Hari M. Srivastava, Bhawna Gupta, Mohammad Idris Qureshi, Mohd Shaid Baboo
Owing to the remarkable success of the hypergeometric functions of one variable, the authors present a study of some families of hypergeometric functions of two or more variables. These functions include (for example) the Kampé de Fériet-type hypergeometric functions in two variables and Srivastava’s general hypergeometric function in three variables. The main aim of this paper is to provide several (presumably new) transformation and summation formulas for appropriately specified members of each of these families of hypergeometric functions in two and three variables. The methodology and techniques, which are used in this paper, are based upon the evaluation of some definite integrals involving logarithmic functions in terms of Riemann’s zeta function, Catalan’s constant, polylogarithm functions, and so on.
{"title":"Some summation theorems and transformations for hypergeometric functions of Kampé de Fériet and Srivastava","authors":"Hari M. Srivastava, Bhawna Gupta, Mohammad Idris Qureshi, Mohd Shaid Baboo","doi":"10.1515/gmj-2023-2114","DOIUrl":"https://doi.org/10.1515/gmj-2023-2114","url":null,"abstract":"Owing to the remarkable success of the hypergeometric functions of one variable, the authors present a study of some families of hypergeometric functions of two or more variables. These functions include (for example) the Kampé de Fériet-type hypergeometric functions in two variables and Srivastava’s general hypergeometric function in three variables. The main aim of this paper is to provide several (presumably new) transformation and summation formulas for appropriately specified members of each of these families of hypergeometric functions in two and three variables. The methodology and techniques, which are used in this paper, are based upon the evaluation of some definite integrals involving logarithmic functions in terms of Riemann’s zeta function, Catalan’s constant, polylogarithm functions, and so on.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077828","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we prove that, for β∈ℂ{betain{mathbb{C}}}, every α∈ℂ{alphain{mathbb{C}}} has at most finitely many (possibly none at all) representations of the form α=dnβn+dn-1βn-1+…+d0{alpha=d_{n}beta^{n}+d_{n-1}beta^{n-1}+dots+d_{0}} with nonnegative integers n,dn,dn-1,…,d0{n,d_{n},d_{n-1},dots,d_{0}} if and only if β is a transcendental number or an algebraic number which has a conjugate over ℚ{{mathbb{Q}}} (possibly β itself) in the real interval
在本文中，我们证明了对于 β∈ ℂ {betain{mathbb{C}}} ，每个 α∈ ℂ {alphain{mathbb{C}} 都有最有限多个（可能没有一个）"α"。 每个 α∈ ℂ {alphain{mathbb{C}}} 最多有有限多个（可能根本没有）形式为 α = d n β n + d n - 1 β n - 1 + ... 的表示。+ d 0 {alpha=d_{n}beta^{n}+d_{n-1}beta^{n-1}+dots+d_{0}} 为非负整数 n , d n , d n - 1 , ..., d 0 {n,d_{n},d_{n-1},（dots,d_{0}}当且仅当β 是一个超越数或代数数，它在ℚ {{mathbb{Q}}}（可能是 β 本身）的实区间 ( 1 , ∞ ) {(1,infty)} 上有一个共轭。这里非难的部分是要证明，对于每个代数数 β 和它在 ℂ ∖ ( 1 , ∞ ) {{mathbb{C}}setminus(1,infty)} 中的所有共轭数，都有α∈ ℚ ( β ) {alphain{mathbb{Q}}(beta)} 无穷多个这样的表示。在一种特殊情况下，当 β 是二次代数数时，卡拉和津杜尔卡最近证明了这一点。
