  We obtain Musielak Orlicz bumps conditions on a pair of weights for the
boundedness of Calder\'on Zygmund operators and their commutators between
variable Lebesgue spaces with different weights. The symbols of the commutators
belong to a wider class of functions. We also give Bloom type estimates for
commutators of Calder\'on Zygmund and fractional integral operators in the
variable Lebesgue context. The techniques involved in both type of results are
related with the theory of sparse domination.

Analysis of PDEs