Mixed A1A1–A∞A∞ bounds for fractional integrals

For a fractional integral operator IαIα we show that ‖Iαf‖Lq(wq)≤c[w]A∞,q1/p′[w]A1,q1/q‖f‖Lp(wp) where [w]A∞,q≡supQ1wq(Q)∫QM(χQwq)dx is Wilson A∞,qA∞,q’s constant which is smaller than the A1,qA1,q constant.We also prove a weak type (1,1)(1,1) bound for a couple of weights of the form (w,Mαw)(w,Mαw) assuming that w∈A∞w∈A∞.