An absolutely convergent multi-indexed power series is holomorphic and differentiates termwise

Fix $\theta$ with $0<\theta <1$. For $z\in {\bar{\mathrm{\Delta }}}_{\theta r}(a)$ the hypothesis and [L14] give $\mathrm{\mid }{c}_{\alpha }(z-a{)}^{\alpha }\mathrm{\mid }\le M{\prod }_{k<m}{\theta }^{{\alpha }_k}$; the box sums of the right side are $M{\prod }_{k<m}{\sum }_{j\le N}{\theta }^j\le M(1-\theta {)}^{-m}$ by [L14] and [L6], and every finite subset of ${\mathbb{N}}^m$ lies in a box, so [L6] makes the majorant series convergent and [L1] and [L5] give absolute and uniform convergence on ${\bar{\mathrm{\Delta }}}_{\theta r}(a)$. Since each $z\in {\mathrm{\Delta }}_r(a)$ lies in some such closed polydisc, $g$ is defined on ${\mathrm{\Delta }}_r(a)$. This is claim 1.

For $0<\theta <{\theta }_1<1$ and every $k<m$ the series ${\sum }_{\alpha }{\alpha }_k{\theta }^{\mathrm{\mid }\alpha \mathrm{\mid }-1}$ converges: by [L7] the sequence $j(\theta \mathrm{/}{\theta }_1{)}^{j-1}$ is null, hence bounded by some $K_{\theta }$, so $j{\theta }^{j-1}\le K_{\theta }{\theta }_1^{j-1}$ and [L6] with [L14] bounds the box sums of ${\sum }_{\alpha }{\alpha }_k{\theta }^{\mathrm{\mid }\alpha \mathrm{\mid }-1}$ by $K_{\theta }(1-{\theta }_1{)}^{-1}(1-\theta {)}^{-(m-1)}$; [L6] then gives convergence.

Fix $\theta$ with $0<\theta <1$, a point $z^0\in {\bar{\mathrm{\Delta }}}_{\theta r}(a)$ and ${\theta }^{\mathrm{\prime }}$ with $\theta <{\theta }^{\mathrm{\prime }}<1$; write $w=z^0-a$, so $\mathrm{\mid }{w}_k\mathrm{\mid }\le \theta r_k$. For $h\in {\mathbb{C}}^m$ with $h\mathrm{\ne }0$ put $\eta ={\max }_{k<m}\mathrm{\mid }{h}_k\mathrm{\mid }\mathrm{/}{r}_k$ and assume $\eta \le ({\theta }^{\mathrm{\prime }}-\theta )\mathrm{/}2$, which holds for all small $\parallel h\parallel$ because $\mathrm{\mid }{h}_k\mathrm{\mid }\le \parallel h\parallel$; then $z^0+h\in {\bar{\mathrm{\Delta }}}_{{\theta }^{\mathrm{\prime }}r}(a)\subseteq {\mathrm{\Delta }}_r(a)$ by [L14].

For each $\alpha$ let ${\varphi }_{\alpha }(\tau )={\prod }_{k<m}(w_k+\tau h_k{)}^{{\alpha }_k}$, a polynomial in $\tau$ of degree at most $\mathrm{\mid }\alpha \mathrm{\mid }$ by [L9] and [L13], hence entire, with ${\varphi }_{\alpha }(\tau )={\sum }_{j\le \mathrm{\mid }\alpha \mathrm{\mid }}{\varphi }_{\alpha ,j}{\tau }^j$ and ${\varphi }_{\alpha ,j}={\varphi }_{\alpha }^{(j)}(0)\mathrm{/}j!$ by [L10] and [L11]. In particular ${\varphi }_{\alpha ,0}=w^{\alpha }$ and, by the product rule of [L10] and an induction on the number of factors ([L12]), ${\varphi }_{\alpha ,1}={\sum }_{k<m}{\alpha }_k{w}^{\alpha -e_k}{h}_k$, terms with ${\alpha }_k=0$ being $0$.

The series $b_k(z^0)={\sum }_{\alpha }{\alpha }_k{c}_{\alpha }{w}^{\alpha -e_k}$ converges absolutely: $\mathrm{\mid }{\alpha }_k{c}_{\alpha }{w}^{\alpha -e_k}\mathrm{\mid }\le M{r}_k^{-1}{\alpha }_k{\theta }^{\mathrm{\mid }\alpha \mathrm{\mid }-1}$ by the hypothesis and [L14], and step 1.2 makes that majorant summable.

Put $T=({\theta }^{\mathrm{\prime }}-\theta )\mathrm{/}\eta$, so $T\ge 2$ by step 1.3. For $\mathrm{\mid }\tau \mathrm{\mid }\le T$ and every $k$, $\mathrm{\mid }{w}_k+\tau h_k\mathrm{\mid }\le \theta r_k+T\mathrm{\mid }{h}_k\mathrm{\mid }\le {\theta }^{\mathrm{\prime }}{r}_k$ by [L14], so $\mathrm{\mid }{\varphi }_{\alpha }(\tau )\mathrm{\mid }\le {\prod }_{k<m}({\theta }^{\mathrm{\prime }}{r}_k{)}^{{\alpha }_k}$ there. Applying [L8] to ${\varphi }_{\alpha }$ on the disc of radius $T$ gives $\mathrm{\mid }{\varphi }_{\alpha ,j}\mathrm{\mid }\le {\prod }_{k<m}({\theta }^{\mathrm{\prime }}{r}_k{)}^{{\alpha }_k}{T}^{-j}$.

