Local Cauchy transform with smooth parameters

Apply [F1] on the manifold $\mathbb{C}$ to $K=D_k^{\mathrm{\prime }}$ and $W=D_k$, obtaining $\chi \in C_c^{\mathrm{\infty }}(D_k)$ equal to $1$ on an open neighborhood of $D_k^{\mathrm{\prime }}$. Compact containment lets us choose an open coordinate disc $D_k^{\mathrm{\prime }\mathrm{\prime }}$ containing $D_k^{\mathrm{\prime }}$ with closure inside that neighborhood. For fixed $z^{\mathrm{\prime }}=(z_1,\dots ,{\widehat{z}}_k,\dots ,z_n)$, extend $h(z^{\mathrm{\prime }},\zeta )=\chi (\zeta )g(z^{\mathrm{\prime }},\zeta )$ by zero from $D_k$ to $\mathbb{C}$; the extension is smooth and supported in the fixed compact set $K_0=\mathrm{supp}\chi$.

In the local integral change variables $\eta =\zeta -z_k$ and write $\tilde{h}(z,\eta )=h(z^{\mathrm{\prime }},z_k+\eta )$, so $T_{k}g(z)=\frac{1}{2\pi i}{\int }_{\mathbb{C}}\tilde{h}(z,\eta ){\eta }^{-1}d\eta \wedge d\bar{\eta }$. For any compact parameter set $Q⋐P^{\mathrm{\prime }\mathrm{\prime }}$, the support of $\tilde{h}$ and all its real parameter derivatives lies in a common disk $\mathrm{\mid }\eta \mathrm{\mid }\le R$, since $K_0$ and the $z_k$-projection of $Q$ are compact. Also ${\int }_{\mathrm{\mid }\eta \mathrm{\mid }\le R}\mathrm{\mid }\eta {\mathrm{\mid }}^{-1}dA(\eta )=2\pi R$, so these integrals converge absolutely. For each real-coordinate multi-index $\alpha$ set $F_{\alpha }(z)=\frac{1}{2\pi i}{\int }_{\mathbb{C}}(D_z^{\alpha }\tilde{h})(z,\eta ){\eta }^{-1}d\eta \wedge d\bar{\eta }$. Fix a real parameter coordinate $t$. The fundamental theorem of calculus writes the difference between the difference quotient of $D_z^{\alpha }\tilde{h}$ in $t$ and $D_t{D}_z^{\alpha }\tilde{h}$ as an average of increments of the latter derivative; on $Q\times \{\mathrm{\mid }\eta \mathrm{\mid }\le R\}$ their supremum tends to zero with the increment by uniform continuity on a slightly larger compact set. Thus $\mathrm{\mid }[F_{\alpha }(z+s{e}_t)-F_{\alpha }(z)]\mathrm{/}s-F_{\alpha +e_t}(z)\mathrm{\mid }\le C{\omega }_{\alpha }(\mathrm{\mid }s\mathrm{\mid }){\int }_{\mathrm{\mid }\eta \mathrm{\mid }\le R}\mathrm{\mid }\eta {\mathrm{\mid }}^{-1}dA(\eta )\to 0$, where ${\omega }_{\alpha }(\delta )\to 0$ is that uniform-continuity modulus and $C$ accounts for the fixed form factor. The same estimate gives continuity of each $F_{\alpha }$, and induction proves $D_z^{\alpha }{T}_{k}g=F_{\alpha }$ for every $\alpha$, hence $T_{k}g\in C^{\mathrm{\infty }}(P^{\mathrm{\prime }\mathrm{\prime }})$.

The transformed formula in step 1.2 and [F4] identify ${\mathrm{\partial }}_{{\bar{z}}_k}{T}_{k}g$ with the integral of ${\mathrm{\partial }}_{\bar{\zeta }}h(z^{\mathrm{\prime }},\zeta )\mathrm{/}(\zeta -z_k)$ against $(2\pi i{)}^{-1}d\zeta \wedge d\bar{\zeta }$. For each fixed $z^{\mathrm{\prime }}$, choose a bounded disc $E\subset \mathbb{C}$ containing both $\mathrm{supp}h(z^{\mathrm{\prime }},\cdot )$ and $D_k^{\mathrm{\prime }\mathrm{\prime }}$; the slice is zero near $\mathrm{\partial }E$. Full AC and the Cauchy–Pompeiu premise are both in [F2], so [F3] on $E$ has zero boundary term and gives this integral equal to $h(z^{\mathrm{\prime }},z_k)=g(z)$ on $P^{\mathrm{\prime }\mathrm{\prime }}$, because $\chi =1$ there. Thus ${\mathrm{\partial }}_{{\bar{z}}_k}{T}_{k}g=g$.

For $\mathrm{\ell }\mathrm{\ne }k$, the cutoff $\chi (\zeta )$ is independent of $z_{\mathrm{\ell }}$, so parameter differentiation in step 1.2 and [F4] give ${\mathrm{\partial }}_{{\bar{z}}_{\mathrm{\ell }}}{T}_{k}g=T_k({\mathrm{\partial }}_{{\bar{z}}_{\mathrm{\ell }}}g)$, with the single cutoff factor already included in $T_k$. If ${\mathrm{\partial }}_{{\bar{z}}_{\mathrm{\ell }}}g=0$, this is zero, proving preservation of each selected equation.

If $g\in C_c^{\mathrm{\infty }}({\mathbb{C}}^n)$, omit the cutoff and put $h=g$. On any compact parameter set, compact support of $g$ and boundedness of $z_k$ again place the translated numerator and every derivative in a common bounded $\eta$-disc. The uniform-continuity estimate of step 1.2 therefore proves that the global integral defines a smooth function.

For fixed values of the other variables, the slice $g(z^{\mathrm{\prime }},\cdot )$ is compactly supported. Its translated parameter derivative is $T_k({\mathrm{\partial }}_{{\bar{z}}_k}g)$, and the Cauchy–Pompeiu argument of step 2.1, using the AC premise and formula [F2, F3], gives ${\mathrm{\partial }}_{{\bar{z}}_k}{T}_{k}g=g$. For every $\mathrm{\ell }\mathrm{\ne }k$, the same differentiation calculation as in step 2.2 gives ${\mathrm{\partial }}_{{\bar{z}}_{\mathrm{\ell }}}{T}_{k}g=T_k({\mathrm{\partial }}_{{\bar{z}}_{\mathrm{\ell }}}g)$.