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

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Harmonic Cauchy estimates in supremum norm

Statement

Assume Countable Choice and $n \geq 2$ . Let $u$ be real or complex harmonic on an open $Ω \subseteq R^{n}$ with $B_{r} (x) ⋐ Ω$ , $r > 0$ , and let $α$ be a multi-index. Then $∣ D^{α} u (x) ∣ \leq C_{n, α}^{'} r^{- ∣ α ∣} \sup_{B_{r} (x)} ∣ u ∣,$ with $C_{n, α}^{'}$ depending only on $n$ and $α$ .

Facts & Assumptions

Given: Countable Choice, an integer $n \geq 2$ , an open set $Ω$ , a harmonic $u$ on $Ω$ , $x \in Ω$ , $r > 0$ with $\overline{B_{r} (x)} \subset Ω$ , and a multi-index $α$ .

[F1]

Under these hypotheses, $∣ D^{α} u (x) ∣ \leq C_{n, α} r^{- n - ∣ α ∣} \int_{B_{r} (x)} ∣ u (y) ∣ d y$ with $C_{n, α}$ independent of $u, x, r, Ω$ (Interior derivative estimates for harmonic functions).

[F2]

For $n \geq 1$ and $r > 0$ , $∣ B_{r} ∣ = ω_{n - 1} r^{n} / n$ , finite and positive (Sphere and ball measures scale in Rn).

[F3]

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

Proof

technique · direct

1.1givenF3

Work under [F3] and set $M : = \sup_{B_{r} (x)} ∣ u ∣ \in [0, + \infty]$ . Since $∣ u ∣$ is continuous and $B_{r} (x)$ is bounded, the integral $\int_{B_{r} (x)} ∣ u ∣$ is defined in $[0, + \infty]$ .

2.1step 1.1F2algebra

If $M < + \infty$ , then $∣ u (y) ∣ \leq M$ for every $y \in B_{r} (x)$ , so by monotonicity of the integral $\int_{B_{r} (x)} ∣ u ∣ \leq M ∣ B_{r} (x) ∣ = M ω_{n - 1} r^{n} / n$ by [F2].

3.1step 2.1F1F2algebra

Substituting step 2.1 into [F1] gives $∣ D^{α} u (x) ∣ \leq C_{n, α} r^{- n - ∣ α ∣} \cdot M ω_{n - 1} r^{n} / n = (C_{n, α} ω_{n - 1} / n) r^{- ∣ α ∣} M$ , so the stated estimate holds with $C_{n, α}^{'} : = C_{n, α} ω_{n - 1} / n$ , a constant depending only on $n$ and $α$ .

4.1step 1.1step 3.1cases∎

If $M = + \infty$ the right-hand side of the stated inequality is $+ \infty$ while $∣ D^{α} u (x) ∣$ is a finite real number, so the inequality holds trivially; for the local applications of this estimate one always takes a compactly contained ball on which $∣ u ∣$ , being continuous, is bounded, so the case $M = + \infty$ never carries mathematical content.

Depends on

Used by

Dependency tree · two levels

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Sources

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