Some remarks on extrapolation with "flat" weights
  We prove an extrapolation result for general operators under some weak assumptions on the boundedness of the operator. In particular, we show that if the operator is weakly bounded on some L^{p_{0}}(w), for all "flat" weights, w in A_{p_{0}}, 1 < p_{0} <\infty, then for p in some small neighborhood around p_{0}, and all "flat" A_{p} weights, w, the operator is weakly bounded on L^{p}(w), and as a result we get strong type estimates for the operator. 
Nicholas Boros, Nikolaos Pattakos and Alexander Volberg
math.CA