Abstract. We characterize the class of weights related to the boundedness of maximal operators associated to a Young function η in the context of variable Lebesgue spaces. Fractional versions of these results are also obtained by means of a weighted Hedberg type inequality. These results are new even in the classical Lebesgue spaces. We also deal with Wiener's type inequalities for the mentioned operators in the variable context. As applications of the strong type results for the maximal operators, we derive weighted boundedness properties for a large class of operators controlled by them.

We give sufficient conditions on variable exponent functions p : R n → [1, ∞) for which the higher-order Riesz transforms, associated with the Ornstein-Uhlenbeck semigroup, are bounded on L p(•) (R n , dγ ), where γ denotes the Gaussian measure.

We prove boundedness results for integral operators of fractional type and their higher order commutators between weighted spaces, including L p -L q , L p -BM O and L p -Lipschitz estimates. The kernels of such operators satisfy certain size condition and a Lipschitz type regularity, and the symbol of the commutator belongs to a Lipschitz class. We also deal with commutators of fractional type operators with less regular kernels satisfying a Hörmander's type inequality. As far as we know, these last results are new even in the unweighted case. Moreover, we give a characterization result involving symbols of the commutators and continuity results for extreme values of p.