Bojan Kuzma, Chi-Kwong Li, Edward Poon, Sushil Singla
Let $1 leq k < n$ be integers. Two $n times n$ matrices $A$ and $B$ form a parallel pair with respect to the $k$-numerical radius $w_k$ if $w_k(A + mu B) = w_k(A) + w_k(B)$ for some scalar $mu$ with $|mu| = 1$; they form a TEA (triangle equality attaining) pair if the preceding equation holds for $mu = 1$. We classify linear bijections on $mathbb M_n$ and on $mathbb H_n$ which preserve parallel pairs or TEA pairs. Such preservers are scalar multiples of $w_k$-isometries, except for some exceptional maps on $mathbb H_n$ when $n=2k$.
{"title":"Linear preservers of parallel matrix pairs with respect to the $k$-numerical radius","authors":"Bojan Kuzma, Chi-Kwong Li, Edward Poon, Sushil Singla","doi":"arxiv-2408.16066","DOIUrl":"https://doi.org/arxiv-2408.16066","url":null,"abstract":"Let $1 leq k < n$ be integers. Two $n times n$ matrices $A$ and $B$ form a\u0000parallel pair with respect to the $k$-numerical radius $w_k$ if $w_k(A + mu B)\u0000= w_k(A) + w_k(B)$ for some scalar $mu$ with $|mu| = 1$; they form a TEA\u0000(triangle equality attaining) pair if the preceding equation holds for $mu =\u00001$. We classify linear bijections on $mathbb M_n$ and on $mathbb H_n$ which\u0000preserve parallel pairs or TEA pairs. Such preservers are scalar multiples of\u0000$w_k$-isometries, except for some exceptional maps on $mathbb H_n$ when\u0000$n=2k$.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208196","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}
Weak R-duals, a generalization of R-duals, were recently introduced; for which duality relations were established. In this paper, we consider the problem of characterizing a given frame sequence to be a weak R-dual of a given frame. Further, we apply these characterization results to the Gabor frame setting and prove that the weak R-duality of the adjoint Gabor system leads to the R-duality of the same, thereby indicating an approach to answer the famous problem of the adjoint Gabor system being an R-dual of a given Gabor frame.
弱 R 对偶是对 R 对偶的一种概括，最近才被提出，并为此建立了对偶关系。在本文中，我们考虑了将给定帧序列表征为给定帧的弱 R 对偶的问题。此外，我们还将这些表征结果应用于 Gabor 帧集，并证明邻接 Gabor 系统的弱 R 对偶性会导致相同的 R 对偶性，从而指明了一种方法来回答邻接 Gabor 系统是给定 Gabor 帧的 R 对偶这一著名问题。
{"title":"Characterizations of weak R-duality and its application to Gabor frames","authors":"Himanshi Bansal, P. Devaraj, S. Arati","doi":"arxiv-2408.14952","DOIUrl":"https://doi.org/arxiv-2408.14952","url":null,"abstract":"Weak R-duals, a generalization of R-duals, were recently introduced; for\u0000which duality relations were established. In this paper, we consider the\u0000problem of characterizing a given frame sequence to be a weak R-dual of a given\u0000frame. Further, we apply these characterization results to the Gabor frame\u0000setting and prove that the weak R-duality of the adjoint Gabor system leads to\u0000the R-duality of the same, thereby indicating an approach to answer the famous\u0000problem of the adjoint Gabor system being an R-dual of a given Gabor frame.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"317 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226624","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}
