Separate holomorphy forces local boundedness on smaller polydiscs

We prove a stronger local claim by induction on $m$: every separately holomorphic function on ${\mathrm{\Delta }}_R^m$ is holomorphic on a neighborhood of each point of ${\mathrm{\Delta }}_R^m$. Once that is known, the displayed boundedness follows, because the compact set ${\bar{\mathrm{\Delta }}}_r^m$ is covered by finitely many such holomorphic neighborhoods and each holomorphic function is bounded on a smaller closed polydisc inside its neighborhood.

Base case $m=1$: separate holomorphy is ordinary one-variable holomorphy by [L1], so the local claim and the boundedness statement are immediate. Assume now that the local claim is known in dimension $m-1$, and prove it in dimension $m\ge 2$.

Fix $p\in {\mathrm{\Delta }}_R^m$. Choose $\rho >0$ with $0<\rho <R$ and ${\bar{\mathrm{\Delta }}}_{2\rho }^m(p)\subseteq {\mathrm{\Delta }}_R^m$. After translating and rescaling each coordinate disc, it is enough to prove that a separately holomorphic function on ${\mathrm{\Delta }}_2^m$ is holomorphic in a neighborhood of the origin. Write $z=(z^{\mathrm{\prime }},w)$ with $z^{\mathrm{\prime }}\in {\mathbb{C}}^{m-1}$ and $w\in \mathbb{C}$.

Box claim. If a closed real box $K=I_1\times \cdots \times I_d\subseteq {\mathbb{R}}^d$ is covered by countably many closed sets $F_n$, then some $F_n$ contains a smaller closed real box with nondegenerate sides. We prove this by induction on $d$. For $d=1$ it is [L2]. Assume the claim in dimension $d-1$. Write $K=I\times K^{\mathrm{\prime }}$. Enumerate the closed subboxes of $K^{\mathrm{\prime }}$ with rational endpoints in the coordinates of $K^{\mathrm{\prime }}$ as $Q_1,Q_2,\dots$. For each pair $(n,j)$ let $E_{n,j}:=\{x\in I:\{x\}\times Q_j\subseteq F_n\}\mathrm{.}$ Each $E_{n,j}$ is closed. Fix $x\in I$. The sections $F_n(x):=\{y\in K^{\mathrm{\prime }}:(x,y)\in F_n\}$ are closed and cover $K^{\mathrm{\prime }}$, so the induction hypothesis in dimension $d-1$ gives some $n$ such that $F_n(x)$ contains a smaller closed box; shrinking slightly if needed, that box contains a rational-endpoint subbox $Q_j$. Hence $x\in E_{n,j}$. So the countable family $E_{n,j}$ covers $I$, and [L2] gives one pair $(n,j)$ for which $E_{n,j}$ contains a nondegenerate closed subinterval $J$. Then $J\times Q_j\subseteq F_n$, proving the claim.

Whenever $0\le d<1<\eta$, one can choose radii $d<s<r_1<1<r_2<\eta$.

For each positive integer $B$, define ${\mathrm{\Omega }}_B:=\{z^{\mathrm{\prime }}\in {\bar{\mathrm{\Delta }}}_1^{m-1}:\mathrm{\mid }f(z^{\mathrm{\prime }},w)\mathrm{\mid }\le B\text{ for every }\mathrm{\mid }w\mathrm{\mid }\le 1\}\mathrm{.}$ For fixed $w$ with $\mathrm{\mid }w\mathrm{\mid }\le 1$, the induction hypothesis applied to $z^{\mathrm{\prime }}\mapsto f(z^{\mathrm{\prime }},w)$ makes that function holomorphic, hence continuous, on ${\mathrm{\Delta }}_2^{m-1}$. Therefore each set $\{z^{\mathrm{\prime }}:\mathrm{\mid }f(z^{\mathrm{\prime }},w)\mathrm{\mid }\le B\}$ is closed, and so every ${\mathrm{\Omega }}_B$ is closed. Also ${\bigcup }_{B\ge 1}{\mathrm{\Omega }}_B={\bar{\mathrm{\Delta }}}_1^{m-1}$, because for fixed $z^{\mathrm{\prime }}$ the slice $w\mapsto f(z^{\mathrm{\prime }},w)$ is holomorphic on ${\mathrm{\Delta }}_2$ and therefore bounded on $\mathrm{\mid }w\mathrm{\mid }\le 1$.

