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 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.

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−α f p as p → r + (1 < r < ∞). The study has been motivated by current investigations of convolution maximal functions in stochastic analysis, where the problem occurs for r = 2. We also touch the problem of comparison of results in various scales of spaces.

We introduce and investigate the grand Orlicz spaces and the grand Lorentz-Orlicz spaces. An application to the problem of global integrability of the Jacobian of orientation preserving mappings is given.