{"title":"Representations of a number in an arbitrary base with unbounded digits","authors":"ArtÅ«ras Dubickas","doi":"10.1515/gmj-2023-2118","DOIUrl":"https://doi.org/10.1515/gmj-2023-2118","url":null,"abstract":"In this paper, we prove that, for <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>β</m:mi> <m:mo>∈</m:mo> <m:mi>ℂ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2118_eq_0137.png\" /> <jats:tex-math>{betain{mathbb{C}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, every <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mi>ℂ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2118_eq_0122.png\" /> <jats:tex-math>{alphain{mathbb{C}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> has at most finitely many (possibly none at all) representations of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>α</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:msup> <m:mi>β</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo></m:mo> <m:msup> <m:mi>β</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mi mathvariant=\"normal\">…</m:mi> <m:mo>+</m:mo> <m:msub> <m:mi>d</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2118_eq_0119.png\" /> <jats:tex-math>{alpha=d_{n}beta^{n}+d_{n-1}beta^{n-1}+dots+d_{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with nonnegative integers <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>d</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>d</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>d</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2118_eq_0231.png\" /> <jats:tex-math>{n,d_{n},d_{n-1},dots,d_{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> if and only if β is a transcendental number or an algebraic number which has a conjugate over <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℚ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2118_eq_0255.png\" /> <jats:tex-math>{{mathbb{Q}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (possibly β itself) in the real interval <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let A and B be two associative rings, let I be an ideal of B and let f∈Hom(A,B){finmathrm{Hom}(A,B)}. In this paper, we give a complete description of generalized derivations over A⋈fI{Abowtie^{f}I}. Furthermore, when A is prime or semi-prime, we give several identities on generalized derivations which provide the commutativity of A⋈fI{Abowtie^{f}I}.
让 A 和 B 是两个关联环，让 I 是 B 的一个理想，让 f ∈ Hom ( A , B ) {finmathrm{Hom}(A,B)} 。在本文中，我们将完整地描述 A ⋈ f I {Abowtie^{f}I} 上的广义推导。此外，当 A 是质数或半质数时，我们给出了关于广义推导的几个同素异形，这些同素异形提供了 A ⋈ f I {Abowtie^{f}I} 的交换性。
{"title":"Generalized derivations over amalgamated algebras along an ideal","authors":"Brahim Boudine, Mohammed Zerra","doi":"10.1515/gmj-2023-2108","DOIUrl":"https://doi.org/10.1515/gmj-2023-2108","url":null,"abstract":"Let <jats:italic>A</jats:italic> and <jats:italic>B</jats:italic> be two associative rings, let <jats:italic>I</jats:italic> be an ideal of <jats:italic>B</jats:italic> and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mi>Hom</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2108_eq_0167.png\" /> <jats:tex-math>{finmathrm{Hom}(A,B)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we give a complete description of generalized derivations over <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>A</m:mi> <m:msup> <m:mo>⋈</m:mo> <m:mi>f</m:mi> </m:msup> <m:mi>I</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2108_eq_0101.png\" /> <jats:tex-math>{Abowtie^{f}I}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Furthermore, when <jats:italic>A</jats:italic> is prime or semi-prime, we give several identities on generalized derivations which provide the commutativity