Envelopes of holomorphy for bounded holomorphic functions

Abstract: Some problems concerning holomorphic continuation of the class of bounded holo­morphic functions from bounded domains in Cn that are domains of holomorphy are solved. A bounded domain of holomorphy Q in C2 with nonschlicht H°°- envelope of holomorphy is constructed and it is shown that there is a point in D for which Glea­son’s Problem for H°°(Q) cannot be solved. Furthermore a proof of the existence of a bounded domain of holomorphy in C2 for which the volume of the H°°- envelope of holomorphy is infinite is given. The idea of the proof is to put a family of so-called ”Sibony domains” into the unit bidisk by a packing procedure and patch them together by thin neighbourhoods of suitably chosen curves.If H°°(Q) is the Banach algebra of bounded holomorphic functions on a bounded domain lì in Cn and if p is a point in 12, then the following problem is known as Gleason’s Problem for                :Is the maximal ideal in H°°(£l) consisting of functions vanishing at p generatedby (Zl -Pl) ,            ,   (Zn   -    Pn) ?A sufficient condition for solving Gleason’s Problem for 77°° (lì) for all points in lì is given. In particular, this condition is fulfilled by a convex domain Q with Lipi+C- boundary (0 < e < 1) and thus generalizes a theorem of S.L.Leibenson. It is also proved that Gleason’s Problem can be solved for all points in certain unions of two polydisks in C2. One of the ideas in the methods of proof is integration along specific polygonal lines.Certain properties of some open sets defined by global plurisubharmonic func­tions in Cn are studied. More precisely, the sets Du = {z 6 Cn : u(z) < 0} andEh = {{z) w) G Cn X C : h(z,w) < 1} are considered where ti is a plurisubharmonic function of minimal growth and h ^ 0 is a non-negative homogeneous plurisubharmonic function. (That is, the functions u and h belong to the classes L(Cn) and 77+(Cn x C) respectively.) It is examined how the fact that Eh and the connected components of Du are H°°-domains of holomorphy is related to the structure of the set of disconti­nuity points of the global defining functions and to polynomial convexity. Moreover it is studied whether these notions are preserved under a certain bijective mapping from L(Cn) to H+(Cn x C). Two counterexamples are given which show that polynomial convexity is not preserved under this bijection. It is also proved, for example, that if Du is bounded and if the set of discontinuity points of u is pluripolar then Du is of type H°°.A survey paper on general properties of envelopes of holomorphy is included. In particular, the paper treats aspects of the theory for the bounded holomorphic functions. The results for the bounded holomorphic functions are compared with the corresponding ones for the holomorphic functions.