- In [ 8 ] , p. 201, P. LELONQ has shown the following theorem: "Let X be a domain in C", n 2 3, and k a fixed integer, 2 5 k 5 n -1. Then X is STEIN ii and only if lor
extra assumptiong on X are needed, see e.g. GRAUERT-REMMERT, [6], p. 158, th. 1). Its equivalence with the classical LEVI problem for pgeudoconvex domains in C" was also shown. Thege results have been elgo proved independently by Hrro-TOMATU [7]. In recent years there has been some interest in connection with the gtill unsolved general LEvI-type problems for complex spaces with singularities, that is finding sufficient conditions for an open subrjpace of a STEIN space to be STEIN, (see Sm's survey, [12], and the paper of FORNAESS-NARASIMBAN, [3]). I n this latter paper mme of the urigolved cases are proved supposing the validity of a condition of the kind considered by LELONG.
In the context of a general complex @pace, such a condition can be formulated aa follows: "Let X be an open subset of the Smm space S oi bounded dimension. Suppose that for every f , f E r ( S , Os), which a' s non constant on each irreducible component of positive dimension of S , we have: Y := (zero set of I } n X is STEIN". We can look then a t the problem : "Let X, S , f and Y be as above. Under which additional amumptions is X STEIN?". Several assumptions have been introduced to give an answer to the problem, namely: X is locally STEIN and H1(X, 0,) = 0, (FORNAESS-NARASW-
W, [3]); X is locally STEIN and H 1 ( X , Ox) of finite dimension (or H I ( X , Ox) HAUS-DORFF), (JENNANE [S]); H l ( X , 0,) of finite dimension, ( U m c o [l]
). A motivation for studying this condition lies in the fact that it is inductively satisfied in the two unsolved general LEVI problems, the locally STEIN and the union problem, (see [ 121 for their descript,ion), but unfortunately nothing is known about H 1 ( X , 0,) in duch cases. Since in both problems the HAUSDORFF part of 0,) is known to be vanishing,
(SILVA [ll] and CASSA [2] resp.), we gubgtitute for the cohomological condition on X, the following :
"the restriction map P ( X , 0,) -+ F( Y , 0,) has dense inlage", *) Work partially supported by MPI.