Generalized maximal functions and the control of related operators on weighted Musielak-Orlicz spaces

BERNARDIS, ANA; DALMASSO, ESTEFANÍA; PRADOLINI, GLADIS

Antequera

Congreso; VI International Course of Mathematical Analysis in Andalucía; 2014

We characterize the class of weights related to the boundedness of maximal operators associated to Young functions of LlogL type in the context of variable Lebesgue spaces and we give sufficient conditions for more general Young functions. Fractional versions of these results are also obtained by means of a weighted Hedberg type inequality in the variable context. These results are new even in the classical Lebesgue spaces. We also deal with Wiener?s type inequalities for the mentioned operators in the spirit of the corresponding result proved in [CU-F] for the Hardy-Littlewood maximal operator. 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, such as singular and fractional integrals with kernels satisfying certain H"ormander type condition and their commutators. References [CU-F] D. Cruz-Uribe and A. Fiorenza, $Llog L$ results for the maximal operator in variable $L^{p}$ spaces, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2631-2647.