We prove new Fourier restriction estimates to the unit sphere $mathbb{S}^{d-1}$ on the class of $O(d−k) times O(k)$-symmetric functions, for every $d ge 4$ and $2 le k le d-2$. As an application, we establish the existence of maximizers for the endpoint Tomas–Stein inequality within that class. Moreover, we construct examples showing that the range of Lebesgue exponents in our estimates is sharp in the Tomas–Stein regime.
我们证明了在$O(d−k) times O(k)$对称函数类上，对于每一个$d ge 4$和$2 le k le d-2$，单位球$mathbb{S}^{d-1}$的新的傅里叶限制估计。作为一个应用，我们在该类中建立了端点Tomas-Stein不等式的极大值存在性。此外，我们构造的例子表明，在我们的估计中，勒贝格指数的范围在托马斯-斯坦政权中是尖锐的。
{"title":"The Tomas–Stein inequality under the effect of symmetries","authors":"Rainer Mandel, D. O. Silva","doi":"10.5445/IR/1000134152","DOIUrl":"https://doi.org/10.5445/IR/1000134152","url":null,"abstract":"We prove new Fourier restriction estimates to the unit sphere $mathbb{S}^{d-1}$ on the class of $O(d−k) times O(k)$-symmetric functions, for every $d ge 4$ and $2 le k le d-2$. As an application, we establish the existence of maximizers for the endpoint Tomas–Stein inequality within that class. Moreover, we construct examples showing that the range of Lebesgue exponents in our estimates is sharp in the Tomas–Stein regime.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73747058","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 new extension of Pitt's theorem for compact operators between quasi-Banach lattices, which permits to describe unconditional bases of finite direct sums of Banach spaces $mathbb{X}_{1}oplusdotsoplusmathbb{X}_{n}$ as direct sums of unconditional bases of its summands. The general splitting principle we obtain yields, in particular, that if each $mathbb{X}_{i}$ has a unique unconditional basis (up to equivalence and permutation), then $mathbb{X}_{1}oplus cdotsoplusmathbb{X}_{n}$ has a unique unconditional basis too. Among the novel applications of our techniques to the structure of Banach and quasi-Banach spaces we have that the space $ell_2oplus mathcal{T}^{(2)}$ has a unique unconditional basis.
{"title":"Uniqueness of unconditional basis of $ell _{2}oplus mathcal {T}^{(2)}$","authors":"F. Albiac, J. L. Ansorena","doi":"10.1090/PROC/15670","DOIUrl":"https://doi.org/10.1090/PROC/15670","url":null,"abstract":"We provide a new extension of Pitt's theorem for compact operators between quasi-Banach lattices, which permits to describe unconditional bases of finite direct sums of Banach spaces $mathbb{X}_{1}oplusdotsoplusmathbb{X}_{n}$ as direct sums of unconditional bases of its summands. The general splitting principle we obtain yields, in particular, that if each $mathbb{X}_{i}$ has a unique unconditional basis (up to equivalence and permutation), then $mathbb{X}_{1}oplus cdotsoplusmathbb{X}_{n}$ has a unique unconditional basis too. Among the novel applications of our techniques to the structure of Banach and quasi-Banach spaces we have that the space $ell_2oplus mathcal{T}^{(2)}$ has a unique unconditional basis.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86670029","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}
The global existence and stability of the solution to the delay differential equation (*)$dot{u} = A(t)u + G(t,u(t-tau)) + f(t)$, $tge 0$, $u(t) = v(t)$, $-tau le tle 0$, are studied. Here $A(t):mathcal{H}to mathcal{H}$ is a closed, densely defined, linear operator in a Hilbert space $mathcal{H}$ and $G(t,u)$ is a nonlinear operator in $mathcal{H}$ continuous with respect to $u$ and $t$. We assume that the spectrum of $A(t)$ lies in the half-plane $Re lambda le gamma(t)$, where $gamma(t)$ is not necessarily negative and $|G(t,u)| le alpha(t)|u|^p$, $p>1$, $tge 0$. Sufficient conditions for the solution to the equation to exist globally, to be bounded and to converge to zero as $t$ tends to $infty$, under the non-classical assumption that $gamma(t)$ can take positive values, are proposed and justified.
