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.     

Tractable embeddings of Besov spaces into small Lebesgue spaces

This paper deals with dimension‐controllable (tractable) embeddings of Besov spaces on n‐dimensional torus into small Lebesgue spaces. Our techniques rely on the approximation structure of Besov spaces, extrapolation properties of small Lebesgue spaces and interpolation.     

Rearrangement transformations on general measure spaces

For a general set transformation R between two measure spaces, we define the rearrangement of a measurable function by means of the Layer's cake formula. We study some functional properties of the Lorentz spaces defined in terms of R, giving a unified approach to the classical rearrangement, Steiner's symmetrization, the multidimensional case, and the discrete setting of trees.     

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.     

Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces

We consider the Hardy-Littlewood maximal operator M on Musielak-Orlicz Spaces Lφ(Rd). We give a necessary condition for the continuity of M on Lφ(Rd) which generalizes the concept of Muckenhoupt classes. In the special case of generalized Lebesgue spaces Lp(⋅)(Rd) we show that this condition is also sufficient. Moreover, we show that the condition is “left-open” in the sense that not only M but also Mq is continuous for some q>1, where .     

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.     

Note on Nikodym maximal in the variable Lebesgue spaces

In this paper we consider the k‐plane Nikodym maximal estimates in the variable Lebesgue spaces . We first formulate the problem about the boundedness of the k‐plane Nikodym maximal and show that the maximal estimate in is equivalent to that in for . So, the optimal Nikodym maximal estimate in follows from Cordoba's estimate.     

Relative rearrangement and Lebesgue spaces with variable exponent

We apply the techniques of monotone and relative rearrangements to the nonrearrangement invariant spaces L^p()(Ω) with variable exponent. In particular, we show that the maps uL^p()(Ω)→k(t)u_*L^p_*()(0,measΩ) and uL^p()(Ω)→u_*L^p*()(0,measΩ) are locally -Hölderian (u_* (resp. p^*) is the decreasing (resp. increasing) rearrangement of u (resp. p)). The pointwise relations for the relative rearrangement are applied to derive the Sobolev embedding with eventually discontinuous exponents.     

Multiplication operators on vector measure Orlicz spaces

Let m be a countably additive vector measure with values in a real Banach space X, and let L¹(m) and L_w(m) be the spaces of functions which are, correspondingly, integrable and weakly integrable with respect to m. Given a Young's function Φ, we consider the vector measure Orlicz spaces L^Φ(m) and L^Φ_w(m) and establish that the Banach space of multiplication operators going from W = L^Φ(m) into Y = L¹ (m) is M = L^Ψ_w (m) with an equivalent norm; here Ψ is the conjugated Young's function for Φ. We also prove that when W = L^Φ_w(m), Y = L¹(m) we have M = L^Ψ_w (m), and when W = L^Φ_w(m), Y = L¹(m) we have M = L^Ψ (m).     

Grand and small norms in Lebesgue spaces

We present a new approach to the definition of grand and small Lebesgue spaces. This in particular allows to include into consideration the extreme exponents and . Basically, the study of the extreme exponent case is the main result of the article. However, we expect that our general construction for the norms in grand and small Lebesgue spaces will imply further studies of the spaces and operators in these spaces in such a general setting.     

Nuclear embeddings in function spaces

Let be the usual Besov spaces in bounded Lipschitz domains Ω in (bounded intervals if ). The paper clarifies under which conditions the continuous embedding between two such spaces with is nuclear.     

Two-weighted estimates for some integral transforms in Lebesgue spaces with mixed norm and imbedding theorems

Two-weighted inequalities are proved for anisotropic potentials. These estimates are used to obtain refinements of the well-known imbedding theorems in the scale of weighted Lebesgue spaces.     

Complex interpolation of Orlicz spaces with respect to a vector measure

We apply the Calderón interpolation methods to Orlicz and weakly Orlicz function spaces with respect to a Banach‐space‐valued measure defined on a σ‐algebra. The results we obtain generalize those in the case of Banach lattices of p‐integrable and weakly p‐integrable functions with respect to such a vector measure.     

Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces

Necessary and sufficient conditions are given for the fractional integral operator to be bounded from weighted strong and weak spaces within the range into suitable weighted and Lipschitz spaces. We also characterize the weights for which can be extended to a bounded operator from weighted into a weighted Lipschitz space of order . Finally, under an additional assumption on the weight, we obtain necessary and sufficient conditions for the boundedness of between weighted Lipschitz 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.