Introduction to Grand Lebesgue Space

What is/are Grand Lebesgue?

Grand Lebesgue - 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"> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> 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<p≤1). ^[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]

Generalized Grand Lebesgue

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> 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]

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]

grand lebesgue space

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"> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> 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<p≤1). ^[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]