研究了时滞微分方程(*)$dot{u} = A(t)u + G(t,u(t-tau)) + f(t)$, $tge 0$, $u(t) = v(t)$, $-tau le tle 0$解的全局存在性和稳定性。这里$A(t):mathcal{H}to mathcal{H}$是Hilbert空间$mathcal{H}$中的一个封闭的、密集定义的线性算子，$G(t,u)$是$mathcal{H}$中关于$u$和$t$的连续的非线性算子。我们假设$A(t)$的光谱位于半平面$Re lambda le gamma(t)$，其中$gamma(t)$不一定是负的，并且$|G(t,u)| le alpha(t)|u|^p$, $p>1$, $tge 0$。在$gamma(t)$可以取正值的非经典假设下，提出并证明了方程解全局存在、有界并在$t$趋于$infty$时收敛于零的充分条件。
{"title":"Stability of solutions to some abstract evolution equations with delay","authors":"N. S. Hoang, A. Ramm","doi":"10.47443/cm.2021.0004","DOIUrl":"https://doi.org/10.47443/cm.2021.0004","url":null,"abstract":"The global existence and stability of the solution to the delay differential equation (*)$dot{u} = A(t)u + G(t,u(t-tau)) + f(t)$, $tge 0$, $u(t) = v(t)$, $-tau le tle 0$, are studied. Here $A(t):mathcal{H}to mathcal{H}$ is a closed, densely defined, linear operator in a Hilbert space $mathcal{H}$ and $G(t,u)$ is a nonlinear operator in $mathcal{H}$ continuous with respect to $u$ and $t$. We assume that the spectrum of $A(t)$ lies in the half-plane $Re lambda le gamma(t)$, where $gamma(t)$ is not necessarily negative and $|G(t,u)| le alpha(t)|u|^p$, $p>1$, $tge 0$. Sufficient conditions for the solution to the equation to exist globally, to be bounded and to converge to zero as $t$ tends to $infty$, under the non-classical assumption that $gamma(t)$ can take positive values, are proposed and justified.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85405567","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 three new examples of twisted Hilbert spaces by considering properties that are "close" to Hilbert. We denote them $Z(mathcal J)$, $Z(mathcal S^2)$ and $Z(mathcal T_s^2)$. The first space is asymptotically Hilbertian but not weak Hilbert. On the opposite side, $Z(mathcal S^2)$ and $Z(mathcal T_s^2)$ are not asymptotically Hilbertian. Moreover, the space $Z(mathcal T_s^2)$ is a HAPpy space and the technique to prove it gives a "twisted" version of a theorem of Johnson and Szankowski (Ann. of Math. 176:1987--2001, 2012). This is, we can construct a nontrivial twisted Hilbert space such that the isomorphism constant from its $n$-dimensional subspaces to $ell_2^n$ grows to infinity as slowly as we wish when $nto infty$.
{"title":"Some more twisted Hilbert spaces","authors":"Daniel Morales, J. Su'arez","doi":"10.5186/aasfm.2021.4653","DOIUrl":"https://doi.org/10.5186/aasfm.2021.4653","url":null,"abstract":"We provide three new examples of twisted Hilbert spaces by considering properties that are \"close\" to Hilbert. We denote them $Z(mathcal J)$, $Z(mathcal S^2)$ and $Z(mathcal T_s^2)$. The first space is asymptotically Hilbertian but not weak Hilbert. On the opposite side, $Z(mathcal S^2)$ and $Z(mathcal T_s^2)$ are not asymptotically Hilbertian. Moreover, the space $Z(mathcal T_s^2)$ is a HAPpy space and the technique to prove it gives a \"twisted\" version of a theorem of Johnson and Szankowski (Ann. of Math. 176:1987--2001, 2012). This is, we can construct a nontrivial twisted Hilbert space such that the isomorphism constant from its $n$-dimensional subspaces to $ell_2^n$ grows to infinity as slowly as we wish when $nto infty$.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87661192","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 : 2020-12-02DOI: 10.22541/au.163257138.88871318/v1
Ashish Pathak, Dileep Kumar
Using the theory of continuous Bessel wavelet transform in $L^2 (mathbb{R})$-spaces, we established the Parseval and inversion formulas for the $L^{p,sigma}(mathbb{R}^+)$- spaces. We investigate continuity and boundedness properties of Bessel wavelet transform in Besov-Hankel spaces. Our main results: are the characterization of Besov-Hankel spaces by using continuous Bessel wavelet coefficient.
{"title":"Besov-Hankel norms in terms of the continuous Bessel wavelet transform","authors":"Ashish Pathak, Dileep Kumar","doi":"10.22541/au.163257138.88871318/v1","DOIUrl":"https://doi.org/10.22541/au.163257138.88871318/v1","url":null,"abstract":"Using the theory of continuous Bessel wavelet transform in $L^2\u0000(mathbb{R})$-spaces, we established the Parseval and\u0000inversion formulas for the\u0000$L^{p,sigma}(mathbb{R}^+)$-\u0000spaces. We investigate continuity and boundedness properties of Bessel\u0000wavelet transform in Besov-Hankel spaces. Our main results: are the\u0000characterization of Besov-Hankel spaces by using continuous Bessel\u0000wavelet coefficient.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81093276","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 : 2020-11-30DOI: 10.22130/SCMA.2021.140176.874
H. Labrigui, S. Kabbaj
In this work, we introduce a new concept of integral $K$-operator frame for the set of all adjointable operators from Hilbert $C^{ast}$-modules $mathcal{H}$ to it self noted $End_{mathcal{A}}^{ast}(mathcal{H}) $. We give some propertis relating some construction of integral $K$-operator frame and operators preserving integral $K$-operator frame and we establish some new results.
