  This paper studies the dynamics of two incompressible immiscible fluids in 2D
modeled by the inhomogeneous Navier-Stokes equations. We prove that if
initially the viscosity contrast is small then there is global-in-time
regularity. This result has been proved recently in [32] for $H^{5/2}$ Sobolev
regularity of the interface. Here we provide a new approach which allows to
obtain preservation of the natural $C^{1+\gamma}$ H\"older regularity of the
interface for all $0<\gamma<1$. Our proof is direct and allows for low Sobolev
regularity of the initial velocity without any extra technicality. It uses new
quantitative harmonic analysis bounds for $C^{\gamma}$ norms of even singular
integral operators on characteristic functions of $C^{1+\gamma}$ domains [21].