Apply the box claim to the real box $[-1\mathrm{/}\sqrt{2},1\mathrm{/}\sqrt{2}{]}^{2m-2}\subseteq {\bar{\mathrm{\Delta }}}_1^{m-1}$ and the closed cover $({\mathrm{\Omega }}_B{)}_{B\ge 1}$. We obtain some $B_0$ and a nondegenerate closed real box contained in ${\mathrm{\Omega }}_{B_0}$. Inside its relative interior choose a closed complex polydisc ${\bar{\mathrm{\Delta }}}_{2\epsilon }^{m-1}(a^{\mathrm{\prime }})$ for some $a^{\mathrm{\prime }}$ and some $\epsilon >0$. Its centre satisfies $d:={\max }_{1\le j<m}\mathrm{\mid }{a}_j^{\mathrm{\prime }}\mathrm{\mid }<1,$ and $\mathrm{\mid }f(z^{\mathrm{\prime }},w)\mathrm{\mid }\le B_0\text{for }{z}^{\mathrm{\prime }}\in {\bar{\mathrm{\Delta }}}_{2\epsilon }^{m-1}(a^{\mathrm{\prime }}),\mathrm{\mid }w\mathrm{\mid }\le 1.$ Hence $f$ is bounded on the product polydisc $E:={\mathrm{\Delta }}_{2\epsilon }^{m-1}(a^{\mathrm{\prime }})\times {\mathrm{\Delta }}_1$.

By [L3], the separately holomorphic and bounded function $f$ is jointly holomorphic on $E$. Put $\zeta =z^{\mathrm{\prime }}-a^{\mathrm{\prime }}$ and retain the notation $f(\zeta ,w)$ after this translation. The original first-variable domain contains the centred polydisc ${\mathrm{\Delta }}_{\eta }^{m-1}$, where $\eta :=2-d>1$, while the original target $z^{\mathrm{\prime }}=0$ now has coordinate $\zeta =-a^{\mathrm{\prime }}$. Thus $f$ is separately holomorphic on ${\mathrm{\Delta }}_{\eta }^{m-1}\times {\mathrm{\Delta }}_2$ and jointly holomorphic on ${\mathrm{\Delta }}_{2\epsilon }^{m-1}\times {\mathrm{\Delta }}_1$. Choose radii $d<s<r_1<1<r_2<\eta$, which is possible because $d<1<\eta$.

For each multi-index $\alpha \in {\mathbb{N}}^{m-1}$, define $c_{\alpha }(w):=\frac{1}{\alpha !}{\mathrm{\partial }}_{\zeta }^{\alpha }f(0,w)(\mathrm{\mid }w\mathrm{\mid }<1),$ using the jointly holomorphic function from step 4.1. By [L4], every $c_{\alpha }$ is holomorphic on ${\mathrm{\Delta }}_1$. For fixed $w$ with $\mathrm{\mid }w\mathrm{\mid }<1$, the induction hypothesis makes $\zeta \mapsto f(\zeta ,w)$ holomorphic on ${\mathrm{\Delta }}_{\eta }^{m-1}$, so these $c_{\alpha }(w)$ are exactly its Taylor coefficients at $0$. Since $f$ is bounded by $B_0$ on the larger product ${\mathrm{\Delta }}_{2\epsilon }^{m-1}\times {\mathrm{\Delta }}_1$, [L5] applied at the strictly smaller radius $\epsilon$ gives $\mathrm{\mid }{c}_{\alpha }(w)\mathrm{\mid }\le B_0{\epsilon }^{-\mathrm{\mid }\alpha \mathrm{\mid }}(\mathrm{\mid }w\mathrm{\mid }<1)\mathrm{.}$ For each nonzero multi-index $\alpha$ with $c_{\alpha }\equiv ̸0$, define $u_{\alpha }(w):=\mathrm{\mid }\alpha {\mathrm{\mid }}^{-1}\log \mathrm{\mid }{c}_{\alpha }(w)\mathrm{\mid }$. By [L6], each such $u_{\alpha }$ is subharmonic on ${\mathrm{\Delta }}_1$. If $c_{\alpha }\equiv 0$, then the term $c_{\alpha }(w){\zeta }^{\alpha }$ vanishes identically and is already harmless for the later power-series tail estimate. The displayed estimate gives a uniform upper bound for the whole family $(u_{\alpha }{)}_{\alpha \mathrm{\ne }0,c_{\alpha }\equiv ̸0}$.