Hence $R_{\alpha }:={\varphi }_{\alpha }(1)-{\varphi }_{\alpha ,0}-{\varphi }_{\alpha ,1}={\sum }_{2\le j\le \mathrm{\mid }\alpha \mathrm{\mid }}{\varphi }_{\alpha ,j}$ satisfies $\mathrm{\mid }{R}_{\alpha }\mathrm{\mid }\le {\prod }_{k<m}({\theta }^{\mathrm{\prime }}{r}_k{)}^{{\alpha }_k}{\sum }_{j\ge 2}{T}^{-j}\le {\displaystyle \frac{2}{T^2}}{\prod }_{k<m}({\theta }^{\mathrm{\prime }}{r}_k{)}^{{\alpha }_k}$, using $T\ge 2$ and [L6]. With $T^{-2}={\eta }^2({\theta }^{\mathrm{\prime }}-\theta {)}^{-2}$ this is ${\displaystyle \frac{2{\eta }^2}{({\theta }^{\mathrm{\prime }}-\theta {)}^2}}{\prod }_{k<m}({\theta }^{\mathrm{\prime }}{r}_k{)}^{{\alpha }_k}$.

Summing against the coefficients, the hypothesis on $\mathrm{\mid }{c}_{\alpha }\mathrm{\mid }$ gives ${\sum }_{\alpha }\mathrm{\mid }{c}_{\alpha }{R}_{\alpha }\mathrm{\mid }\le {\displaystyle \frac{2{\eta }^{2}M}{({\theta }^{\mathrm{\prime }}-\theta {)}^2}}{\sum }_{\alpha }{\prod }_{k<m}{\theta }^{\mathrm{\prime }{\alpha }_k}={\displaystyle \frac{2{\eta }^{2}M}{({\theta }^{\mathrm{\prime }}-\theta {)}^2(1-{\theta }^{\mathrm{\prime }}{)}^m}}$ by [L6] and [L14]; since $\eta \le \parallel h\parallel \mathrm{/}{\min }_{k<m}{r}_k$, this is at most a constant times $\parallel h{\parallel }^2$.

By steps 2.1, 5.1 and 2.2, and by [L1] and [L4] which allow the absolutely convergent series to be split term by term, $g(z^0+h)-g(z^0)-{\sum }_{k<m}{b}_k(z^0)h_k={\sum }_{\alpha }{c}_{\alpha }{R}_{\alpha }$, whose modulus is $O(\parallel h{\parallel }^2)$ and therefore $o(\parallel h\parallel )$. The map $h\mapsto {\sum }_{k<m}{b}_k(z^0)h_k$ is $\mathbb{C}$-linear by [L3] and [L15], so [L2] makes $g$ complex differentiable at $z^0$ with that differential, and ${\mathrm{\partial }}_{z_k}g(z^0)=b_k(z^0)$ by [L3]. As $\theta <1$ and $z^0$ were arbitrary, this is claim 2.

For claim 3 fix $k<m$ and $\theta$ with $0<\theta <1$, and re-index the derived series by $\beta =\alpha -e_k$, so its coefficient at $\beta$ is $({\beta }_k+1)c_{\beta +e_k}$, of modulus at most $M({\beta }_k+1){\theta }^{\mathrm{\mid }\beta \mathrm{\mid }+1}{\prod }_{l<m}(\theta r_l{)}^{-{\beta }_l}{r}_k^{-1}$ by the hypothesis and [L14]. By [L7] the numbers $({\beta }_k+1){\theta }^{\mathrm{\mid }\beta \mathrm{\mid }+1}$ are bounded by a constant $M^{\mathrm{\prime }\mathrm{\prime }}$, so the derived coefficients satisfy a bound of the same shape with polyradius $\theta r$ and constant $M^{\mathrm{\prime }\mathrm{\prime }}M\mathrm{/}{r}_k$.

Iterating step 7.1 and step 6.1, an induction on $\mathrm{\mid }\beta \mathrm{\mid }$ ([L12]) shows that every ${\mathrm{\partial }}_z^{\beta }g$ exists on ${\mathrm{\Delta }}_r(a)$ and is the termwise $\beta$-fold derived series, whose coefficient at $\alpha \ge \beta$ is $({\prod }_{k<m}{\alpha }_k({\alpha }_k-1)\cdots ({\alpha }_k-{\beta }_k+1))c_{\alpha }$ and which vanishes unless $\alpha \ge \beta$ componentwise. Evaluating at $z=a$, [L13] kills every monomial $(z-a{)}^{\alpha -\beta }$ except the one with $\alpha =\beta$, whose coefficient is $\beta !c_{\beta }$ by [L11]; so ${\mathrm{\partial }}_z^{\beta }g(a)=\beta !c_{\beta }$.