Bochner–Martinelli on the unit ball

The unit ball is bounded with smooth boundary, $0\in B$, and the constant function $1$ is holomorphic and smooth on $\bar{B}$; the full-AC hypothesis supplies the stated premise of [F2] by [F3]. Applying [F2] at $z=0$ to $f=1$ gives $1={\int }_{\mathrm{\partial }B}{\mathrm{\Omega }}_n(\zeta ,0)-{\int }_B\bar{\mathrm{\partial }}1\wedge {\mathrm{\Omega }}_n(\zeta ,0)={\int }_{\mathrm{\partial }B}{\mathrm{\Omega }}_n(\zeta ,0)$, since the holomorphic case in [F2] has zero interior term.

Fix $j$ and define $R_t(\zeta )=e^{it}\zeta$ on $\mathrm{\partial }B$; by the explicit kernel [F1], each coefficient gains $e^{-it}$ while its wedge part, with $n$ factors $d{\zeta }_k$ and $n-1$ factors $d{\bar{\zeta }}_k$, gains $e^{it}$, so $R_t^{\ast }{\mathrm{\Omega }}_n(\zeta ,0)={\mathrm{\Omega }}_n(\zeta ,0)$ and $R_t^{\ast }({\zeta }_j{\mathrm{\Omega }}_n(\zeta ,0))=e^{it}{\zeta }_j{\mathrm{\Omega }}_n(\zeta ,0)$.

The restriction of $R_t$ is an orientation-preserving diffeomorphism of the oriented sphere $\mathrm{\partial }B$: its ambient real determinant is $1$ and it carries outward radial normals to outward radial normals. The form ${\zeta }_j{\mathrm{\Omega }}_n(\zeta ,0)$ is smooth on the compact sphere, hence compactly supported there. With $I_j={\int }_{\mathrm{\partial }B}{\zeta }_j{\mathrm{\Omega }}_n(\zeta ,0)$, [F4] and step 1.2 give $I_j={\int }_{\mathrm{\partial }B}{R}_t^{\ast }({\zeta }_j{\mathrm{\Omega }}_n)=e^{it}{I}_j$. Taking $t=\pi$ yields $I_j=-I_j$, hence $I_j=0$. This calculation also covers $n=1$, since the kernel has one holomorphic differential and no antiholomorphic differentials in that case.

The integrals therefore equal $1(0)=1$ and $z_j(0)=0$ for the constant and coordinate functions, respectively; both are holomorphic on $B$, so these explicit values agree with the holomorphic cases of [F2]. [F2, step 1.1, step 2.1, given, algebra] $\mathrm{\square }$