Fix $w\in {\mathrm{\Delta }}_1$. The Cauchy estimates [L5] applied to the holomorphic function $\zeta \mapsto f(\zeta ,w)$ on ${\mathrm{\Delta }}_{\eta }^{m-1}$, with the strictly smaller radius $r_2$, show that $\mathrm{\mid }{c}_{\alpha }(w)\mathrm{\mid }\le M(w)r_2^{-\mathrm{\mid }\alpha \mathrm{\mid }}$ for some finite constant $M(w)$ and every nonzero multi-index $\alpha$. Therefore ${\mathrm{lim\hspace{0.17em}sup}}_{\begin{array}{c}\mathrm{\mid }\alpha \mathrm{\mid }\to \mathrm{\infty }\\ c_{\alpha }\equiv ̸0\end{array}}{u}_{\alpha }(w)\le -\log r_2\mathrm{.}$

Let $S:=\{\alpha \in {\mathbb{N}}^{m-1}\setminus \{0\}:c_{\alpha }\equiv ̸0\}\mathrm{.}$ If $S$ is finite, the required tail estimate is immediate. Otherwise enumerate it as $(\alpha (j){)}_{j\ge 1}$ with nondecreasing degrees and put $v_j=u_{\alpha (j)}$. By steps 5.1 and 6.1, the subharmonic functions $v_j$ have a common upper bound $A$ on ${\mathrm{\Delta }}_1$ and satisfy ${\mathrm{lim\hspace{0.17em}sup}}_j{v}_j(w)\le C:=-\log r_2$ pointwise.

We claim that for every compact $K⋐{\mathrm{\Delta }}_1$ and every $\delta >0$, $v_j\le C+\delta$ on $K$ for all sufficiently large $j$. Otherwise choose $j_k\uparrow \mathrm{\infty }$ and $q_k\in K$ with $v_{j_k}(q_k)>C+\delta$, and pass to a subsequence with $q_k\to q\in K$. Choose $t>0$ with $\overline{D(q,t)}\subseteq {\mathrm{\Delta }}_1$, discard finitely many terms so $q_k\in D(q,t)$, and put $P_k(\theta ):=P((q_k-q)\mathrm{/}t,e^{i\theta })$. By the definition of $S$ and [L9], no selected coefficient can vanish on a nonempty open subset of ${\mathrm{\Delta }}_1$. More specifically for the boundary argument, $v_{j_k}$ cannot be identically $-\mathrm{\infty }$ on the circle: if it were, every constant harmonic function $-n$ would majorize its boundary values, so [L10] would give $v_{j_k}(q_k)\le -n$ for every $n$, contradicting the finite strict lower bound just chosen. For $\zeta$ on the circle define ${\varphi }_{k,n}(\zeta ):={\sup }_{\eta \in \mathrm{\partial }D(q,t)}(v_{j_k}(\eta )-n\mathrm{\mid }\zeta -\eta \mathrm{\mid })\mathrm{.}$ Compactness and the common upper bound make each ${\varphi }_{k,n}$ finite and continuous, and upper semicontinuity gives ${\varphi }_{k,n}\downarrow v_{j_k}$ pointwise. Harmonic comparison, their Poisson extensions, and monotone convergence in [L7] therefore give $v_{j_k}(q_k)\le \frac{1}{2\pi }{\int }_0^{2\pi }{P}_k(\theta )v_{j_k}(q+t{e}^{i\theta })d\theta \mathrm{.}$ The kernels $P_k$ tend uniformly to $1$. The functions $P_k(\theta )(A-v_{j_k}(q+t{e}^{i\theta }))$ are nonnegative and measurable, so Fatou's lemma [L7] and the pointwise limsup bound make the lower limit of their normalized integrals at least $A-C$. Since each $P_k$ has normalized integral $1$, the preceding inequality gives ${\mathrm{lim\hspace{0.17em}sup}}_k{v}_{j_k}(q_k)\le C$, a contradiction.

