Mixed – bounds for fractional integrals

For a fractional integral operator I α we show that ‖ I α f ‖ L q ( w q ) ≤ c [ w ] A ∞ , q 1 / p ′ [ w ] A 1 , q 1 / q ‖ f ‖ L p ( w p ) where [ w ] A ∞ , q ≡ sup Q 1 w q ( Q ) ∫ Q M ( χ Q w q ) d x is Wilson A ∞ , q ’s constant which is smaller than the A 1 , q constant. o prove a weak type ( 1 , 1 ) bound for a couple of weights of the form ( w , M α w ) assuming that w ∈ A ∞ .