Interior C^{1,\alpha} regularity for infinity-harmonic functions

1 · The essentials

The manuscript studies a class of functions called infinity-harmonic functions in spaces with three or more dimensions. It claims that, away from the edge of their domain, their slopes change continuously and obey a quantitative bound. More specifically, the change in slope between nearby points is controlled by a positive power of their distance. The same power works for all bounded functions in this class in a given dimension. On the ball of half the radius, both the size of the slope and its variation are controlled by how much the function varies across the original unit ball. The power is not given explicitly, and the manuscript makes no claim about behavior at the boundary. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a landscape whose slope you inspect well away from its outer edge. The claimed estimate resembles a rule saying that the landscape's total height range limits both how steep the interior can be and how quickly its slope can change; this picture is only an analogy, since the stated result concerns dimensions three and higher.

Why it matters

The claim gives more than continuity of the slope: it supplies a common quantitative control for bounded infinity-harmonic functions in each dimension covered. In particular, the manuscript states that infinity-harmonic functions are locally continuously differentiable on every open subset of $R^{d}$ for $d \geq 3$ .

2 · Going further

The precise claim is a uniform interior $C^{1, α_{d}}$ estimate for bounded infinity-harmonic functions in dimensions $d \geq 3$ . There is an exponent $α_{d} \in (0, 1/3]$ , depending only on dimension, such that the gradient's supremum norm and its $α_{d}$ -Hölder seminorm on $B_{1/2}$ are bounded by a dimension-dependent constant times the oscillation of the function on $B_{1}$ .

In notation, the claimed estimate has the form $∥\nabla u ∥_{L^{\infty} (B_{1/2})} + [\nabla u]_{C^{0, α_{d}} (B_{1/2})} \leq C_{d} osc_{B_{1}} u$ . Here the oscillation is the supremum minus the infimum of the function on the unit ball. The Hölder seminorm measures changes in the gradient relative to the distance between points raised to the power $α_{d}$ . Thus the estimate controls both the gradient's magnitude and its continuity inside a smaller ball.

The supplied abstract states the estimate and its local $C^{1}$ consequence, but does not describe the argument behind them. It also supplies no prior-results discussion or companion manuscripts, so neither an argument outline nor a relationship to companion results can be established from this source. The exponent is not explicit, and the abstract expressly makes no endpoint or boundary regularity claim.

Useful notions

Gradients as measures of slope
Balls and interior regions
Supremum, infimum, and oscillation
Continuity and Hölder continuity

What was known before / what is new

Before

The supplied text does not identify earlier regularity results or give a historical account of the problem. It therefore does not establish which parts were previously known.

New

The manuscript presents a uniform interior $C^{1, α_{d}}$ estimate for bounded infinity-harmonic functions in every dimension $d \geq 3$ . Its claimed control uses constants and a positive exponent depending only on dimension, with the function's oscillation supplying the size bound. It also states local $C^{1}$ regularity as a consequence.

Limits and assumptions

The quantitative claim assumes a bounded infinity-harmonic function and applies in dimensions $d \geq 3$ , controlling the half unit ball using oscillation on the unit ball. The exponent lies in $(0, 1/3]$ but is not explicit; this does not assert that the endpoint $1/3$ is attained. No endpoint or boundary regularity claim is made, and the source does not identify additional open problems. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Gradient: The collection of first derivatives describing a function's slope in each coordinate direction.
Interior regularity: Control of a function's smoothness away from the boundary of its domain.
Oscillation: The supremum of a function minus its infimum on the region being considered.
Hölder seminorm: A measure of differences between values divided by the distance between their points raised to a specified positive power.
$C^{1, α}$ regularity: Continuous differentiability with first derivatives whose changes are controlled by distance raised to the power $α$ .

Keywords : Infinity-harmonic functions · Interior regularity · Hölder estimates

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves uniform interior $C^{1, α_{d}}$ regularity for bounded infinity-harmonic functions in every dimension d ≥ 3, for a positive exponent depending only on dimension. The gradient's supremum norm and Hölder seminorm on the half unit ball are bounded by a dimension-dependent constant times the oscillation on the unit ball. The exponent is not explicit.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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Uniform Interior $C^{1, α}$ Estimates for Infinity-Harmonic Functions

04/10/2026

To be verified

We prove a uniform interior $C^{1, α_{d}}$ estimate for bounded infinity-harmonic functions in every dimension d ≥ 3. For some $α_{d} \in (0, 1/3]$ depending only on the dimension, the gradient's supremum norm and α_d-Hölder seminorm on B_1/2 are bounded by a dimension-dependent constant times the oscillation on B₁. The exponent is not explicit, and no endpoint or boundary regularity claim is made. In particular, infinity-harmonic functions are locally C¹ on every open subset of ℝ^d, for d ≥ 3.

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