Harmonic Quasiconformal Mappings

• Harmonic quasiconformal mappings are sense-preserving homeomorphisms expressed as h(z)+overline{g(z)} that combine harmonicity with controlled geometric distortion.
• They integrate elliptic regularity with quasiconformal bounds, yielding key results in boundary smoothness, extension criteria, and functional inequalities.
• Applications include sharp coefficient estimates, Hardy space inclusions, and precise growth theorems relevant to planar and higher-dimensional mapping theories.

A harmonic quasiconformal mapping is a sense-preserving homeomorphism $f: D \to f(D) \subset \widehat{\mathbb{C}}$, expressible as $f(z) = h(z) + \overline{g(z)}$ where $h$ and $g$ are analytic and the complex dilatation $\omega(z) = g'(z)/h'(z)$ satisfies $|\omega(z)| < 1$. The interplay between harmonicity and quasiconformality in the planar and higher-dimensional settings has led to a broad suite of regularity, extension, and function space inclusion results, grounded in analytic subordination, boundary regularity, and sharp functional inequalities.

1. Structural and Analytical Foundations

Harmonic quasiconformal mappings generalize conformal and holomorphic mappings by combining elliptic regularity (the harmonic property $\Delta f = 0$) with the geometric distortion control given by quasiconformal theory ($|\omega(z)| \le k < 1$, $K = (1+k)/(1-k)$). The Jacobian $J_f(z) = |h'(z)|^2 - |g'(z)|^2 > 0$ guarantees sense-preserving injectivity. The harmonic mapping class $S_{\mathcal{H}}(K)$ is typically normalized via $f(0)=0$, $f_z(0)=1$, $f_{\bar z}(0)=0$ (Wang et al., 2024).

Boundary regularity of the domain and codomain critically influences interior regularity: Dini-smoothness and $C^{1,\mu}$ boundaries yield optimal Lipschitz properties (Kalaj, 2014, Kalaj, 2011), while $C^1$ boundaries support global Hölder continuity but not in general Lipschitz behavior (Kalaj, 2020).

2. Pre-Schwarzian and Schwarzian Derivatives

Analysis of harmonic quasiconformal mappings revolves around two key functionals:

• Pre-Schwarzian: $$P_f(z) = (\log J_f(z))_z = \frac{h''(z)}{h'(z)} - \frac{\omega'(z)\,\overline{\omega(z)}}{1-|\omega(z)|^2}$$;
• Schwarzian: $S_f(z) = (\log J_f)_{zz} - \frac12[(\log J_f)_z]^2$.

Norms $\|P_f\| = \sup_{z\in\mathbb{D}}|P_f(z)|(1-|z|^2)$ and $\|S_f\| = \sup_{z\in\mathbb{D}}|S_f(z)|(1-|z|^2)^2$ provide affine- and linear-invariant control for univalence and extension criteria (Wang et al., 2024, Wang et al., 2021, Efraimidis et al., 2021, Hernández et al., 2014, Efraimidis, 2020).

3. Sharp Growth, Hardy Spaces, and Extremal Functions

Chuaqui–Hernández–Martín (Wang et al., 2024) characterize the order (sharp growth exponent) of families of harmonic $K$-quasiconformal mappings with Schwarzian norm bound $\lambda$ as

$$\alpha(\mathcal{S}_{\mathcal{H}}(K, \lambda)) = \sqrt{1 + \frac{\lambda}{2} + \frac{1}{2}\left(\frac{K-1}{K+1}\right)^2} + \frac{K-1}{2(K+1)}$$

This is realized by an explicit harmonic Koebe function, constructed via shearing the classical analytic Koebe map with $\omega(z)=k\,z$, $k = (K-1)/(K+1)$. Taylor coefficients $A(n,k), B(n,k)$ yield sharp bounds and conjectures for the analytic and co-analytic series expansions.

For Hardy space membership, combining the Astala–Koskela $p<1/(2K)$ bound and the order-to-Hardy principle yields precise $h^p$ inclusion ranges for these mapping families, which are sharp with respect to both the class order and boundary regularity. Major subclass results (convex, starlike, close-to-convex) and explicit $p$-ranges are established using extremal mean inequalities for $h', g'$ and the Baernstein star-function framework (Das et al., 8 May 2025).

