We prove that every Banach space admitting a Gateaux smooth norm and containing a complemented copy of $ell_1$ has an equivalent renorming which is simultaneously G^ateaux smooth and octahedral. This is a partial solution to a problem from the early nineties.
{"title":"Octahedrality and Gâteaux smoothness","authors":"Ch. Cobollo, P. Hájek","doi":"arxiv-2408.03737","DOIUrl":"https://doi.org/arxiv-2408.03737","url":null,"abstract":"We prove that every Banach space admitting a Gateaux smooth norm and\u0000containing a complemented copy of $ell_1$ has an equivalent renorming which is\u0000simultaneously G^ateaux smooth and octahedral. This is a partial solution to a\u0000problem from the early nineties.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"194 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930633","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}
A collectively $sigma$-Levi set of operators is a generalization of the $sigma$-Levi operator. By use of collective order convergence, we investigate relations between collectively $sigma$-Levi and collectively compact sets of operators.
{"title":"On collectively $σ$-Levi sets of operators","authors":"Eduard Emelyanov","doi":"arxiv-2408.03686","DOIUrl":"https://doi.org/arxiv-2408.03686","url":null,"abstract":"A collectively $sigma$-Levi set of operators is a generalization of the\u0000$sigma$-Levi operator. By use of collective order convergence, we investigate\u0000relations between collectively $sigma$-Levi and collectively compact sets of\u0000operators.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"304 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930554","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 investigate a class of Fourier integral operators with weakened symbols, which satisfy a multi-parameter differential inequality in $R^n$. We establish that these operators retain the classical $L^p$ boundedness and the $H^1$ to $L^1$ boundedness. Notably, the Hardy space considered here is the traditional single-parameter Hardy space rather than a product Hardy space.
{"title":"Boundedness of New Type Fourier Integral Operators with Product Structure","authors":"Chaoqiang Tan, Zipeng Wang","doi":"arxiv-2408.03211","DOIUrl":"https://doi.org/arxiv-2408.03211","url":null,"abstract":"We investigate a class of Fourier integral operators with weakened symbols,\u0000which satisfy a multi-parameter differential inequality in $R^n$. We establish\u0000that these operators retain the classical $L^p$ boundedness and the $H^1$ to\u0000$L^1$ boundedness. Notably, the Hardy space considered here is the traditional\u0000single-parameter Hardy space rather than a product Hardy space.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930559","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}
Rhaly operators, as generalizations of the Ces`aro operator, are studied from the standpoint of view of spectral theory and invariant subspaces, extending previous results by Rhaly and Leibowitz to a framework where generalized Ces`aro operators arise naturally.
{"title":"Rhaly operators: more on generalized Cesàro operators","authors":"Eva A. Gallardo-Gutiérrez, Jonathan R. Partington","doi":"arxiv-2408.03182","DOIUrl":"https://doi.org/arxiv-2408.03182","url":null,"abstract":"Rhaly operators, as generalizations of the Ces`aro operator, are studied\u0000from the standpoint of view of spectral theory and invariant subspaces,\u0000extending previous results by Rhaly and Leibowitz to a framework where\u0000generalized Ces`aro operators arise naturally.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930564","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}
Ehsan Ameli, Ali Akbar Arefijamaal, Fahimeh Arabyani Neyshaburi
Computing the excess as a method of measuring the redundancy of frames was recently introduced to address certain issues in frame theory. In this paper, the concept of excess for the fusion frame setting is studied. Initially, a local approach is presented to determine exactly which part of each subspace should be considered as redundancy. Then, several explicit methods are provided to compute the excess of fusion frames and their $Q$-duals. In particular, some upper bounds for the excess of $Q$-dual fusion frames are established. It turns out that each fusion frame and its $Q$-dual may not necessarily have the same excess. Along the way, unlike ordinary frames, it follows that for every $n in Bbb{N}$, we can provide a fusion frame together an its $Q$-dual such that the difference of their excess is $n$. Furthermore, the connection between the excess of fusion frames and their orthogonal complement fusion frames are completely characterized. Finally, several examples are exhibited to confirm the obtained results.
