations is different from corresponding results for\u0000anisotropic Triebel-Lizorkin function spaces.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208189","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 $ mathbb{B}(mathscr{H})$ represent the $C^*$-algebra, which consists of all bounded linear operators on $mathscr{H},$ and let $N ( .) $ be a norm on $ mathbb{B}(mathscr{H})$. We define a norm $w_{(N,e)} (. , . )$ on $ mathbb{B}^2(mathscr{H})$ by $$ w_{(N,e)}(B,C)=underset{|lambda_1|^2+lambda_2|^2leq1}sup underset{thetainmathbb{R}}sup Nleft(Re left(e^{itheta}(lambda_1B+lambda_2C)right)right),$$ for every $B,Cinmathbb{B}(mathscr{H})$ and $lambda_1,lambda_2inmathbb{C}.$ We investigate basic properties of this norm and prove some bounds involving it. In particular, when $N( .)$ is the Hilbert-Schmidt norm, we prove some Hilbert-Schmidt Euclidean operator radius inequalities for a pair of bounded linear operators.
让 $ mathbb{B}(mathscr{H})$ 表示 $C^*$-代数，它由 $mathscr{H} 上的所有有界线性算子组成，并让 $N ( .) $ 是 $mathbb{B}(mathscr{H})$ 上的一个规范。我们定义一个在 $mathbb{B}(mathscr{H} $ 上的规范 $w_{(N,e)} (. , .)$ on $mathbb{B}^2(mathscr{H})$ by $$w_{(N,e)}(B. C)=underset{B、C)=underset{|lambda_1|^2+lambda_2|^2leq1}supunderset{thetainmathbb{R}}sup Nleft(Releft(e^{itheta}(lambda_1B+lambda_2C)right)right)、$$ for every$B,Cinmathbb{B}(mathscr{H})$ and $lambda_1,lambda_2inmathbb{C}.特别是，当 $N( .)$ 是希尔伯特-施密特规范时，我们证明了一对有界线性算子的一些希尔伯特-施密特欧几里得算子半径不等式。
{"title":"Generalized Euclidean operator radius inequalities of a pair of bounded linear operators","authors":"Suvendu Jana","doi":"arxiv-2409.02235","DOIUrl":"https://doi.org/arxiv-2409.02235","url":null,"abstract":"Let $ mathbb{B}(mathscr{H})$ represent the $C^*$-algebra, which consists of\u0000all bounded linear operators on $mathscr{H},$ and let $N ( .) $ be a norm on $\u0000mathbb{B}(mathscr{H})$. We define a norm $w_{(N,e)} (. , . )$ on $\u0000mathbb{B}^2(mathscr{H})$ by $$\u0000w_{(N,e)}(B,C)=underset{|lambda_1|^2+lambda_2|^2leq1}sup\u0000underset{thetainmathbb{R}}sup Nleft(Re\u0000left(e^{itheta}(lambda_1B+lambda_2C)right)right),$$ for every\u0000$B,Cinmathbb{B}(mathscr{H})$ and $lambda_1,lambda_2inmathbb{C}.$ We\u0000investigate basic properties of this norm and prove some bounds involving it.\u0000In particular, when $N( .)$ is the Hilbert-Schmidt norm, we prove some\u0000Hilbert-Schmidt Euclidean operator radius inequalities for a pair of bounded\u0000linear operators.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208174","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 paper, we use the Mordukhovich derivatives to precisely find the covering constants for the metric projection operator onto nonempty closed and convex subsets in uniformly convex and uniformly smooth Banach spaces. We consider three cases of the subsets: closed balls in uniformly convex and uniformly smooth Banach spaces, closed and convex cylinders in l, and the positive cone in L, for some p. By using Theorem 3.1 in [2] and as applications of covering constants obtained in this paper, we prove the solvability of some stochastic fixed-point problems. We also provide three examples with specific solutions of stochastic fixed-point problems.
在本文中，我们利用莫尔杜霍维奇导数精确地找到了均匀凸和均匀光滑巴拿赫空间中的非空闭凸子集上的度量投影算子的覆盖常数。我们考虑了子集的三种情况：均匀凸和均匀光滑巴拿赫空间中的闭球、l 中的闭凸圆柱体和 L 中的正圆锥（对于某些 p）。通过使用 [2] 中的定理 3.1 以及本文中得到的覆盖常数的应用，我们证明了一些随机定点问题的可解性。我们还提供了三个具体解决随机定点问题的例子。
{"title":"Covering Constants for Metric Projection Operator with Applications to Stochastic Fixed-Point Problems","authors":"Jinlu Li","doi":"arxiv-2409.01511","DOIUrl":"https://doi.org/arxiv-2409.01511","url":null,"abstract":"In this paper, we use the Mordukhovich derivatives to precisely find the\u0000covering constants for the metric projection operator onto nonempty closed and\u0000convex subsets in uniformly convex and uniformly smooth Banach spaces. We\u0000consider three cases of the subsets: closed balls in uniformly convex and\u0000uniformly smooth Banach spaces, closed and convex cylinders in l, and the\u0000positive cone in L, for some p. By using Theorem 3.1 in [2] and as applications\u0000of covering constants obtained in this paper, we prove the solvability of some\u0000stochastic fixed-point problems. We also provide three examples with specific\u0000solutions of stochastic fixed-point problems.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208201","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 that the key property in models of Nonlinear Elasticity which corresponds to the non-interpenetration of matter, i.e. injectivity a.e., can be achieved in the class of weak limits of homeomorphisms under very minimal assumptions. Let $Omegasubseteq mathbb{R}^n$ be a domain and let $p>leftlfloorfrac{n}{2}rightrfloor$ for $ngeq 4$ or $pgeq 1$ for $n=2,3$. Assume that $f_kin W^{1,p}$ is a sequence of homeomorphisms such that $f_krightharpoonup f$ weakly in $W^{1,p}$ and assume that $J_f>0$ a.e. Then we show that $f$ is injective a.e.