Fix $0<\sigma <1$. Apply step 8.1 to $K={\bar{\mathrm{\Delta }}}_{\sigma }$ and $\delta =\log (r_2\mathrm{/}{r}_1)>0$. For all sufficiently large $j$, $v_j(w)\le -\log r_1(\mathrm{\mid }w\mathrm{\mid }\le \sigma ),$ or equivalently $\mathrm{\mid }{c}_{\alpha (j)}(w)\mathrm{\mid }{r}_1^{\mathrm{\mid }\alpha (j)\mathrm{\mid }}\le 1$.

Fix such a $\sigma$. If $S$ is finite, then ${\sum }_{\alpha }{c}_{\alpha }(w){\zeta }^{\alpha }$ is a finite sum in $\zeta$. Otherwise step 9.1 dominates its tail on ${\bar{\mathrm{\Delta }}}_s^{m-1}\times {\bar{\mathrm{\Delta }}}_{\sigma }$ by the convergent product-geometric majorant ${\sum }_{\alpha }(s\mathrm{/}{r}_1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}$; the finitely many low-degree terms are harmless and the coefficients outside $S$ vanish. Thus the series converges locally uniformly. Every partial sum is holomorphic, so [L8] gives a jointly holomorphic limit $G(\zeta ,w)$ on ${\mathrm{\Delta }}_s^{m-1}\times {\mathrm{\Delta }}_{\sigma }$.

On the open set ${\mathrm{\Delta }}_{\min (\epsilon ,s)}^{m-1}\times {\mathrm{\Delta }}_{\sigma }$, the Taylor expansion of the jointly holomorphic function from step 4.1 is exactly the series defining $G$. Thus $G=f$ on that nonempty open set. For fixed $w\in {\mathrm{\Delta }}_{\sigma }$, both $\zeta \mapsto G(\zeta ,w)$ and $\zeta \mapsto f(\zeta ,w)$ are holomorphic on ${\mathrm{\Delta }}_s^{m-1}$ and agree on a nonempty open subset, so [L9] gives equality on all of ${\mathrm{\Delta }}_s^{m-1}$. Hence $f=G$ on ${\mathrm{\Delta }}_s^{m-1}\times {\mathrm{\Delta }}_{\sigma }$, and $f$ is jointly holomorphic there.

Because ${\max }_j\mathrm{\mid }-a_j^{\mathrm{\prime }}\mathrm{\mid }=d<s$ and $0<\sigma$, the translated coordinates of the original target, $(-a^{\mathrm{\prime }},0)$, lie in the product from step 11.1. Thus that step proves the required local holomorphicity at the original origin in dimension $m$. By the reductions in steps 1.1 and 1.3, every point of ${\mathrm{\Delta }}_R^m$ has a holomorphic neighborhood. Therefore $f$ is locally bounded on ${\mathrm{\Delta }}_R^m$, and in particular bounded on every smaller closed polydisc ${\bar{\mathrm{\Delta }}}_r^m$. This closes the induction.