4. Extension Criteria and Schwarzian-Based Results

Multiple univalence and quasiconformal extension theorems are predicated on the smallness of the Schwarzian norm:

• Disk case: For $\|S_f\|$ below an absolute threshold $\delta_0$, $f$ is univalent and extends quasiconformally to $\widehat{\mathbb{C}}$; the extension's maximal dilatation is $K = (1+t)/(1-t)$ where $\|S_f\| \le t\,\delta_0$ and $\sup|\omega|<1$ (Hernández et al., 2014, Efraimidis et al., 2021).
• Planar domain case: For any uniform domain, there exists $c>0$ (depending only on the domain's hyperbolic geometry and maximal dilatation bound) such that $\|S_f\|_D \le c$ implies injectivity. For multiply connected domains with quasicircle boundaries, quasiconformal extension is obtained under the same small Schwarzian criterion (Efraimidis, 2021, Efraimidis, 2020).
• Explicit formulas: Two Ahlfors–Weill-type extension formulas for harmonic mappings with small Schwarzian norm generalize the classic holomorphic case to the harmonic setting, with explicit dependence on $P_h$, $P_f$, and $\omega$ (Efraimidis et al., 2021).

Quasiconformal extension mechanisms in higher dimensions, such as the harmonic quasi-isometric extension to hyperbolic space in the resolution of the Schoen conjecture, align via analogous tension field estimates and maximal distortion control (Markovic, 2013).

5. Boundary Regularity, Lipschitz and Hölder Properties

Boundary smoothness directly impacts interior regularity:

• Dini-smooth boundaries: Any harmonic quasiconformal mapping between Dini-smooth Jordan domains is Lipschitz—a sharp result, as the Dini condition is optimal for conformal and minimal surface parametrizations (Kalaj, 2014).
• $C^{1,\mu}$ boundaries: Harmonic quasiconformal mappings between $C^{1,\mu}$ surfaces are globally Lipschitz, with explicit constants depending only on $K$, $\mu$, and geometric boundary data (Kalaj, 2011). Co-Lipschitz properties (bi-Lipschitz) are proved in the planar case for Lyapunov domains ($C^{1,\mu}$ Jordan domains), resolving long-standing regularity questions (Mateljević, 2018, Kalaj, 2010).
• $C^1$ boundaries: Only global Hölder continuity of every exponent $\alpha<1$ holds, with optimality evidenced by examples where Lipschitz continuity fails (Kalaj, 2020).
• Hölder boundary data: For arbitrary domains with uniformly perfect "nonthin" boundary portions, boundary $\alpha$-Hölder regularity implies interior $\alpha$-Hölder continuity, extending results under weaker regularity than previously considered (Arsenović et al., 2010).

6. Geometric, Coefficient, and Function-Theoretic Results

Extremal function theory and subclass analysis is advanced via:

• Coefficient bounds: Sharp inequalities for analytic and co-analytic Taylor coefficients in subclasses defined by close-to-convexity, starlikeness, or prescribed subordination relations (e.g., $1+z h''/h' \prec (1-(2\alpha-1)z)/(1-z)$) (Wang et al., 2020).
• Integral representations: All such mappings admit nontrivial integral representations via Schwarz function techniques, encoding the analytic and co-analytic structure (Wang et al., 2020).
• Growth and area theorems: Optimal two-sided growth bounds and area formulas explicitly incorporate the interaction of the analytic and co-analytic components and the dilatation parameter (Wang et al., 2020).
• Partial sums and close-to-convexity: Exact radii where the partial sums of harmonic mappings retain close-to-convexity are calculated via extremal functions (Wang et al., 2020).

7. Topological and Functional-Analytic Aspects

The space $HQ(\mathbb{D})$ of harmonic quasiconformal automorphisms is shown to have nuanced topological properties in the uniform topology:

• Separability, path-connectivity, absence of isolated points.
• Noncompactness and incompleteness: Even in the disk case, sequences of harmonic QC maps can converge uniformly to harmonic limits failing quasiconformality (e.g., Cantor-modified boundary data) (Biersack, 2023).

These results frame the broader moduli-theoretic structure of harmonic QC mappings and their automorphism group, highlight the subtleties of function space closure, and suggest further investigations into relationships with Teichmüller theory.

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For further developments, see (Wang et al., 2024, Hernández et al., 2014, Efraimidis et al., 2021, Das et al., 8 May 2025, Efraimidis, 2021, Kalaj, 2020, Efraimidis, 2020, Kalaj, 2011, Arsenović et al., 2010, Kalaj, 2010, Kalaj, 2014).