Grand Lebesgue
大勒贝格

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Grand Lebesgue 大勒贝格

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Dec 02, 2024

Grand Lebesgue(大勒贝格)研究综述

Grand Lebesgue 大勒贝格 - We introduce a new class of quasi-Banach spaces as an extension of the classical Grand Lebesgue Spaces for ‘‘small’’values of the parameter, and we investigate some its properties, in particular, completeness, fundamental function, operators estimates, Boyd indices, contraction principle, tail behavior, dual space, generalized triangle and quadrilateral constants and inequalities. ^[1] org/1998/Math/MathML"> Δ 2 condition near the origin, then This result permits to clarify the assumptions on the increasing function against the Lebesgue norm in the definition of generalized grand Lebesgue spaces and to sharpen and simplify the statements of some known results concerning these spaces. ^[2] We prove that the Grand Lebesgue Space, builded on a unimodular locally compact topological group, forms a Banach algebra relative to the convolution. ^[3] The goal of this paper is to prove the boundedness of fundamental integral operators of Harmonic Analysis in generalized weighted grand Lebesgue spaces $$L^{p),\varphi }_w$$ L w p ) , φ defined on domains in $${\mathbb {R}}^n$$ R n without assuming that the underlying measure $$\mu $$ μ is doubling. ^[4] In this paper, we prove the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces Lp)(0,1)(0 ^[5] In the present paper, we discuss generalized grand Lebesgue spaces on homogeneous Lie groups. ^[6] The approximation properties of the matrix transforms, constructed via lower triangular matrices, satisfying some additional conditions, in the generalized grand Lebesgue spaces with variable exponent are studied and the appropriate rates of approximation are estimated. ^[7] weighted spaces, variable exponent and grand Lebesgue spaces. ^[8] The proof generalizes and makes sharp an equivalence previously known only in the particular case when $$\psi $$ψ is a power; such case had a relevant role for the study of grand Lebesgue spaces. ^[9] In this paper, the weighted grand Lebesgue spaces with mixed-norms are introduced and boundedness criteria in these spaces of strong maximal functions and Riesz transforms are presented. ^[10] Applications to Matsaev ideals, Grand Lebesgue spaces, Bourgain-Brezis-Mironescu-Maz'ya-Shaposhnikova limits, as well as a new vector valued extrapolation theorems, are provided. ^[11] For Hausdorff operator of general type defined on p-adic linear space $\mathbb{Q}_p^n$ℚpn, we give sufficient conditions of its boundedness in weighted Lebesgue and grand Lebesgue spaces. ^[12] Weighted grand Lebesgue spaces with mixed norms are introduced, and criteria for the boundedness of strong maximal functions and Riesz transforms in these spaces are given. ^[13]
我们引入了一类新的准巴拿赫空间，作为经典大勒贝格空间的扩展，用于参数的“小”值，我们研究了它的一些性质，特别是完整性、基本函数、算子估计、博伊德指数，收缩原理，尾部行为，对偶空间，广义三角形和四边形常数和不等式。 ^[1] org/1998/Math/MathML"> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> 条件接近原点，则该结果允许澄清广义大勒贝格空间定义中针对勒贝格范数的递增函数的假设和锐化和简化关于这些空间的一些已知结果的陈述。 ^[2] 我们证明了建立在单模局部紧拓扑群上的 Grand Lebesgue 空间形成了相对于卷积的 Banach 代数。 ^[3] 本文的目的是证明在广义加权大勒贝格空间 $$L^{p),\varphi }_w$$ L w p) , φ 中定义在 $${\ mathbb {R}}^n$$ R n 不假设基础度量 $$\mu $$ μ 加倍。 ^[4] 在本文中，我们证明了大勒贝格空间 Lp)(0,1)(0<p≤1) 中单调函数的 Hardy 算子的有界性。 ^[5] 在本文中，我们讨论了齐次李群上的广义大勒贝格空间。 ^[6] 研究了具有变指数的广义大勒贝格空间中通过下三角矩阵构造的矩阵变换的逼近性质，并估计了适当的逼近率。 ^[7] 加权空间、变指数空间和大 Lebesgue 空间。 ^[8] 该证明概括并明确了先前仅在 $$\psi $$ψ 是幂的特定情况下才知道的等价；这种情况对研究大勒贝格空间具有相关作用。 ^[9] 本文介绍了具有混合范数的加权大 Lebesgue 空间，并给出了这些强极大函数空间和 Riesz 变换的有界准则。 ^[10] 提供了对 Matsaev 理想、Grand Lebesgue 空间、Bourgain-Brezis-Mironescu-Maz'ya-Shaposhnikova 极限以及新的向量值外推定理的应用。 ^[11] 对于定义在p进线性空间$\mathbb{Q}_p^n$ℚpn上的一般类型的Hausdorff算子，我们给出了其在加权勒贝格空间和大勒贝格空间中有界的充分条件。 ^[12] 引入了具有混合范数的加权大 Lebesgue 空间，并给出了这些空间中强极大函数和 Riesz 变换的有界准则。 ^[13]

