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Hölder regularity and nonvanishing Jacobian of the normalized Beltrami solution

Statement

Assume the Axiom of Choice. It implies Countable Choice (The Axiom of Choice, AC implies DC implies countable choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $m \geq 0$ be an integer, let $0 < α < 1$ , and let $0 \leq κ < 1$ . Let $μ$ be a Beltrami coefficient on the Riemann sphere with $∥ μ ∥_{\infty} \leq κ$ (Measurable Beltrami coefficients and measurable conformal structures), and let $f$ be the solution normalized by $f (0) = 0$ , $f (1) = 1$ , and $f (\infty) = \infty$ (The measurable Riemann mapping theorem on the sphere, The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity, The Riemann sphere is the published one-point compactification of the complex plane). Let $U \subseteq \hat{C}$ be open and suppose $μ$ is of class $C^{m, α}$ chartwise on $U$ (Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains, The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity). Then:

(i) Regularity. The coordinate expression of $f$ is locally of class $C^{m + 1, α}$ : for every source and target holomorphic chart pair, the expression $ϕ_{t} \circ f \circ ϕ_{s}^{- 1}$ is $C_{l o c}^{m + 1, α}$ wherever defined over $U$ .

(ii) Positive Jacobian and local diffeomorphism. At every $x \in U$ , the real Jacobian determinant of the coordinate expression in any such chart pair is positive. Thus its differential is invertible, and $f$ is a local $C^{m + 1, α}$ diffeomorphism at every point of $U$ .

(iii) Global case. If $μ$ is of class $C^{m, α}$ chartwise on all of $\hat{C}$ , then $f$ is a $C^{m + 1, α}$ diffeomorphism of the sphere chartwise, and so is $f^{- 1}$ .

No Sobolev bootstrap of unspecified order is asserted. For arbitrary weak solutions without the global homeomorphism hypothesis, the injectivity and positive-Jacobian conclusions do not follow from Weak solutions factor holomorphically in Hölder coordinates.

Facts & Assumptions

Given: AC; $m \geq 0$ an integer; $0 < α < 1$ ; $0 \leq κ < 1$ ; a sphere Beltrami coefficient $μ$ with $∥ μ ∥_{\infty} \leq κ$ ; its normalized global solution $f$ ; and an open set $U$ on which $μ$ has chartwise class $C^{m, α}$ .

[F1]

AC implies Countable Choice, which is required by the measurable coefficient, local coordinate, and weak factorization interfaces (The Axiom of Choice, AC implies DC implies countable choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F2]

The sphere coefficient and weak equation have chartwise pullback laws; in any source and target holomorphic charts, the coordinate expression of a weak solution satisfies the plane Beltrami equation with the source-chart coefficient (Measurable Beltrami coefficients and measurable conformal structures, Weak solutions of the Beltrami equation, The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity, The Riemann sphere is the published one-point compactification of the complex plane, A complex domain is a nonempty connected open subset of $C$ ).

[F3]

Under AC the global theorem supplies the normalized sphere homeomorphism and its weak Beltrami equation (The measurable Riemann mapping theorem on the sphere). The present proof consumes its existence, normalization, homeomorphism and weak-equation conclusions, not its separate coefficient-ratio conclusion.

[F4]

A continuous chart representative with essential norm at most $κ$ satisfies $∣ ν (z) ∣ \leq κ$ at every point: any strict violation would persist on an open disk of positive area. At each point, the local coordinate lemma then supplies a nondegenerate $C^{m + 1, α}$ Beltrami coordinate $Φ$ whose inverse has the same regularity (Measurable Beltrami coefficients and measurable conformal structures, Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains, Euclidean balls have positive finite Lebesgue measure, Nondegenerate local Hölder coordinates for a Hölder coefficient).

[F5]

Every $W_{l o c}^{1, 2}$ weak solution factors almost everywhere as $H \circ Φ$ for a holomorphic $H$ in these coordinates and has a local $C^{m + 1, α}$ representative; this factorization does not itself assert injectivity or a nonzero Jacobian (Weak solutions factor holomorphically in Hölder coordinates).

[F6]

If two continuous functions on a planar open set agree almost everywhere, then they agree everywhere: a nonzero difference at a point persists on a small open disk, which has positive area (Euclidean balls have positive finite Lebesgue measure).

[F7]

An injective holomorphic function on a complex domain has nowhere-zero derivative and a holomorphic inverse onto its open image (An injective holomorphic map has no critical point and is biholomorphic onto its image, Biholomorphic maps between complex domains).

[F8]

For a $C^{1}$ plane map $G$ , expanding the Wirtinger formulas gives $\det D G = ∣ G_{z} ∣^{2} - ∣ G_{\overset{ˉ}{z}} ∣^{2}$ ; for holomorphic $H$ , this is $∣ H^{'} ∣^{2}$ . The real chain rule and determinant multiplicativity give $\det D (H \circ Φ) = (\det D H \circ Φ) \det D Φ$ (The Wirtinger derivatives $\partial_{z} f$ and $\partial_{\overset{ˉ}{z}} f$ , and antiholomorphic functions, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive, The Jacobian determinant of a holomorphic map is $∣ f^{'} ∣^{2}$ and is positive exactly where $f^{'} \neq 0$ , The Jacobian determinant of a square-dimensional $C^{1}$ map is the determinant of its Jacobian matrix, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ ).