of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>A</m:mi> <m:msup> <m:mo>⋈</m:mo> <m:mi>f</m:mi> </m:msup> <m:mi>I</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2108_eq_0101.png\" /> <jats:tex-math>{Abowtie^{f}I}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let ℒ2=(-Δ)2+V2{mathcal{L}_{2}=(-Delta)^{2}+V^{2}} be the Schrödinger-type operator on ℝn{mathbb{R}^{n}} (n≥5{ngeq 5}), let Hℒ21(ℝn){H^{1}_{mathcal{L}_{2}}(mathbb{R}^{n})} be the Hardy space related to ℒ2{mathcal{L}_{2}}, and let BMOθ(ρ){mathrm{BMO}_{theta}(rho)}
设 ℒ 2 = ( - Δ ) 2 + V 2 {mathcal{L}_{2}=(-Delta)^{2}+V^{2}} 是ℝ n 上的薛定谔型算子 {mathbb{R}^{n}}( n≥ 5 {ngeq 5} ) ( n ≥ 5 {ngeq 5} ), 让 H ℒ 2 1 ( ℝ n ) {H^{1}_{mathcal{L}_{2}}}(mathbb{R}^{n})} 是与ℒ 2 {mathcal{L}_{2}} 相关的哈代空间。 让 BMO θ ( ρ ) {mathrm{BMO}_{theta}(rho)} 是 Bongioanni、Harboure 和 Salinas 引入的 BMO 型空间。本文将研究换向器 [ b , T α , β , j ] {[b,T_{alpha,beta,j}]} 的有界性。 T α , β , j = V 2 α ∇ j ℒ 2 - β {T_{alpha,beta,j}=V^{2alpha}nabla^{j}mathcal{L}_{2}^{-beta}} , j = 1 , 2 , 3 {j=1,2,3} , 并且 b∈ BMO θ ( ρ ) {binmathrm{BMO}_{theta}(rho)} 。这里，0 < α ≤ 1 - j 4 {0<alphaleq 1-frac{j}{4}} 。 , j 4 < β ≤ 1 {frac{j}{4}<betaleq 1} , β -α = j{4}. , β - α = j 4 {beta-alpha=frac{j}{4}} 非负电势 V 同时属于反向荷尔德类 RH s {mathrm{RH}_{s}} ，s ≥ n 2 {sgeqfrac{n}{2}} 和与 ( - Δ ) 2 {(-Delta)^{2}} 相关的高斯类。 .得到 [ b , T α , β , j ] {[b,T_{alpha,beta,j}]} 的 L p {L^{p}} 有界性，同时证明 [ b , T α , β , j ] {[b、T_{alpha,beta,j}]} 从 H ℒ 2 1 ( ℝ n ) {H^{1}_{mathcal{L}_{2}}(mathbb{R}^{n})} 到弱 L 1 ( ℝ n ) {L^{1}(mathbb{R}^{n})} 是有界的。
Alexander Domoshnitsky, Elnatan Berenson, Shai Levi, Elena Litsyn
A version of the Floquet theory for first order delay differential equations is proposed. Formula of solutions representation is obtained. On this basis, the stability of first order delay differential equations is studied. An analogue of the classical integral Lyapunov–Zhukovskii test of stability is proved. New, in comparison with all known, tests of the exponential stability are obtained on the basis of the Floquet theory. A possibility to achieve the exponential stability is connected with oscillation of solutions.
{"title":"Floquet theory and stability for a class of first order differential equations with delays","authors":"Alexander Domoshnitsky, Elnatan Berenson, Shai Levi, Elena Litsyn","doi":"10.1515/gmj-2023-2119","DOIUrl":"https://doi.org/10.1515/gmj-2023-2119","url":null,"abstract":"A version of the Floquet theory for first order delay differential equations is proposed. Formula of solutions representation is obtained. On this basis, the stability of first order delay differential equations is studied. An analogue of the classical integral Lyapunov–Zhukovskii test of stability is proved. New, in comparison with all known, tests of the exponential stability are obtained on the basis of the Floquet theory. A possibility to achieve the exponential stability is connected with oscillation of solutions.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077885","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let ω be a doubling weight and 0<p≤q<∞{0<pleq q<infty}. The essential norm of Riemann–Stieltjes operator Tg{T_{g}} from the weighted Bergman space Aωp{A^{p}_{omega}} to Aωq{A^{q}_{omega}} was investigated in the unit ball of ℂn{mathbb{C}^{n}}.
设 ω 为加倍权重，且 0 < p ≤ q < ∞ {0<pleq q<infty} 。在 ℂ n {mathbb{C}^{n} 的单位球中研究了从加权伯格曼空间 A ω p {A^{p}_{omega}} 到 A ω q {A^{q}_{omega}} 的黎曼-斯蒂尔杰斯算子 T g {T_{g}} 的基本规范。｝ .
