Hartogs–Bochner extension theorem

Remarks

The proof uses the Bochner–Martinelli formula. The Bochner–Martinelli integral of $f$ over $\partial D$ defines a function $F_-$ on $D$ and a function $F_+$ on $\C^n \setminus \bar D$. The tangential Cauchy–Riemann equations make both functions holomorphic, and the jump formula gives $F_- - F_+ = f$ on $\partial D$. The function $F_+$ tends to $0$ at infinity. On each complex line that misses $\bar D$ it is an entire function of one variable tending to $0$, hence zero by Liouville's theorem. These lines fill a nonempty open set. Since $\C^n \setminus \bar D$ is connected, $F_+ = 0$ by the identity theorem, so $F = F_-$ has boundary values $f$.

Example: if $f$ is the restriction to $\partial D$ of a function holomorphic near $\bar D$, then $f$ satisfies the tangential Cauchy–Riemann equations and $F$ is that function.

The hypothesis $n \ge 2$ is necessary. For $n = 1$ the tangential condition is empty. The function $f(z) = \bar z$ on the unit circle has no holomorphic extension to the disc that is continuous up to the boundary. Indeed $\int_{\abs{z} = 1} \bar z \, dz = 2\pi i \ne 0$, whereas Cauchy's theorem would make the integral of such an extension zero.

Going deeper

For a student

The theorem is a boundary version of Hartogs's extension theorem. Applied to the restriction of a holomorphic function defined near $\partial D$, it gives the extension of that function to $D$. It is the starting point of the theory of CR functions on real hypersurfaces and of their holomorphic extension.