Many results for Banach spaces also hold for quasi-Banach spaces. One important such example is results depending on the Baire Category Theorem (BCT). We use the BCT to explore Lions problem for a quasi-Banach couple $(A_0, A_1)$. Lions problem, posed in 1960's, is to prove that different parameters $(theta,p)$ produce different interpolation spaces $(A_0, A_1)_{theta, p}$. We first establish conditions on $A_0$ and $A_1$ so that interpolation spaces of this couple are strictly intermediate spaces between $A_0+A_1$ and $A_0cap A_1$. This result, together with a reiteration theorem, gives a partial solution to Lions problem for quasi-Banach couples. We then apply our interpolation result to (partially) answer a question posed by Pietsch. More precisely, we show that if $pneq p^*$ the operator ideals $mathcal{L}^{(a)}_{p,q}(X,Y)$, $mathcal{L}^{(a)}_{p^*,q^*}(X,Y)$ generated by approximation numbers are distinct. Moreover, for any fixed $p$, either all operator ideals $mathcal{L}^{(a)}_{p,q}(X,Y)$ collapse into a unique space or they are pairwise distinct. We cite counterexamples which show that using interpolation spaces is not appropriate to solve Pietsch's problem for operator ideals based on general $s$-numbers. However, the BCT can be used to prove a lethargy result for arbitrary $s$-numbers which guarantees that, under very minimal conditions on $X,Y$, the space $mathcal{L}^{(s)}_{p,q}(X,Y)$ is strictly embedded into $mathcal{L}^{mathcal{A}}(X,Y)$. The paper is dedicated to the memory of Prof. A. Pietsch, who passed away recently.
{"title":"Using the Baire Category Theorem to Explore Lions Problem for Quasi-Banach Spaces","authors":"A. G. Aksoy, J. M. Almira","doi":"arxiv-2408.14893","DOIUrl":"https://doi.org/arxiv-2408.14893","url":null,"abstract":"Many results for Banach spaces also hold for quasi-Banach spaces. One\u0000important such example is results depending on the Baire Category Theorem\u0000(BCT). We use the BCT to explore Lions problem for a quasi-Banach couple $(A_0,\u0000A_1)$. Lions problem, posed in 1960's, is to prove that different parameters\u0000$(theta,p)$ produce different interpolation spaces $(A_0, A_1)_{theta, p}$.\u0000We first establish conditions on $A_0$ and $A_1$ so that interpolation spaces\u0000of this couple are strictly intermediate spaces between $A_0+A_1$ and $A_0cap\u0000A_1$. This result, together with a reiteration theorem, gives a partial\u0000solution to Lions problem for quasi-Banach couples. We then apply our\u0000interpolation result to (partially) answer a question posed by Pietsch. More\u0000precisely, we show that if $pneq p^*$ the operator ideals\u0000$mathcal{L}^{(a)}_{p,q}(X,Y)$, $mathcal{L}^{(a)}_{p^*,q^*}(X,Y)$ generated by\u0000approximation numbers are distinct. Moreover, for any fixed $p$, either all\u0000operator ideals $mathcal{L}^{(a)}_{p,q}(X,Y)$ collapse into a unique space or\u0000they are pairwise distinct. We cite counterexamples which show that using\u0000interpolation spaces is not appropriate to solve Pietsch's problem for operator\u0000ideals based on general $s$-numbers. However, the BCT can be used to prove a\u0000lethargy result for arbitrary $s$-numbers which guarantees that, under very\u0000minimal conditions on $X,Y$, the space $mathcal{L}^{(s)}_{p,q}(X,Y)$ is\u0000strictly embedded into $mathcal{L}^{mathcal{A}}(X,Y)$. The paper is dedicated\u0000to the memory of Prof. A. Pietsch, who passed away recently.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208205","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 review the basic properties of paired operators and their adjoints, the transposed paired operators, with particular reference to commutation relations, and we study the properties of their kernels, bringing out their similarities and also, somewhat surprisingly, their stark differences. Various notions expressing different invariance properties are also reviewed and we extend to paired operators some known invariance results.
