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We give the following, equivalent, explicit expressions for the norms of the small and grand Lebesgue spaces, which depend only on the non-decreasing rearrangement (we assume here that the underlying measure space has measure 1): f L (p ≈ [f * (s)] p ds 1 p (1 < p < ∞). Similar results are proved for the generalized small and grand spaces.

We prove that if the exponent function p(·) satisfies log-Hölder continuity conditions locally and at infinity, then the fractional maxi-We also prove a weak-type inequality corresponding to the weak (1, n/(n − α)) inequality for M α . We build upon earlier work on the Hardy-Littlewood maximal operator by Cruz-Uribe, Fiorenza and Neugebauer [3]. As a consequence of these results for M α , we show that the fractional integral operator I α satisfies the same norm inequalities. These in turn yield a generalization of the Sobolev embedding theorem to variable L p spaces.

We prove weighted strong and weak-type norm inequalities for the Hardy–Littlewood maximal operator on the variable Lebesgue space Lp(·). Our results generalize both the classical weighted norm inequalities on Lp and the more recent results on the boundedness \ud of the maximal operator on variable Lebesgue spaces

Abstract. We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0, 1) ⊂ R, and the maximal function is localized in (0, 1). Moreover, we prove that the inequality M f p),w ≤ c f p),w holds with some c independent of f iff w belongs to the well known Muckenhoupt class Ap, and therefore iff M f p,w ≤ c f p,w for some c independent of f .Some results of similar type are discussed for the case of small Lebesgue spaces.

1In this paper, we show that the interpolation spaces between Grand, small or classical Lebesgue are so called Lorentz-Zygmund spaces or more generally GΓ-spaces. As a direct consequence of our results any Lorentz-Zygmund space L a,r (Log L) β , is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces provided that 1 < a < ∞, β = 0. The method consists in computing the so called K-functional of the interpolation space and in identifying the associated norm.

We prove that the commutator [b, Iα], b ∈ BMO, Iα the fractional integral operator, satisfies the sharp, modular weak-type inequality where B(t) = t log(e + t) and Ψ(t) = [t log(e + t α/n )] n/(n−α). These commutators were first considered by Chanillo, and our result complements his. The heart of our proof consists of the pointwise inequality,where M # is the sharp maximal operator, and M α,B is a generalization of the fractional maximal operator in the scale of Orlicz spaces. Using this inequality we also prove one-weight inequalities for the commutator; to do so we prove one and two-weight norm inequalities for M α,B which are of interest in their own right.

We consider a generalized version of the small Lebesgue spaces, introduced in [5] as the associate spaces of the grand Lebesgue spaces. We find a simplified expression for the norm, prove relevant properties, compute the fundamental function and discuss the comparison with the Orlicz spaces.