We establish necessary and sufficient conditions for a weight v guaranteeing the boundedness of multilinear Riemann-Liouville operators from ∏ j = 1 m L p j to L v q .

In this paper, we consider axisymmetric solutions of inhomogeneous incompressible Navier-Stokes system under Dirichlet boundary conditions in the exterior of a cylinder. We establish the global well-posedness of strong solutions and obtain uniform estimates for the gradient of the density as well as algebraic decay rates for spatial gradients of the velocity. Note that the initial density can contain vacuum states and the initial values do not need to satisfy compatibility condition.