{"title":"Kernels of paired operators and their adjoints","authors":"M. Cristina Câmara, Jonathan R. Partington","doi":"arxiv-2408.14120","DOIUrl":"https://doi.org/arxiv-2408.14120","url":null,"abstract":"We review the basic properties of paired operators and their adjoints, the\u0000transposed paired operators, with particular reference to commutation\u0000relations, and we study the properties of their kernels, bringing out their\u0000similarities and also, somewhat surprisingly, their stark differences. Various\u0000notions expressing different invariance properties are also reviewed and we\u0000extend to paired operators some known invariance results.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208202","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}
M. G. Cabrera-Padilla, A. Jiménez-Vargas, A. Keten Çopur
Given an open subset $U$ of a complex Banach space $E$, a weight $v$ on $U$, and a complex Banach space $F$, let $mathcal{H}^infty_v(U,F)$ denote the Banach space of all weighted holomorphic mappings $fcolon Uto F$, under the weighted supremum norm $left|fright|_v:=supleft{v(x)left|f(x)right|colon xin Uright}$. In this paper, we introduce and study the classes of weighted holomorphic mappings $mathcal{H}^infty_{vmathcal{K}_{p}}(U,F)$ (resp., $mathcal{H}^infty_{vmathcal{K}_{wp}}(U,F)$ and $mathcal{H}^infty_{vmathcal{K}_{up}}(U,F)$) for which the set $(vf)(U)$ is relatively $p$-compact (resp., relatively weakly $p$-compact and relatively unconditionally $p$-compact). We prove that these mapping classes are characterized by $p$-compact (resp., weakly $p$-compact and unconditionally $p$-compact) linear operators defined on a Banach predual space of $mathcal{H}^infty_v(U)$ by linearization. We show that $mathcal{H}^infty_{vmathcal{K}_{p}}$ (resp., $mathcal{H}^infty_{vmathcal{K}_{wp}}$ and $mathcal{H}^infty_{vmathcal{K}_{up}}$) is a Banach ideal of weighted holomorphic mappings which is generated by composition with the ideal of $p$-compact (resp., weakly $p$-compact and unconditionally $p$-compact) linear operators and contains the Banach ideal of all right $p$-nuclear weighted holomorphic mappings. We also prove that these weighted holomorphic mappings can be factorized through a quotient space of $l_{p^*}$, and $finmathcal{H}^infty_{vmathcal{K}_{p}}(U,F)$ (resp., $finmathcal{H}^infty_{vmathcal{K}_{up}}(U,F))$ if and only if its transposition $f^t$ is quasi $p$-nuclear (resp., quasi unconditionally $p$-nuclear).
{"title":"Weighted holomorphic mappings associated with p-compact type sets","authors":"M. G. Cabrera-Padilla, A. Jiménez-Vargas, A. Keten Çopur","doi":"arxiv-2408.14459","DOIUrl":"https://doi.org/arxiv-2408.14459","url":null,"abstract":"Given an open subset $U$ of a complex Banach space $E$, a weight $v$ on $U$,\u0000and a complex Banach space $F$, let $mathcal{H}^infty_v(U,F)$ denote the\u0000Banach space of all weighted holomorphic mappings $fcolon Uto F$, under the\u0000weighted supremum norm\u0000$left|fright|_v:=supleft{v(x)left|f(x)right|colon xin Uright}$.\u0000In this paper, we introduce and study the classes of weighted holomorphic\u0000mappings $mathcal{H}^infty_{vmathcal{K}_{p}}(U,F)$ (resp.,\u0000$mathcal{H}^infty_{vmathcal{K}_{wp}}(U,F)$ and\u0000$mathcal{H}^infty_{vmathcal{K}_{up}}(U,F)$) for which the set $(vf)(U)$ is\u0000relatively $p$-compact (resp., relatively weakly $p$-compact and relatively\u0000unconditionally $p$-compact). We prove that these mapping classes are\u0000characterized by $p$-compact (resp., weakly $p$-compact and unconditionally\u0000$p$-compact) linear operators defined on a Banach predual space of\u0000$mathcal{H}^infty_v(U)$ by linearization. We show that\u0000$mathcal{H}^infty_{vmathcal{K}_{p}}$ (resp.,\u0000$mathcal{H}^infty_{vmathcal{K}_{wp}}$ and\u0000$mathcal{H}^infty_{vmathcal{K}_{up}}$) is a Banach ideal of weighted\u0000holomorphic mappings which is generated by composition with the ideal of\u0000$p$-compact (resp., weakly $p$-compact and