Weighted norm inequalities for integral and maximal operators on variable Lebesgue spaces

BERNARDIS, ANA; DALMASSO, ESTEFANÍA; PRADOLINI, GLADIS

Santander

Congreso; XV School of Mathematics "Lluís Santaló". Dyadic Harmonic Analysis and related topics; 2014

Universidad Internacional Menéndez Pelayo

We give weighted boundedness results on variable Lebesgue spaces $L^{p(\cdot)}$ for certain maximal operators associated to a Young function. It is known that this maximal functions control a large class of singular and fractional integrals of convolution type and their commutators, whose kernels satisfy a generalized H\"ormander type condition. In this direction, we characterize the class of weights for which those maximal operators are bounded in $L^{p(\cdot)}$ when the Young function is of $L\log L$ type 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. This is a joint work with Ana Bernardis and Gladis Pradolini.