Extrapolation and weighted norm inequalities in the variable Lebesgue spaces

We extend the theory of Rubio de Francia extrapolation, including off-diagonal, limited range and A ∞ A_∞ extrapolation, to the weighted variable Lebesgue spaces L p ( ⋅ ) ( w ) Lp(· )(w) . As a consequence we are able to show that a number of different operators from harmonic analysis are bounded on these spaces. The proofs of our extrapolation results are developed in a way that outlines a general approach to proving extrapolation theorems on other Banach function spaces.