unconditionally $p$-compact) linear\u0000operators and contains the Banach ideal of all right $p$-nuclear weighted\u0000holomorphic mappings. We also prove that these weighted holomorphic mappings\u0000can be factorized through a quotient space of $l_{p^*}$, and\u0000$finmathcal{H}^infty_{vmathcal{K}_{p}}(U,F)$ (resp.,\u0000$finmathcal{H}^infty_{vmathcal{K}_{up}}(U,F))$ if and only if its\u0000transposition $f^t$ is quasi $p$-nuclear (resp., quasi unconditionally\u0000$p$-nuclear).","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208200","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 recontextualize the theory of matrix weights within the setting of Banach lattices. We define an intrinsic notion of directional Banach function spaces, generalizing matrix weighted Lebesgue spaces. Moreover, we prove an extrapolation theorem for these spaces based on the boundedness of the convex-set valued maximal operator. We also provide bounds and equivalences related to the convex body sparse operator. Finally, we introduce a weak-type analogue of directional Banach function spaces. In particular, we show that the weak-type boundedness of the convex-set valued maximal operator on matrix weighted Lebesgue spaces is equivalent to the matrix Muckenhoupt condition, with equivalent constants.
{"title":"A lattice approach to matrix weights","authors":"Zoe Nieraeth","doi":"arxiv-2408.14666","DOIUrl":"https://doi.org/arxiv-2408.14666","url":null,"abstract":"In this paper we recontextualize the theory of matrix weights within the\u0000setting of Banach lattices. We define an intrinsic notion of directional Banach\u0000function spaces, generalizing matrix weighted Lebesgue spaces. Moreover, we\u0000prove an extrapolation theorem for these spaces based on the boundedness of the\u0000convex-set valued maximal operator. We also provide bounds and equivalences\u0000related to the convex body sparse operator. Finally, we introduce a weak-type\u0000analogue of directional Banach function spaces. In particular, we show that the\u0000weak-type boundedness of the convex-set valued maximal operator on matrix\u0000weighted Lebesgue spaces is equivalent to the matrix Muckenhoupt condition,\u0000with equivalent constants.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208199","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 paper presents a point-free version of the Lebesgue integral for simple functions on $sigma$-locales. It describes the integral with respect to a measure defined on the coframe of all $sigma$-sublocales, moving beyond the constraints of Boolean algebras. It also extends the notion of integrable function, usually reserved for measurable functions, to localic general functions.
{"title":"Lebesgue integration on $σ$-locales: simple functions","authors":"Raquel Bernardes","doi":"arxiv-2408.13911","DOIUrl":"https://doi.org/arxiv-2408.13911","url":null,"abstract":"This paper presents a point-free version of the Lebesgue integral for simple\u0000functions on $sigma$-locales. It describes the integral with respect to a\u0000measure defined on the coframe of all $sigma$-sublocales, moving beyond the\u0000constraints of Boolean algebras. It also extends the notion of integrable\u0000function, usually reserved for measurable functions, to localic general\u0000functions.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226429","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 $X$ be a Banach function space over the unit circle such that the Riesz projection $P$ is bounded on $X$ and let $H[X]$ be the abstract Hardy space built upon $X$. We show that the essential norm of the Toeplitz operator $T(a):H[X]to H[X]$ coincides with $|a|_{L^infty}$ for every $ain C+H^infty$ if and only if the essential norm of the backward shift operator $T(mathbf{e}_{-1}):H[X]to H[X]$ is equal to one, where $mathbf{e}_{-1}(z)=z^{-1}$. This result extends an observation by B"ottcher, Krupnik, and Silbermann for the case of classical Hardy spaces.
