Some New Weighted Estimates on Product Spaces

We complete our theory of weighted L-p (w(1)) x L-q (w(2)) -> L-r (w(1)(r/p) w(2)(r/q)) estimates for bilinear bi-parameter Calderon-Zygmund operators under the assumption that w(1) is an element of A(p) and w(2 )is an element of A(q) are bi-parameter weights. This is done by lifting a previous restriction on the class of singular integrals by extending a classical result of Muckenhoupt and Wheeden regarding weighted BMO spaces to the product BMO setting. We use this extension of the Muckenhoupt-Wheeden result also to generalise some two-weight commutator estimates from biparameter to multi-parameter. This gives a fully satisfactory Bloom-type upper estimate for [T-1, [T-2, ...[b,T-k]]], where each T-i can be a completely general multi-parameter Calderon-Zygmund operator.