On defining functions for unbounded pseudoconvex domains

We introduce the notion of the kernel K(G) of an arbitrary
domain G in a complex manifold X as the set of all points where every bounded above plurisubharmonic function on G fails to be strictly plurisubharmonic. We show that for every strictly pseudoconvex domain G with smooth boundary in a complex manifold X there exists a global
defining function that is strictly plurisubharmonic precisely in the complement of K(G). We then investigate properties of the kernel. Among the other results we prove 1-pseudoconcavity of the kernel, and we show that in general the kernel does not posses any analytic structure.