Generalized Grand Lebesgue 广义大勒贝格

org/1998/Math/MathML"> Δ 2 condition near the origin, then This result permits to clarify the assumptions on the increasing function against the Lebesgue norm in the definition of generalized grand Lebesgue spaces and to sharpen and simplify the statements of some known results concerning these spaces. ^[1] In the present paper, we discuss generalized grand Lebesgue spaces on homogeneous Lie groups. ^[2] The approximation properties of the matrix transforms, constructed via lower triangular matrices, satisfying some additional conditions, in the generalized grand Lebesgue spaces with variable exponent are studied and the appropriate rates of approximation are estimated. ^[3]
org/1998/Math/MathML"> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> 条件接近原点，则该结果允许澄清广义大勒贝格空间定义中针对勒贝格范数的递增函数的假设和锐化和简化关于这些空间的一些已知结果的陈述。 ^[1] 在本文中，我们讨论了齐次李群上的广义大勒贝格空间。 ^[2] 研究了具有变指数的广义大勒贝格空间中通过下三角矩阵构造的矩阵变换的逼近性质，并估计了适当的逼近率。 ^[3]

Weighted Grand Lebesgue

The goal of this paper is to prove the boundedness of fundamental integral operators of Harmonic Analysis in generalized weighted grand Lebesgue spaces $$L^{p),\varphi }_w$$ L w p ) , φ defined on domains in $${\mathbb {R}}^n$$ R n without assuming that the underlying measure $$\mu $$ μ is doubling. ^[1] In this paper, the weighted grand Lebesgue spaces with mixed-norms are introduced and boundedness criteria in these spaces of strong maximal functions and Riesz transforms are presented. ^[2] Weighted grand Lebesgue spaces with mixed norms are introduced, and criteria for the boundedness of strong maximal functions and Riesz transforms in these spaces are given. ^[3]
本文的目的是证明在广义加权大勒贝格空间 $$L^{p),\varphi }_w$$ L w p) , φ 中定义在 $${\ mathbb {R}}^n$$ R n 不假设基础度量 $$\mu $$ μ 加倍。 ^[1] 本文介绍了具有混合范数的加权大 Lebesgue 空间，并给出了这些强极大函数空间和 Riesz 变换的有界准则。 ^[2] 引入了具有混合范数的加权大 Lebesgue 空间，并给出了这些空间中强极大函数和 Riesz 变换的有界准则。 ^[3]

