Jonathan proposes a simple modification to the priority search tree which 
goes as follows: 

A J-priority search tree is a range tree on X, a max-heap and min-heap on Y. 

Suppose the J-priority search tree contains $n$ 2-D points,
what is the space complexity of this structure?

Compare cost of doing a 2D range query ([BX:EX], [BY:EY]) on 
J-priority search tree to that of the range priority tree. 
An example illustrating the worse case scenario with suffice.