When does the class [A→B] consist of continuous domains?

Given classes of domains (or topological spaces) A and B, when are all function spaces [A→B] again continuous domains? The principle result of this paper is that for A either all compact and core compact spaces or only the single domain consisting of a decreasing sequence with two lower bounds, then the largest B consists of all continuous domains such that ↓x is a sup-semilattice for each x. We also establish an analogue for L-domains.