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. A new proof is given of the extrapolation theorem of J. L. Rubiode Francia [7]. Unlike the proof in [7], the proof presented here is independentof the theory of vector valued inequalities. I am indebted to J. L. Rubio de Francia for calling my attention to the following:THEOREM. Let T be a sublinear operator defined on a class of measurablefunctions in Rn. Suppose that, for some po with 1 , and for every weight win the class Ap0 of B. Muckenhoupt (see [1 and 5]), T satisfies an inequality (1) |r„ \Tf(x)rw(x)dx 0, defineG = {M(g1^tw)w~1y, where M stands for the Hardy-Littlewood maximal operator.Then

An extrapolation theorem in the theory of 𝐴_{𝑝} weights