Inverse volume inequalities for matrix weights

For weights in the matricial Muckenhoupt classes we investigate a number of properties analogous to properties which hold in the scalar Muckenhoupt classes. In contrast to the scalar case we exhibit for each p, 1 < p < , a matrix weight W A p,q \ p <p A p ,q . We also give a necessary and sufficient condition on W in A p,q , a "reverse inverse volume inequality", to ensure that W is in A p ,q for some p < p.