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Well I was thinking on this problem, any idea to progress will be gladly accepted, Let $f:\Omega\rightarrow V$ be a biholomorphic map where $\Omega$ and $V$ are bounded open set in $\mathbb{C}^n$ with $\mathbb{C}^w$ boundary, Does $f$ extend continously to the boundary? for n=1 and not $\mathbb{C}^w$ boundary I have got the answer from thomas here.

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Where is "here"? – Georges Elencwajg Jun 3 '12 at 16:21
Ah, thank you, Mex. – Georges Elencwajg Jun 3 '12 at 17:30
:) :) :) :) :) :) :) – La Belle Noiseuse Jun 3 '12 at 17:56
up vote 2 down vote accepted

It's true if $\Omega$ and $V$ are strictly pseudoconvex with $C^\infty$ boundary. It is also true for real-analytic boundaries if the domains are pseudoconvex.

For the non-pseudoconvex case, I need to check some references, but I'm pretty sure the problem has not been solved in full generality. I'm not aware of any counterexamples (for real analytic boundary.)

Added: You should check out Fefferman, C.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math.26, 1-65 (1974)

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Extension for non-pseudo-convex domains (real analytic boundary) in 2 dimensions:… (Diederich and Pinchuk). – user31373 Jun 3 '12 at 16:53
@LeonidKovalev Thanks for the reference, I had a vague recollection that it was known for $n = 2$. – mrf Jun 3 '12 at 21:39

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