{"title":"Weak limits of Sobolev homeomorphisms are one to one","authors":"Ondřej Bouchala, Stanislav Hencl, Zheng Zhu","doi":"arxiv-2409.01260","DOIUrl":"https://doi.org/arxiv-2409.01260","url":null,"abstract":"We prove that the key property in models of Nonlinear Elasticity which\u0000corresponds to the non-interpenetration of matter, i.e. injectivity a.e., can\u0000be achieved in the class of weak limits of homeomorphisms under very minimal\u0000assumptions. Let $Omegasubseteq mathbb{R}^n$ be a domain and let\u0000$p>leftlfloorfrac{n}{2}rightrfloor$ for $ngeq 4$ or $pgeq 1$ for\u0000$n=2,3$. Assume that $f_kin W^{1,p}$ is a sequence of homeomorphisms such that\u0000$f_krightharpoonup f$ weakly in $W^{1,p}$ and assume that $J_f>0$ a.e. Then we\u0000show that $f$ is injective a.e.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208177","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 address the existence of non-trivial closed invariant subspaces of operators $T$ on Banach spaces whenever their square $T^2$ have or, more generally, whether there exists a polynomial $p$ with $mbox{deg}(p)geq 2$ such that the lattice of invariant subspaces of $p(T)$ is non-trivial. In the Hilbert space setting, the $T^2$-problem was posed by Halmos in the seventies and in 2007, Foias, Jung, Ko and Pearcy conjectured it could be equivalent to the emph{Invariant Subspace Problem}.
{"title":"A note on a Halmos problem","authors":"Maximiliano Contino, Eva Gallardo-Gutierrez","doi":"arxiv-2409.01167","DOIUrl":"https://doi.org/arxiv-2409.01167","url":null,"abstract":"We address the existence of non-trivial closed invariant subspaces of\u0000operators $T$ on Banach spaces whenever their square $T^2$ have or, more\u0000generally, whether there exists a polynomial $p$ with $mbox{deg}(p)geq 2$\u0000such that the lattice of invariant subspaces of $p(T)$ is non-trivial. In the\u0000Hilbert space setting, the $T^2$-problem was posed by Halmos in the seventies\u0000and in 2007, Foias, Jung, Ko and Pearcy conjectured it could be equivalent to\u0000the emph{Invariant Subspace Problem}.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208181","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}
{"title":"A class of Berezin-type operators on weighted Fock spaces with $A_{infty}$-type weights","authors":"Jiale Chen","doi":"arxiv-2409.01132","DOIUrl":"https://doi.org/arxiv-2409.01132","url":null,"abstract":"Let $0<alpha,beta,t<infty$ and $mu$ be a positive Borel measure on\u0000$mathbb{C}^n$. We consider the Berezin-type operator\u0000$S^{t,alpha,beta}_{mu}$ defined by\u0000$$S^{t,alpha,beta}_{mu}f(z):=left(int_{mathbb{C}^n}e^{-frac{beta}{2}|z-u|^2}|f(u)|^te^{-frac{alpha\u0000t}{2}|u|^2}dmu(u)right)^{1/t},quad zinmathbb{C}^n.$$ We completely\u0000characterize the boundedness and compactness of $S^{t,alpha,beta}_{mu}$ from\u0000the weighted Fock space $F^p_{alpha,w}$ into the Lebesgue space $L^q(wdv)$ for\u0000all possible indices, where $w$ is a weight on $mathbb{C}^n$ that satisfies an\u0000$A_{infty}$-type condition. This solves an open problem raised by Zhou, Zhao\u0000and Tang [Banach J. Math. Anal. 18 (2024), Paper No. 20]. As an application, we\u0000obtain the description of the boundedness and compactness of Toeplitz-type\u0000operators acting between weighted Fock spaces induced by $A_{infty}$-type\u0000weights.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208183","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}
Fusion frames are widely studied for their applications in recovering signals from large data. These are proved to be very useful in many areas, such as, distributed processing, wireless sensor networks, packet encoding. Inspired by the work of Bemrose et al.cite{Be16}, this paper delves into the properties and characterizations of weaving fusion frames.
{"title":"A note on weaving fusion frames","authors":"Animesh Bhandari","doi":"arxiv-2409.01288","DOIUrl":"https://doi.org/arxiv-2409.01288","url":null,"abstract":"Fusion frames are widely studied for their applications in recovering signals\u0000from large data. These are proved to be very useful in many areas, such as,\u0000distributed processing, wireless sensor networks, packet encoding. Inspired by\u0000the work of Bemrose et al.cite{Be16}, this paper delves into the properties\u0000and characterizations of weaving fusion frames.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208178","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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