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

Musielak Orlicz bumps and Bloom type estimates for commutators of  Calder\'on Zygmund and fractional integral operators on variable Lebesgue  spaces via sparse operators