[F9]

Composing a local $C^{m + 1, α}$ diffeomorphism with a holomorphic local diffeomorphism preserves the $C^{m + 1, α}$ class: repeated chain and product rules give finite sums, and the mean-value theorem makes the smooth factors locally Lipschitz so the top-order Hölder bound is preserved (Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains, $C^{k}$ maps and multi-index derivative notation in Euclidean space, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

Proof

technique · factor the normalized homeomorphic solution through a nondegenerate local coordinate and use injectivity of the holomorphic factor

1.1F2F3given

Fix $x_{0} \in U$ . By [F3], $f$ is a homeomorphism and a weak solution. Choose a source holomorphic chart $ϕ_{s}$ around $x_{0}$ and a target holomorphic chart $ϕ_{t}$ around $f (x_{0})$ . Continuity of $f$ lets us shrink the source neighborhood so its image lies in the target chart. In these coordinates write $F : = ϕ_{t} \circ f \circ ϕ_{s}^{- 1}$ and let $ν$ be the source-chart expression of $μ$ . By [F2], $F \in W_{l o c}^{1, 2}$ solves $F_{\overset{ˉ}{z}} = ν F_{z}$ ; it is continuous and injective, and $ν$ is $C^{m, α}$ near $z_{0} : = ϕ_{s} (x_{0})$ .

2.1F1F4F5F6step 1.1

By [F4], choose a nondegenerate $C^{m + 1, α}$ coordinate $Φ$ on a neighborhood of $z_{0}$ . Shrink to a connected disk $D$ inside that neighborhood and the source chart domain. Applying [F5] to $F ∣_{D}$ gives a holomorphic $H$ on the complex domain $Φ (D)$ such that $F = H \circ Φ$ almost everywhere on $D$ , and $H \circ Φ$ is locally $C^{m + 1, α}$ . Both $F$ and $H \circ Φ$ are continuous. By [F6], their almost-everywhere equality is pointwise equality on $D$ . This proves the local regularity in (i) near $x_{0}$ .

3.1F4F7F8F9step 2.1

The pointwise identity from step 2.1, injectivity of $F$ , and injectivity of $Φ$ imply that $H$ is injective on $Φ (D)$ . By [F7], $H^{'}$ is nowhere zero and $H^{- 1}$ is holomorphic on $H (Φ (D))$ . The real chain rule and [F8] give $J_{F} (z) = \det D (H \circ Φ) (z) = ∣ H^{'} (Φ (z)) ∣^{2} J_{Φ} (z) > 0$ for every $z \in D$ , since $J_{Φ} > 0$ by [F4]. Hence $D F (z)$ is invertible. The local inverse is $Φ^{- 1} \circ H^{- 1}$ ; [F4] and [F9] show it is also $C^{m + 1, α}$ locally. Therefore $F$ is a local $C^{m + 1, α}$ diffeomorphism, proving (ii) near $x_{0}$ .

4.1F2F3step 2.1step 3.1given∎

The point $x_{0}$ and its source and target charts were arbitrary, so steps 2.1 and 3.1 prove (i) and (ii) throughout $U$ , with positive Jacobian in every holomorphic chart pair. If $U = \hat{C}$ , the same local statement holds at every point; the local inverses agree with the global inverse because $f$ is a homeomorphism. Thus $f$ and $f^{- 1}$ are chartwise $C^{m + 1, α}$ , proving (iii).

Source notes

Astala, Clop, Faraco, Jääskeläinen and Koski, Nonlinear Beltrami operators, Schauder estimates and bounds for the Jacobian, was read through the full relevant passages: the Introduction's linear-case regularity statement and Theorem 1.1, Lemma 3.1, and the complete proof of Theorem 1.1. The paper's Theorem 1.1 proves positivity of the Jacobian for its broader nonlinear class under its stated Hölder/Lipschitz condition; its exact- $α$ linear-case comment points to further references. This item does not substitute that source for the higher-order argument: it derives the exact $C^{m + 1, α}$ exponent from the local coordinate and factorization suppliers. Lyubich §14.4 was read in full and treats the real-analytic local case by characteristics; it is context only for the Hölder theorem.

Supplier reconciliation

The stable global theorem supplies the normalized homeomorphic weak solution consumed in step 1.1, and the earlier local coordinate and factorization lemmas supply the regularity and positive-Jacobian conclusions; this proof does not consume the global theorem's separate coefficient-ratio conclusion. The local arguments above retain their own stated hypotheses; this reconciliation is separate from root mathematical decisions and full-run certification.

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Sources

Kari Astala, Albert Clop, Daniel Faraco, Jarmo Jääskeläinen and Aleksis Koski, Nonlinear Beltrami operators, Schauder estimates and bounds for the Jacobian, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), 1543–1559 (standard reference, not scraped)
Mikhail Lyubich, Conformal Geometry and Dynamics of Quadratic Polynomials, vol. I (book draft, Stony Brook) (standard reference, not scraped)