Boundary analogues of Hartog's theorem

The Bochner-Martinelli Integral and Its Applications

“…does not depend on sand h. We only need to prove that F gives a holomorphic extension of f into D \ K. In case (2), this follows because F gives a holomoq)hic…”

“…where da is the surface area element of ar € (generated by Hausdorff measure A2n- 2), and ' t/J = Ida/dEl; since ' t/J i =-0 on ra,b, we obtain that Jar. s~oo Jar,nB (z,r) But in view of Stokes's formula,…”

“…To do this, it suffices to take P = -J(8Ps,t/8(I) d(2 (here we have in mind formal integration with respect to (2). Then…”

“…However, since such a mapping u(z, () does not exist in general, we need to take a mapping s(z, () with values in the tangent bundle T(X) such that s(z, () E TAX) for all (z, () E X x X and such that conditions (1) and (2) hold. But in principle, even such a mapping may not exist (property (1) may fail).…”

“…Therefore the following construction is proposed in [76]. First construct a mapping s(z, () satisfying only condition (2). Then, using Oartan's Theorem B, find a holomorphic function cp(z, () on X x X and an integer K, ~ 0 such that cp(z, z) = 1 for all z E X and cp" (…”