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.     

Potentials with product kernels in grand Lebesgue spaces: One-weight criteria*

We study the boundedness problem for fractional integral operators with product kernels and corresponding strong fractional maximal operators in unweighted and weighted grand Lebesgue spaces. Among other statements, we prove that the one-weight inequality $ {{\left\| {{T_{\alpha }}\left( {f{w^{\alpha }}} \right)} \right\|}_{{L_w^{{q),\theta q/p}}}}}\leqslant c{{\left\| f \right\|}_{{L_w^{{p),\theta }}}}} $ , where q is the Hardy–Littlewood–Sobolev exponent of p, holds for potentials with product kernels T _α if and only if the weight w belongs to the Muckenhoupt class A _1+q/p′ defined with respect to n-dimensional intervals with sides parallel to the coordinate axes. We also provide a motivation of choosing θq/p as the second parameter of the target space.     

Integral operators with variable kernels on weak Hardy spaces

In this note the authors study the mapping properties of a class of integral operators with variable kernels on the weak Hardy spaces.     

Integral intertwining operators and quantum homogeneous spaces

Integral representations of functions on quantum homogeneous spaces are considered. The Dirichlet problem for the quantum ball is solved and a q-analog of the Cauchy-Szeg formula is derived.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 3, pp. 355–363, December, 1995.     

Integral operators with operator-valued kernels

Under fairly mild measurability and integrability conditions on operator-valued kernels, boundedness results for integral operators on Bochner spaces L_p(X) are given. In particular, these results are applied to convolutions operators.     

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.     

Generalized potentials in variable exponent Lebesgue spaces on homogeneous spaces

We consider generalized potential operators with the kernel on bounded quasimetric measure space (X, μ, d) with doubling measure μ satisfying the upper growth condition μB(x, r) ? Kr^N, N ∈ (0, ∞). Under some natural assumptions on a(r) in terms of almost monotonicity we prove that such potential operators are bounded from the variable exponent Lebesgue space L^p(?)(X, μ) into a certain Musielak‐Orlicz space L^p(X, μ) with the N‐function Φ(x, r) defined by the exponent p(x) and the function a(r). A reformulation of the obtained result in terms of the Matuszewska‐Orlicz indices of the function a(r) is also given. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

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.     

Extrapolation results in grand Lebesgue spaces defined on product sets

Extrapolation results in weighted grand Lebesgue spaces defined with respect to product measure \(\mu \times \nu \) on \(X\times Y\), where \((X, d, \mu )\) and \((Y, \rho , \nu )\) are spaces of homogeneous type, are obtained. As applications of the derived results we prove new one-weight estimates for multiple integral operators such as strong maximal, Calderón–Zygmund and fractional integral operators with product kernels in these spaces.     

Weighted inequalities for integral operators with some homogeneous kernels

In this paper we study integral operators of the form

Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

In this paper, we study the boundedness of the Hausdorff operator H_? on the real line R. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space L~p(R)and the Hardy space H~1(R). The key idea is to reformulate H_? as a Calder′on-Zygmund convolution operator,from which its boundedness is proved by verifying the Hrmander condition of the convolution kernel. Secondly,to prove the boundedness on the Hardy space H~p(R) with 0 p 1, we rewrite the Hausdorff operator as a singular integral operator with the non-convolution kernel. This novel reformulation, in combination with the Taibleson-Weiss molecular characterization of H~p(R) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H~1(R).     

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.