Title: A new approach to bounds on mixing Abstract: We consider mixing by incompressible flows. In 2003, Bressan stated a conjecture concerning a bound on the mixing achieved by the flow in terms of an L^1 norm of the velocity field. Existing results in the literature use an L^p norm with p>1. In this paper we introduce a new approach to prove such results. It recovers most of the existing results and offers new perspective on the problem. Our approach makes use of a recent harmonic analysis estimate from Seeger, Smart and Street.