{"title":"Excess of Fusion Frames: A Comprehensive Approach","authors":"Ehsan Ameli, Ali Akbar Arefijamaal, Fahimeh Arabyani Neyshaburi","doi":"arxiv-2408.03179","DOIUrl":"https://doi.org/arxiv-2408.03179","url":null,"abstract":"Computing the excess as a method of measuring the redundancy of frames was\u0000recently introduced to address certain issues in frame theory. In this paper,\u0000the concept of excess for the fusion frame setting is studied. Initially, a\u0000local approach is presented to determine exactly which part of each subspace\u0000should be considered as redundancy. Then, several explicit methods are provided\u0000to compute the excess of fusion frames and their $Q$-duals. In particular, some\u0000upper bounds for the excess of $Q$-dual fusion frames are established. It turns\u0000out that each fusion frame and its $Q$-dual may not necessarily have the same\u0000excess. Along the way, unlike ordinary frames, it follows that for every $n in\u0000Bbb{N}$, we can provide a fusion frame together an its $Q$-dual such that the\u0000difference of their excess is $n$. Furthermore, the connection between the\u0000excess of fusion frames and their orthogonal complement fusion frames are\u0000completely characterized. Finally, several examples are exhibited to confirm\u0000the obtained results.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930562","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 the Gelfand-Shilov setting, the localisation operator $A^{varphi_1,varphi_2}_a$ is equal to the Weyl operator whose symbol is the convolution of $a$ with the Wigner transform of the windows $varphi_2$ and $varphi_1$. We employ this fact, to extend the definition of localisation operators to symbols $a$ having very fast super-exponential growth by allowing them to be mappings from ${mathcal D}^{{M_p}}(mathbb R^d)$ into ${mathcal D}'^{{M_p}}(mathbb R^d)$, where $M_p$, $pinmathbb N$, is a non-quasi-analytic Gevrey type sequence. By choosing the windows $varphi_1$ and $varphi_2$ appropriately, our main results show that one can consider symbols with growth in position space of the form $exp(exp(l|cdot|^q))$, $l,q>0$.
{"title":"Extension of Localisation Operators to Ultradistributional Symbols With Super-Exponential Growth","authors":"Stevan Pilipović, Bojan Prangoski, Đorđe Vučković","doi":"arxiv-2408.02437","DOIUrl":"https://doi.org/arxiv-2408.02437","url":null,"abstract":"In the Gelfand-Shilov setting, the localisation operator\u0000$A^{varphi_1,varphi_2}_a$ is equal to the Weyl operator whose symbol is the\u0000convolution of $a$ with the Wigner transform of the windows $varphi_2$ and\u0000$varphi_1$. We employ this fact, to extend the definition of localisation\u0000operators to symbols $a$ having very fast super-exponential growth by allowing\u0000them to be mappings from ${mathcal D}^{{M_p}}(mathbb R^d)$ into ${mathcal\u0000D}'^{{M_p}}(mathbb R^d)$, where $M_p$, $pinmathbb N$, is a\u0000non-quasi-analytic Gevrey type sequence. By choosing the windows $varphi_1$\u0000and $varphi_2$ appropriately, our main results show that one can consider\u0000symbols with growth in position space of the form $exp(exp(l|cdot|^q))$,\u0000$l,q>0$.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930557","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}
This article generalizes the results of [J. Math. Anal. Appl. 512 (2022), Article No. 126075], which presented a theory of distributions (generalized functions) with a singular curve contained in the domain of the test functions. In this present article we construct a theory of distributions in $mathbb{R}^n$ with a ``thick submanifold'', that is, a new theory of thick distributions in $mathbb{R}^n$ whose domain contains a submanifold on which test functions may be singular.
{"title":"Distributions in spaces with thick submanifolds","authors":"Jiajia Ding, Jasson Vindas, Yunyun Yang","doi":"arxiv-2408.02864","DOIUrl":"https://doi.org/arxiv-2408.02864","url":null,"abstract":"This article generalizes the results of [J. Math. Anal. Appl. 512 (2022),\u0000Article No. 126075], which presented a theory of distributions (generalized\u0000functions) with a singular curve contained in the domain of the test functions.\u0000In this present article we construct a theory of distributions in\u0000$mathbb{R}^n$ with a ``thick submanifold'', that is, a new theory of thick\u0000distributions in $mathbb{R}^n$ whose domain contains a submanifold on which\u0000test functions may be singular.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930634","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 investigate $L^p$ boundedness of the spherical maximal operator $mathfrak{m}^alpha$ of (complex) order $alpha$ on the $n$-dimensional hyperbolic space $mathbb{H}^n$, which was introduced and studied by Kohen [13]. We prove that when $ngeq 2$, for $alphainmathbb{R}$ and $11-n+n/p$ for $1 max {{(2-n)/p}-{1/(p p_n)}, {(2-n)/p} - (p-2)/ [p p_n(p_n-2) ] } $ for $2leq pleq infty$, with $p_n=2(n+1)/(n-1)$ for $ngeq 3$ and $p_n=4$ for $n=2$.
