Generalization of the notion of grand Lebesgue space

We introduce families of weighted grand Lebesgue spaces which generalize weighted grand Lebesgue spaces (known also as Iwaniec-Sbordone spaces). The generalization admits a possibility of expanding usual (weighted) Lebesgue spaces to grand spaces by various ways by means of additional functional parameter. For such generalized grand spaces we prove a theorem on the boundedness of linear operators under the information of their boundedness in ordinary weighted Lebesgue spaces. By means of this theorem we prove boundedness of the Hardy-Littlewood maximal operator and the Calderon-Zygmund singular operators in the weighted grand spaces.     

Grand Bochner–Lebesgue space and its associate space

Our aim is to introduce the grand Bochner–Lebesgue space in the spirit of Iwaniec–Sbordone spaces, also known as grand Lebesgue spaces, and prove some of its properties. We will also deal with the associate space for grand Bochner–Lebesgue spaces.     

Sawyer's duality principle for grand Lebesgue spaces

The aim of this paper is to extend Sawyer's duality principle from the cone of decreasing functions of the Lebesgue space to the cone of decreasing functions of the grand Lebesgue space and to prove the boundedness of classical Hardy operators in the grand Lebesgue spaces.     

On grand Lebesgue spaces on sets of infinite measure

We consider grand Lebesgue spaces on sets of infinite measure and study the dependence of these spaces on the choice of the so‐called. We also consider Mikhlin and Marcinkiewicz theorems on Fourier multipliers in the setting of grand spaces.     

Anisotropic Mixed-Norm Hardy Spaces

We introduce and explore Hardy spaces defined by mixed Lebesgue norms and anisotropic dilations. We prove that the definitions of these spaces in terms of smooth, non-tangential, auxiliary, grand, and Poisson maximal operators coincide. We also study the relation between anisotropic mixed-norm Hardy spaces and mixed-norm Lebesgue spaces.     

Compactness Results for a Class of Limiting Interpolation Methods

We study the behavior of compact operators when we interpolate them by real methods defined through slowly varying functions and rearrangement invariant spaces. We apply these results to prove compactness of certain integral operators acting between grand Lebesgue spaces and between small Lebesgue spaces.

     

Sawyer Duality Principle in Grand Lebesgue Spaces

The Sawyer duality principle is obtained for grand Lebesgue spaces on the unit interval, and the Hardy operators are shown to be bounded in these spaces.     

Weighted Grand Lebesgue Space with a Mixed Norm and Integral Operators

Doklady Mathematics - 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.     

Integral operators on fully measurable weighted grand Lebesgue spaces

In this paper we introduce the weighted version of fully measurable grand Lebesgue spaces and obtain characterizations for the boundedness of maximal operator, Hilbert transform and the Hardy averaging operator on these spaces.     

On Elliptic Homogeneous Differential Operators in Grand Spaces

We give an application of so-called grand Lebesgue and grand Sobolev spaces, intensively studied during last decades, to partial differential equations. In the case of unbounded domains such spaces are defined using so-called grandizers. Under some natural assumptions on the choice of grandizers, we prove the existence, in some grand Sobolev space, of a solution to the equation Pm(D)u(x) = f(x), x ∈ ℝn, m < n, with the right-hand side in the corresponding grand Lebesgue space, where Pm(D) is an arbitrary elliptic homogeneous in the general case we improve some known facts for the fundamental solution of the operator Pm(D): we construct it in the closed form either in terms of spherical hypersingular integrals or in terms of some averages along plane sections of the unit sphere.     

Integral operators with homogeneous kernels in grand Lebesgue spaces

Sufficient conditions on the kernel and the grandizer that ensure the boundedness of integral operators with homogeneous kernels in grand Lebesgue spaces on ? ⁿ as well as an upper bound for their norms are obtained. For some classes of grandizers, necessary conditions and lower bounds for the norm of these operators are also obtained. In the case of a radial kernel, stronger estimates are established in terms of one-dimensional grand norms of spherical means of the function. A sufficient condition for the boundedness of the operator with homogeneous kernel in classical Lebesgue spaces with arbitrary radial weight is obtained. As an application, boundedness in grand spaces of the one-dimensional operator of fractional Riemann–Liouville integration and of a multidimensional Hilbert-type operator is studied.     

Sobolev‐type inequalities for potentials in grand variable exponent Lebesgue spaces

We introduce a new scale of grand variable exponent Lebesgue spaces denoted by . These spaces unify two non‐standard classes of function spaces, namely, grand Lebesgue and variable exponent Lebesgue spaces. The boundedness of integral operators of Harmonic Analysis such as maximal, potential, Calderón–Zygmund operators and their commutators are established in these spaces. Among others, we prove Sobolev‐type theorems for fractional integrals in . The spaces and operators are defined, generally speaking, on quasi‐metric measure spaces with doubling measure. The results are new even for Euclidean spaces.     

Weighted extrapolation in Iwaniec—Sbordone spaces. Applications to integral operators and approximation theory

We prove extrapolation theorems in weighted Iwaniec–Sbordone spaces and apply them to one-weight inequalities for several integral operators of harmonic analysis. In addition, in weighted grand Lebesgue spaces, we establish Bernstein and Nikol’skii type inequalities and prove direct and inverse theorems on the approximation of functions.     

含参数加权极大Lebesgue空间中次线性算子的有界性质

引进一类含参数加权极大Lebesgue空间并得到满足一定尺寸条件的次线性算子在该类空间中的有界性质.特别地,还考虑了该类空间上次线性算子与BMO函数生成交换子的相应有界性质.     

Riemann–Hilbert problem in the class of Cauchy type integrals with densities of grand Lebesgue Spaces

The present paper deals with a solution of the Riemann–Hilbert problem with piecewise continuous coefficients in the class of Cauchy type integrals with densities in grand Lebesgue spaces. Necessary and sufficient solvability condition is established. In solvability case the solutions in explicit form are constructed.     

奇异积分和位势积分交换子在极大Morrey空间上的有界性

设齐次空间(X,ρ,μ)上定义一类极大Morrey空间L~(p),θ,λ)(X,μ).此类极大Morrey空间是经典的Morrey空间和极大Lebesgue空间的推广.本文考虑了C-Z积分算子、位势算子与BMO函数生成的交换子在该类极大Morrey空间上的有界性.事实上,这些结果甚至在一般的欧式空间上也是新颖的.     

Grand Sobolev spaces and their applications in geometric function theory and PDEs

Function spaces that are slightly larger than the Lebesgue L ^p (Ω) spaces (even larger than the Marcinkiewicz L ^p,∞ (Ω) spaces) have been introduced by Iwaniec and Sbordone [Arch. Ration. Mech. Anal. 119 (1992), 129–143] in connection with integrability properties of the Jacobian. These are the grand Lebesgue spaces L ^p)(Ω). In this survey we collect a number of results which prove that these spaces are useful in various classical settings of geometric function theory and partial differential equations (PDEs).     

Embeddings between grand,small, and variable Lebesgue spaces

We give conditions on the exponent function p( · ) that imply the existence of embeddings between the grand, small, and variable Lebesgue spaces. We construct examples to show that our results are close to optimal. Our work extends recent results by the second author, Rakotoson and Sbordone.     

Exponential tail estimates in the law of ordinary logarithm (LOL) for triangular arrays of random variables

We derive exponential bounds for the tail of the distribution of normalized sums of triangular arrays of random variables, not necessarily independent, under the law of ordinary logarithm.

Furthermore, we provide estimates for partial sums of triangular arrays of independent random variables belonging to suitable grand Lebesgue spaces and having heavy-tailed distributions.