{"title":"Essential norm of Riemann–Stieltjes operator on weighted Bergman spaces with doubling weights","authors":"Lian Hu, Songxiao Li, Rong Yang","doi":"10.1515/gmj-2023-2110","DOIUrl":"https://doi.org/10.1515/gmj-2023-2110","url":null,"abstract":"Let ω be a doubling weight and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo>≤</m:mo> <m:mi>q</m:mi> <m:mo><</m:mo> <m:mi mathvariant=\"normal\">∞</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2110_eq_0161.png\" /> <jats:tex-math>{0<pleq q<infty}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The essential norm of Riemann–Stieltjes operator <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>T</m:mi> <m:mi>g</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2110_eq_0210.png\" /> <jats:tex-math>{T_{g}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> from the weighted Bergman space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mi>A</m:mi> <m:mi>ω</m:mi> <m:mi>p</m:mi> </m:msubsup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2110_eq_0172.png\" /> <jats:tex-math>{A^{p}_{omega}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> to <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mi>A</m:mi> <m:mi>ω</m:mi> <m:mi>q</m:mi> </m:msubsup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2110_eq_0173.png\" /> <jats:tex-math>{A^{q}_{omega}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> was investigated in the unit ball of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℂ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2110_eq_0250.png\" /> <jats:tex-math>{mathbb{C}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cristian Conde, Fuad Kittaneh, Hamid Reza Moradi, Mohammad Sababheh
Operator matrices have played a significant role in the study of properties of the numerical radii of Hilbert space operators. This paper presents several new sharp upper bounds for the numerical radii of operator matrices in terms of certain complex combinations. The obtained results reveal many interesting properties of the numerical radius.
{"title":"Numerical radii of operator matrices in terms of certain complex combinations of operators","authors":"Cristian Conde, Fuad Kittaneh, Hamid Reza Moradi, Mohammad Sababheh","doi":"10.1515/gmj-2023-2112","DOIUrl":"https://doi.org/10.1515/gmj-2023-2112","url":null,"abstract":"Operator matrices have played a significant role in the study of properties of the numerical radii of Hilbert space operators. This paper presents several new sharp upper bounds for the numerical radii of operator matrices in terms of certain complex combinations. The obtained results reveal many interesting properties of the numerical radius.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is known that if B and B′{B^{prime}} are translation invariant convex density differentiation bases and the maximal operators associated to them locally majorize each other, then B and B′{B^{prime}} differentiate the integrals of the same class of non-negative functions. We show that under the same conditions it is not possible to assert more about similarity of the differential properties of B and B′{B^{prime}} in view of their positive equivalence.
众所周知，如果 B 和 B ′ {B^{/prime}} 是平移不变的凸密度微分基，并且与它们相关的最大算子在局部上相互大化，那么 B 和 B ′ {B^{/prime} 就微分同一类非负函数的积分。我们证明，在同样的条件下，鉴于 B 和 B ′ {B^{prime} 的正等价性，不可能断言它们的微分性质有更多的相似性。
