  In this paper we consider spaces of bivariate splines of bi-degree (m, n)
with maximal order of smoothness over domains associated to a two-dimensional
grid. We define admissible classes of domains for which suitable combinatorial
technique allows us to obtain the dimension of such spline spaces and the
number of tensor-product B-splines acting effectively on these domains.
Following the strategy introduced recently by Giannelli and Juettler, these
results enable us to prove that under certain assumptions about the
configuration of a hierarchical T-mesh the hierarchical B-splines form a basis
of bivariate splines of bi-degree (m, n) with maximal order of smoothness over
this hierarchical T-mesh. In addition, we derive a sufficient condition about
the configuration of a hierarchical T-mesh that ensures a weighted partition of
unity property for hierarchical B-splines with only positive weights.