grand lebesgue space 大勒贝格空间

We introduce a new class of quasi-Banach spaces as an extension of the classical Grand Lebesgue Spaces for ‘‘small’’values of the parameter, and we investigate some its properties, in particular, completeness, fundamental function, operators estimates, Boyd indices, contraction principle, tail behavior, dual space, generalized triangle and quadrilateral constants and inequalities. ^[1] org/1998/Math/MathML"> Δ 2 condition near the origin, then This result permits to clarify the assumptions on the increasing function against the Lebesgue norm in the definition of generalized grand Lebesgue spaces and to sharpen and simplify the statements of some known results concerning these spaces. ^[2] We prove that the Grand Lebesgue Space, builded on a unimodular locally compact topological group, forms a Banach algebra relative to the convolution. ^[3] The goal of this paper is to prove the boundedness of fundamental integral operators of Harmonic Analysis in generalized weighted grand Lebesgue spaces $$L^{p),\varphi }_w$$ L w p ) , φ defined on domains in $${\mathbb {R}}^n$$ R n without assuming that the underlying measure $$\mu $$ μ is doubling. ^[4] In this paper, we prove the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces Lp)(0,1)(0 ^[5] In the present paper, we discuss generalized grand Lebesgue spaces on homogeneous Lie groups. ^[6] The approximation properties of the matrix transforms, constructed via lower triangular matrices, satisfying some additional conditions, in the generalized grand Lebesgue spaces with variable exponent are studied and the appropriate rates of approximation are estimated. ^[7] weighted spaces, variable exponent and grand Lebesgue spaces. ^[8] The proof generalizes and makes sharp an equivalence previously known only in the particular case when $$\psi $$ψ is a power; such case had a relevant role for the study of grand Lebesgue spaces. ^[9] In this paper, the weighted grand Lebesgue spaces with mixed-norms are introduced and boundedness criteria in these spaces of strong maximal functions and Riesz transforms are presented. ^[10] Applications to Matsaev ideals, Grand Lebesgue spaces, Bourgain-Brezis-Mironescu-Maz'ya-Shaposhnikova limits, as well as a new vector valued extrapolation theorems, are provided. ^[11] For Hausdorff operator of general type defined on p-adic linear space $\mathbb{Q}_p^n$ℚpn, we give sufficient conditions of its boundedness in weighted Lebesgue and grand Lebesgue spaces. ^[12] Weighted grand Lebesgue spaces with mixed norms are introduced, and criteria for the boundedness of strong maximal functions and Riesz transforms in these spaces are given. ^[13]
我们引入了一类新的准巴拿赫空间，作为经典大勒贝格空间的扩展，用于参数的“小”值，我们研究了它的一些性质，特别是完整性、基本函数、算子估计、博伊德指数，收缩原理，尾部行为，对偶空间，广义三角形和四边形常数和不等式。 ^[1] org/1998/Math/MathML"> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> 条件接近原点，则该结果允许澄清广义大勒贝格空间定义中针对勒贝格范数的递增函数的假设和锐化和简化关于这些空间的一些已知结果的陈述。 ^[2] 我们证明了建立在单模局部紧拓扑群上的 Grand Lebesgue 空间形成了相对于卷积的 Banach 代数。 ^[3] 本文的目的是证明在广义加权大勒贝格空间 $$L^{p),\varphi }_w$$ L w p) , φ 中定义在 $${\ mathbb {R}}^n$$ R n 不假设基础度量 $$\mu $$ μ 加倍。 ^[4] 在本文中，我们证明了大勒贝格空间 Lp)(0,1)(0<p≤1) 中单调函数的 Hardy 算子的有界性。 ^[5] 在本文中，我们讨论了齐次李群上的广义大勒贝格空间。 ^[6] 研究了具有变指数的广义大勒贝格空间中通过下三角矩阵构造的矩阵变换的逼近性质，并估计了适当的逼近率。 ^[7] 加权空间、变指数空间和大 Lebesgue 空间。 ^[8] 该证明概括并明确了先前仅在 $$\psi $$ψ 是幂的特定情况下才知道的等价；这种情况对研究大勒贝格空间具有相关作用。 ^[9] 本文介绍了具有混合范数的加权大 Lebesgue 空间，并给出了这些强极大函数空间和 Riesz 变换的有界准则。 ^[10] 提供了对 Matsaev 理想、Grand Lebesgue 空间、Bourgain-Brezis-Mironescu-Maz'ya-Shaposhnikova 极限以及新的向量值外推定理的应用。 ^[11] 对于定义在p进线性空间$\mathbb{Q}_p^n$ℚpn上的一般类型的Hausdorff算子，我们给出了其在加权勒贝格空间和大勒贝格空间中有界的充分条件。 ^[12] 引入了具有混合范数的加权大 Lebesgue 空间，并给出了这些空间中强极大函数和 Riesz 变换的有界准则。 ^[13]