An explicit Koppelman type integral formula on analytic varieties.

Integral formulas with weight factors

“…(2) The case Q = 0 (or G = 1) and Sj = C, -z ; for | C -z\ < small constant, is in [8]. In fact the proof of Theorem 1 will be reduced to this case.…”

An integral formula on submanifolds of domains of $\Bbb C^n$

“…Fir_st we cover M by sufficiently small open sets {U,} (open in C-) so that each M n Uo is a complete intersection . Then the result in [3] gives kernels K9 for which (1 .2) holds in M n U,. Such kernels are not unique in the sense that there are certain choices that can be made, in particular, K9 depends on Hefer decompositions of the holomorphic functions which define M n Uo (as the set of therr common zeros) .…”

“…The purpose of this paper is to construct an analogue of (1. for u E Cho a) (M) and z E M. In [3] we derived an integral formula like (1.2) in the case M is a complete intersection (i.e., if there exist functions hl, . .…”

“…. n dhp :~0 on M); the construction in the general case (Le ., when M is not necessarily a complete intersection) will be based on the result of [3] . A similar construction was carried out by Berndtsson [1] .…”

“…But using a result of Berndtsson [1] we show that such Hefer decompositions can be chosen appropriately so that K9 =K9 onmnu,nu, ; thus we can define a global kernel K9 (by setting K9 =: K9 on M n U,) . Then we have to show that (1 .2) holds; and this is done along the same lines as in [3]] . Here is an outline of this proof in the case u is a holomorphic function on M. In this case we have to show that f CEaU, which holds by the result of [3] (or [2] for that case) since Ko = Kó in Uv n M. Our result gives, in particular, integral formulas for domains D CC X, if X is a Stein manifold, via the theorem that Stein manifolds admit embedding in some ON.…”