Multilinear multipliers and singular integrals with smooth kernels on Hardy spaces

We consider weighted norm inequalities for multilinear multipliers whose symbols satisfy a product-type Hörmander condition. Our approach is to consider a more general family of multilinear singular integral operators associated to a family of smooth kernels that satisfy an L r L^r -Schwartz regularity condition. We give conditions for these operators to satisfy weighted Hardy space estimates and derive our results for multipliers as a special case. As an additional application, we use Rubio de Francia extrapolation to prove multilinear estimates on the variable exponent Hardy spaces.