{"title":"On the essential norms of Toeplitz operators on abstract Hardy spaces built upon Banach function spaces","authors":"Oleksiy Karlovych, Eugene Shargorodsky","doi":"arxiv-2408.13907","DOIUrl":"https://doi.org/arxiv-2408.13907","url":null,"abstract":"Let $X$ be a Banach function space over the unit circle such that the Riesz\u0000projection $P$ is bounded on $X$ and let $H[X]$ be the abstract Hardy space\u0000built upon $X$. We show that the essential norm of the Toeplitz operator\u0000$T(a):H[X]to H[X]$ coincides with $|a|_{L^infty}$ for every $ain\u0000C+H^infty$ if and only if the essential norm of the backward shift operator\u0000$T(mathbf{e}_{-1}):H[X]to H[X]$ is equal to one, where\u0000$mathbf{e}_{-1}(z)=z^{-1}$. This result extends an observation by B\"ottcher,\u0000Krupnik, and Silbermann for the case of classical Hardy spaces.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208206","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, the notion of Fredholm tetrad of closed linear subspaces in a Banach space is introduced. Then the stability of the Fredholm tetrad is proved. After that, the notion of semi-compact perturbation of a closed linear subspace is introduced. Then for a of pair of closed linear subspace of a Banach space such that one is a semi-compact perturbation of the other, it is proved that the relative dimension between them is well-defined. If the perturbation is compact, the relative dimension is stable. Finally the perturbed argumented Morse index is studied.
{"title":"Gaps and relative dimensions","authors":"Chenfeng Liao, Chaofeng Zhu","doi":"arxiv-2408.13837","DOIUrl":"https://doi.org/arxiv-2408.13837","url":null,"abstract":"In this paper, the notion of Fredholm tetrad of closed linear subspaces in a\u0000Banach space is introduced. Then the stability of the Fredholm tetrad is\u0000proved. After that, the notion of semi-compact perturbation of a closed linear\u0000subspace is introduced. Then for a of pair of closed linear subspace of a\u0000Banach space such that one is a semi-compact perturbation of the other, it is\u0000proved that the relative dimension between them is well-defined. If the\u0000perturbation is compact, the relative dimension is stable. Finally the\u0000perturbed argumented Morse index is studied.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208207","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 give a unified proof of the end-point estimates of the totally-geodesic $k$-plane transform of radial functions on spaces of constant curvature. The problem of getting end-point estimates is not new and some results are available in literature. However, these results were obtained independently without much focus on the similarities between underlying geometries. We give a unified proof for the end-point estimates on spaces of constant curvature by making use of geometric ideas common to the spaces of constant curvature, and obtaining a unified formula for the $k$-plane transform of radial functions. We also give some inequalities for certain special functions as an application to one of our lemmata.
{"title":"End-point estimates of the totally-geodesic Radon transform on simply connected spaces of constant curvature: A Unified Approach","authors":"Aniruddha Deshmukh, Ashisha Kumar","doi":"arxiv-2408.13541","DOIUrl":"https://doi.org/arxiv-2408.13541","url":null,"abstract":"In this article, we give a unified proof of the end-point estimates of the\u0000totally-geodesic $k$-plane transform of radial functions on spaces of constant\u0000curvature. The problem of getting end-point estimates is not new and some\u0000results are available in literature. However, these results were obtained\u0000independently without much focus on the similarities between underlying\u0000geometries. We give a unified proof for the end-point estimates on spaces of\u0000constant curvature by making use of geometric ideas common to the spaces of\u0000constant curvature, and obtaining a unified formula for the $k$-plane transform\u0000of radial functions. We also give some inequalities for certain special\u0000functions as an application to one of our lemmata.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142208209","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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