{"title":"The spherical maximal operators on hyperbolic spaces","authors":"Peng Chen, Minxing Shen, Yunxiang Wang, Lixin Yan","doi":"arxiv-2408.02180","DOIUrl":"https://doi.org/arxiv-2408.02180","url":null,"abstract":"In this article we investigate $L^p$ boundedness of the spherical maximal\u0000operator $mathfrak{m}^alpha$ of (complex) order $alpha$ on the\u0000$n$-dimensional hyperbolic space $mathbb{H}^n$, which was introduced and\u0000studied by Kohen [13]. We prove that when $ngeq 2$, for $alphainmathbb{R}$\u0000and $1<p<infty$, if begin{eqnarray*}\u0000|mathfrak{m}^alpha(f)|_{L^p(mathbb{H}^n)}leq C|f|_{L^p(mathbb{H}^n)},\u0000end{eqnarray*} then we must have $alpha>1-n+n/p$ for $1<pleq 2$; or\u0000$alphageq max{1/p-(n-1)/2,(1-n)/p}$ for $2<p<infty$. Furthermore, we\u0000improve the result of Kohen [13, Theorem 3] by showing that\u0000$mathfrak{m}^alpha$ is bounded on $L^p(mathbb{H}^n)$ provided that\u0000$mathop{mathrm{Re}} alpha> max {{(2-n)/p}-{1/(p p_n)}, {(2-n)/p} -\u0000(p-2)/ [p p_n(p_n-2) ] } $ for $2leq pleq infty$, with $p_n=2(n+1)/(n-1)$\u0000for $ngeq 3$ and $p_n=4$ for $n=2$.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930560","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":"The Generalized Grand Wiener Amalgam Spaces and the boundedness of Hardy-Littlewood maximal operators","authors":"A. Turan Gürkanlı","doi":"arxiv-2408.02406","DOIUrl":"https://doi.org/arxiv-2408.02406","url":null,"abstract":"Let $1<p,q<infty$ and let $a(x), b(x)$ be weight functions. In the present\u0000paper we define and investigate some basic properties of generalized grand\u0000Wiener amalgam space $W( L^{p),theta_1}(mathbb R^{n})),\u0000L^{q),theta_2}(mathbb R^{n})),$ where generalized grand Lebesgue spaces\u0000$L_{a}^{p)}(mathbb R^{n})$ and $L_{b}^{q)}(mathbb R^{n}),$ are local and\u0000global components, respectively. Next we study embeddings for these spaces,\u0000also we give some more properties of these spaces. At the end of this work, we\u0000discuss boundedness and unboundedness of the Hardy-Littlewood maximal operator\u0000between some generalized grand Wiener amalgam spaces.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969804","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 $U$ be a bounded open subset of the complex plane and let $A_{alpha}(U)$ denote the set of functions analytic on $U$ that also belong to the little Lipschitz class with Lipschitz exponent $alpha$. It is shown that if $A_{alpha}(U)$ admits a bounded point derivation at $x in partial U$, then there is an approximate Taylor Theorem for $A_{alpha}(U)$ at $x$. This extends and generalizes known results concerning bounded point derivations.
{"title":"Approximate Taylor theorem for analytic Lipschitz functions","authors":"Stephen Deterding","doi":"arxiv-2408.02522","DOIUrl":"https://doi.org/arxiv-2408.02522","url":null,"abstract":"Let $U$ be a bounded open subset of the complex plane and let $A_{alpha}(U)$\u0000denote the set of functions analytic on $U$ that also belong to the little\u0000Lipschitz class with Lipschitz exponent $alpha$. It is shown that if\u0000$A_{alpha}(U)$ admits a bounded point derivation at $x in partial U$, then\u0000there is an approximate Taylor Theorem for $A_{alpha}(U)$ at $x$. This extends\u0000and generalizes known results concerning bounded point derivations.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"193 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930487","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}
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：