{"title":"On the comparison of translation invariant convex differentiation bases","authors":"Irakli Japaridze","doi":"10.1515/gmj-2023-2070","DOIUrl":"https://doi.org/10.1515/gmj-2023-2070","url":null,"abstract":"It is known that if <jats:italic>B</jats:italic> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>B</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2070_eq_0070.png\" /> <jats:tex-math>{B^{prime}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> are translation invariant convex density differentiation bases and the maximal operators associated to them locally majorize each other, then <jats:italic>B</jats:italic> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>B</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2070_eq_0070.png\" /> <jats:tex-math>{B^{prime}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> differentiate the integrals of the same class of non-negative functions. We show that under the same conditions it is not possible to assert more about similarity of the differential properties of <jats:italic>B</jats:italic> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>B</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2070_eq_0070.png\" /> <jats:tex-math>{B^{prime}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in view of their positive equivalence.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the local maximal oscillatory integral operator Tα,β∗(f)(x)=sup0<t<1|∫ℝnei|tξ|α|tξ|βΨ(|tξ|)f^(ξ)e2πi〈x,ξ〉𝑑ξ|,displaystyle T_{alpha,beta}^{ast}(f)(x)=sup_{0<t<1}Bigg{|}int_{mathbb{% R}^{n}}frac{e^{i|txi|^{alpha}}}{|txi|^{beta}}Psi(|txi|)widehat{f}(xi)% e^{2pi ilangle x,xirangle},dxiBigg{|}, where
我们研究局部最大振荡积分算子 T α , β ∗ ( f ) ( x ) = sup 0 < t <；1 | ∫ ℝ n e i | t ξ | α | t ξ | β Ψ ( | t ξ | ) f ^ ( ξ ) e 2 π i 〈 x 、ξ 〉𝑑 ξ | , displaystyle T_{alpha,beta}^{ast}(f)(x)=sup_{0<；t<；1}Bigg{|}int_{mathbb{% R}^{n}}frac{e^{i|txi|^{alpha}}}{|txi|^{beta}}Psi(|txi|)widehat{f}(xi)% e^{2pi ilangle x,其中 α ∈ ( 0 , 1 ) {alphain(0,1)}, β >；0 {beta>0} Ψ 是在原点附近消失的截止函数。首先，在 0 < p < 1 {0<p<1} 的情况下，我们可以得到 H p ( Ψ) 。 我们得到 H p ( ℝ n ) → L p ( ℝ n ) {{{H^{p}}({{mathbb{R}^{n}})}rightarrow{{L^{p}({{mathbb{R}^{n}})}} T α 的有界性、β∗ {T_{alpha,beta}^{ast}} 与 α , β {alpha,beta} 和 p 之间的尖锐关系。然后，利用插值法，当 p > 1 {p>1} 时，我们得到 L p ( ℝ n ) {{{L^{p}({{mathbb{R}^{n}})}}} 对 T α , β∗ {T_{alpha,beta}^{ast}} 的约束性。｝ 这是对凯尼格和斯陶巴赫最新结果的改进。在临界情况 p = 1 {p=1} 和 β = n α 2 {beta=frac{nalpha}{2}} 下，我们证明了 T α , β = n α 2 {beta=frac{nalpha}{2}} 和 β = n α 3 {beta=frac{nalpha}{2}} 我们证明 T α , β ∗ : B q ( ℝ n ) → L 1 , ∞ ( ℝ n ) {T_{alpha,beta}^{ast}:B_{q}({mathbb{R}^{n}})rightarrow L^{1,infty}({% mathbb{R}^{n}}} 其中 B q ( ℝ n ) {B_{q}({mathbb{R}^{n}})} 是 Lu、Taibleson 和 Weiss 为研究 Bochner-Riesz 均值在临界指数处的几乎每次收敛而引入的块空间。作为进一步的应用，我们得到了分数薛定谔算子 { e i t k | △ | α } 组合的收敛速度。 {{e^{itk|triangle|^{alpha}}}} .
The present study introduces the notions of statistical convergence of order α and strong p-Cesàro summability of order α in partial metric spaces. Also, we examine the inclusion relations between these concepts. In addition, we introduce the notion of λ-statistical convergence of order α in partial metric spaces while providing relations linked to these sequence spaces.
{"title":"On statistical convergence of order α in partial metric spaces","authors":"Erdal Bayram, Çiğdem A. Bektaş, Yavuz Altın","doi":"10.1515/gmj-2023-2116","DOIUrl":"https://doi.org/10.1515/gmj-2023-2116","url":null,"abstract":"The present study introduces the notions of statistical convergence of order α and strong <jats:italic>p</jats:italic>-Cesàro summability of order α in partial metric spaces. Also, we examine the inclusion relations between these concepts. In addition, we introduce the notion of λ-statistical convergence of order α in partial metric spaces while providing relations linked to these sequence spaces.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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