The Hardy spaces on tubes over convex cones have an especially rich structure, so that precise results are known concerning the boundary values of Hp functions.
[1] Certain tubes over cones support a Bergman metric in terms of which they become bounded symmetric domains.
Then any element of Cn can be decomposed into real and imaginary parts: Let A be an open subset of Rn.
The tube over A, denoted TA, is the subset of Cn consisting of all elements whose real parts lie in A:[2][a] Suppose that A is a connected open set.
This means that A is an open convex set such that, whenever x lies in A, so does the entire ray from the origin to x. Symbolically, If A is a cone, then the elements of H2(TA) have L2 boundary limits in the sense that[5] exists in L2(B).