{"title":"Integral $K$-Operator Frames for $End_{mathcal{A}}^{ast}(mathcal{H})$","authors":"H. Labrigui, S. Kabbaj","doi":"10.22130/SCMA.2021.140176.874","DOIUrl":"https://doi.org/10.22130/SCMA.2021.140176.874","url":null,"abstract":"In this work, we introduce a new concept of integral $K$-operator frame for the set of all adjointable operators from Hilbert $C^{ast}$-modules $mathcal{H}$ to it self noted $End_{mathcal{A}}^{ast}(mathcal{H}) $. We give some propertis relating some construction of integral $K$-operator frame and operators preserving integral $K$-operator frame and we establish some new results.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85033570","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 : 2020-11-28DOI: 10.2140/INVOLVE.2021.14.349
A. Myers, Muhammadyusuf Odinaev, David Walmsley
Let $H(mathbb{C})$ be the set of all entire functions endowed with the topology of uniform convergence on compact sets. Let $lambda,binmathbb{C}$, let $C_gamma:H(mathbb{C})to H(mathbb{C})$ be the composition operator $C_gamma f(z)=f(lambda z+b)$, and let $D$ be the derivative operator. We extend results on the hypercyclicity of the non-convolution operators $T_{lambda,b}=C_gamma circ D$ by showing that whenever $|lambda|geq 1$, the algebra of operators begin{align*} {psi(T_{lambda,b}): psi(z)in H(mathbb{C}), psi(0)=0 text{ and } psi(T_{lambda,b}) text{ is continuous}} end{align*} and the family of operators begin{align*} {C_gammacircvarphi(D): varphi(z) text{ is an entire function of exponential type with } varphi(0)=0} end{align*} consist entirely of hypercyclic operators (i.e., each operator has a dense orbit).
设$H(mathbb{C})$为紧集上具有一致收敛拓扑的所有完整函数的集合。设$lambda,binmathbb{C}$, $C_gamma:H(mathbb{C})to H(mathbb{C})$是复合运算符$C_gamma f(z)=f(lambda z+b)$, $D$是导数运算符。我们扩展了关于非卷积算子$T_{lambda,b}=C_gamma circ D$的超循环性的结果，证明当$|lambda|geq 1$时，算子的代数begin{align*} {psi(T_{lambda,b}): psi(z)in H(mathbb{C}), psi(0)=0 text{ and } psi(T_{lambda,b}) text{ is continuous}} end{align*}和算子族begin{align*} {C_gammacircvarphi(D): varphi(z) text{ is an entire function of exponential type with } varphi(0)=0} end{align*}完全由超循环算子组成(即每个算子都有一个密集的轨道)。
{"title":"Two families of hypercyclic nonconvolution operators","authors":"A. Myers, Muhammadyusuf Odinaev, David Walmsley","doi":"10.2140/INVOLVE.2021.14.349","DOIUrl":"https://doi.org/10.2140/INVOLVE.2021.14.349","url":null,"abstract":"Let $H(mathbb{C})$ be the set of all entire functions endowed with the topology of uniform convergence on compact sets. Let $lambda,binmathbb{C}$, let $C_gamma:H(mathbb{C})to H(mathbb{C})$ be the composition operator $C_gamma f(z)=f(lambda z+b)$, and let $D$ be the derivative operator. We extend results on the hypercyclicity of the non-convolution operators $T_{lambda,b}=C_gamma circ D$ by showing that whenever $|lambda|geq 1$, the algebra of operators begin{align*} {psi(T_{lambda,b}): psi(z)in H(mathbb{C}), psi(0)=0 text{ and } psi(T_{lambda,b}) text{ is continuous}} end{align*} and the family of operators begin{align*} {C_gammacircvarphi(D): varphi(z) text{ is an entire function of exponential type with } varphi(0)=0} end{align*} consist entirely of hypercyclic operators (i.e., each operator has a dense orbit).","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91391639","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 : 2020-11-18DOI: 10.7494/OPMATH.2021.41.3.283
D. Alpay, P. Jorgensen
We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present ageneral positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysisand metric geometry and provide a number of examples.
{"title":"New characterizations of reproducing kernel Hilbert spaces and applications to metric geometry","authors":"D. Alpay, P. Jorgensen","doi":"10.7494/OPMATH.2021.41.3.283","DOIUrl":"https://doi.org/10.7494/OPMATH.2021.41.3.283","url":null,"abstract":"We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present ageneral positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysisand metric geometry and provide a number of examples.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89893038","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 : 2020-11-11DOI: 10.1007/S43670-021-00011-5
Antonio G. Garc'ia
{"title":"Average sampling in certain subspaces of Hilbert–Schmidt operators on $$L^2(mathbb {R}^d)$$","authors":"Antonio G. Garc'ia","doi":"10.1007/S43670-021-00011-5","DOIUrl":"https://doi.org/10.1007/S43670-021-00011-5","url":null,"abstract":"","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79078563","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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