CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)audited 2026-10-02

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Entire harmonic functions of sublinear growth are constant

Statement

Assume Countable Choice and $n \geq 2$ . Use one-based basis labels $e_{i} : = e_{i - 1}^{c a n}$ for $1 \leq i \leq n$ . Let $u : R^{n} \to R$ or $C$ be harmonic. If $\lim_{R \to \infty} R^{- 1} \sup_{B_{R} (0)} ∣ u ∣ = 0,$ then $u$ is constant.

Facts & Assumptions

Given: Countable Choice, an integer $n \geq 2$ , a harmonic $u$ on all of $R^{n}$ with $\lim_{R \to \infty} R^{- 1} \sup_{B_{R} (0)} ∣ u ∣ = 0$ .

[F1]

Under the compact-ball hypotheses, $∣ D^{α} u (x) ∣ \leq C_{n, α}^{'} r^{- ∣ α ∣} \sup_{B_{r} (x)} ∣ u ∣$ (Harmonic Cauchy estimates in supremum norm).

[F2]

A continuous function on an interval whose derivative vanishes at every interior point is constant there (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant); the chain rule computes the derivative of a restriction to a line (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F3]

Countable Choice ${A C}_{ω}$ is the standing hypothesis (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

1.1givenF3algebra

Work under [F3] and fix $x \in R^{n}$ . For every $R > ∣ x ∣$ put $ρ_{R} : = (R - ∣ x ∣) / 2 > 0$ . If $y \in {\overline{B}}_{ρ_{R}} (x)$ , then $∣ y ∣ \leq ∣ y - x ∣ + ∣ x ∣ \leq ρ_{R} + ∣ x ∣ = (R + ∣ x ∣) / 2 < R$ , so ${\overline{B}}_{ρ_{R}} (x) \subset B_{R} (0)$ .

2.1step 1.1F1algebra

Apply [F1] with radius $ρ_{R}$ and coordinate multi-index $α = e_{i}$ for each $i = 1, \dots, n$ . With $K_{n} : = 2 \sqrt{n} \max_{i} C_{n, e_{i}}^{'}$ , step 1.1 gives $∣ D u (x) ∣ \leq \sqrt{n} \max_{i} ∣ \partial_{i} u (x) ∣ \leq \sqrt{n} \max_{i} C_{n, e_{i}}^{'} ρ_{R}^{- 1} \sup_{B_{ρ_{R}} (x)} ∣ u ∣ \leq K_{n} \frac{R}{R - ∣ x ∣} R^{- 1} \sup_{B_{R} (0)} ∣ u ∣$ for every $R > ∣ x ∣$ .

3.1step 2.1algebra

Letting $R \to \infty$ in step 2.1, the factor $R / (R - ∣ x ∣) \to 1$ and $R^{- 1} \sup_{B_{R} (0)} ∣ u ∣ \to 0$ by hypothesis, so $∣ D u (x) ∣ \leq \lim_{R \to \infty} K_{n} \frac{R}{R - ∣ x ∣} R^{- 1} \sup_{B_{R} (0)} ∣ u ∣ = 0$ ; hence $D u (x) = 0$ for every $x \in R^{n}$ .

4.1step 3.1F2∎

Therefore $u$ is constant: for fixed $x \in R^{n}$ and each coordinate $i$ , if $u$ is real-valued then $t \mapsto u (x + t e_{i})$ has zero derivative for every real $t$ by step 3.1, so it is constant on $R$ by [F2]; if $u$ is complex-valued, apply [F2] separately to the real and imaginary parts of this line restriction, whose derivatives also vanish by step 3.1. Thus each coordinate line restriction is constant, and changing the coordinates one at a time connects any two points of $R^{n}$ , so $u$ has the same value everywhere. The chain rule identifies each line derivative with the corresponding partial derivative. No bounded-Liouville theorem is invoked; the sublinear growth hypothesis is used exactly in step 3.1.

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Sources

Leon Simon, Lectures on PDE (2015 rough draft) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (2014) (standard reference, not scraped)