Harmonic Quasiregular Mappings

• Harmonic quasiregular mappings are generalizations of harmonic and analytic functions that integrate harmonic structure with controlled distortion properties.
• They offer robust frameworks for solving boundary value problems and establishing sharp norm inequalities in diverse function spaces.
• Their applications span higher-dimensional elliptic PDEs, geometric topology, and advanced analysis in function spaces with optimal growth and coefficient estimates.

A harmonic quasiregular mapping is a generalization of analytic and harmonic functions that combines the analytic structure of harmonicity with the distortion properties of quasiregular (or more generally, quasiconformal) mappings. These mappings are central in modern geometric function theory, analysis on manifolds, solution theories for elliptic partial differential equations, and the study of various extremal problems. Harmonic quasiregular mappings exhibit strong rigidity and integrability properties, and have been a focus of profound classical theory and novel advances across both function theory and nonlinear elliptic PDEs. The theory extends naturally to the multiply connected and higher-dimensional settings, with a rich landscape of function space properties, sharp norm inequalities, and boundary value phenomena.

1. Definitions and Fundamental Structure

A complex-valued mapping $f=u+iv$ defined on a domain $\Omega\subset\mathbb{C}$ (or more generally on $\mathbb{R}^n$) is harmonic if $\Delta f=0$ in $\Omega$. Every such $f$ admits a canonical decomposition $f=h+\overline{g}$, with $h$ and $g$ analytic (or more generally, holomorphic in several variables). The notion of quasiregularity is encoded in the pointwise inequality for the complex derivatives $f_z, f_{\bar z}$: $f$ is $K$-quasiregular if

$$|f_z| + |f_{\bar z}| \le K(|f_z| - |f_{\bar z}|) \quad \text{a.e. in }\Omega$$

for some $K\ge 1$. Equivalently, the (analytic) dilatation $\omega = g'/h'$ satisfies $|\omega(z)| \le k < 1$, $k = (K-1)/(K+1)$. The mapping is sense-preserving if the Jacobian $J_f = |f_z|^2 - |f_{\bar z}|^2 > 0$ almost everywhere.

This analytic structure places these mappings in the Sobolev class $W^{1,2}_{\text{loc}}(\Omega)$ and connects naturally to divergence-type elliptic equations via the A-harmonic operator

$\operatorname{div}(A(z)\nabla u(z)) = 0,$

where $u = \operatorname{Re} f$ and $A(z)$ is a symmetric, elliptic matrix determined by $|\mu(z)|$ with $\det A(z) = 1$ and ellipticity ratio $K(z) = (1 + |\mu|)/(1 - |\mu|)$ (Gutlyanskii et al., 2015).

In higher dimensions ($\mathbb{R}^n$, $n\ge2$), a vector-valued mapping $f=(f_1,\ldots,f_n)$ is harmonic if each $f_j$ is harmonic, and quasiregular if $|Df(x)|^n \le K J_f(x)$, where $|Df|$ denotes the operator norm of the derivative matrix and $J_f$ its determinant (Chen et al., 2023, Pankka et al., 2017).

2. Analytic and Boundary Value Theory

The analysis of boundary value problems for harmonic quasiregular mappings closely parallels that of analytic and classical harmonic functions, with substantial technical innovation due to the nonlinearity of the Beltrami equation. The foundational Dirichlet, Neumann, Poincaré, and Hilbert (Riemann–Hilbert) boundary problems can all be formulated and solved in the $K$-quasiregular context (Gutlyanskii et al., 2015). Solutions exist for arbitrary (logarithmic capacity-measurable) boundary data prescribed along classes of tangential ("BS-class") Jordan arcs, rather than only through nontangential limits.

The solution spaces for these boundary value problems are infinite-dimensional, even under arbitrary measurable (with respect to logarithmic capacity) data. Regularity theory guarantees interior Hölder continuity, precise distortion bounds, and the existence of limits along prescribed approach regions at almost every boundary point.

The standard approach employs factorization via conformal and quasiconformal reductions: solving the Beltrami equation with prescribed $\mu$ to obtain a rectification $h$, then solving the analytic or A-harmonic problem in the rectified domain, and finally transporting the solution back to $\Omega$ (Gutlyanskii et al., 2015).

3. Function Space Properties and Norm Inequalities

Harmonic quasiregular mappings exhibit a suite of function space and norm-equivalence properties that mirror the analytic case, but with modified constants and sharper regularity statements.

Hardy, Bergman, and Sobolev/BMO/Besov Scales:

• If $f = u + iv$ is $K$-quasiregular in the unit disk, the inclusion $u\in h^p \implies v\in h^p$ for $p>1$ holds, with sharp constants (Riesz-type theorems) (Chen et al., 2023, Kalaj, 2023, Chen et al., 22 Sep 2025, Sun et al., 3 Jan 2026).
• For $f$ in the harmonic Bergman space $a^p$, $u\in a^p$ implies $v\in a^p$ with $\|v\|_{a^p} \le C_{p,K} \|u\|_{a^p}$, extending the classical Hardy–Littlewood theorem to the harmonic quasiregular setting (Das et al., 25 Jul 2025).
• Möbius-invariant $Q_h$, $F_h$, and non-derivative $M_h$ spaces have been characterized, including sharp $K$-dependent stability: if $u\in Q_h(1,p,\alpha)$, then $v\in Q_h(1,p,\alpha)$ with $\|v\| \le K \|u\|$ (Sun et al., 3 Jan 2026).

Sharpness and Extremizers:

• The constants in these norm inequalities are optimal; extremal mappings of affine type $f(z) = h(z) + \overline{w h(z)}$, $|w| = k$, achieve equality (Sun et al., 3 Jan 2026, Das et al., 25 Jul 2025, Chen et al., 22 Sep 2025).
• For normalized univalent harmonic mappings, precise growth rates of Taylor coefficients and radial means have been established, confirming conjectures on coefficient bounds (Chen, 22 Sep 2025).

Boundary Smoothness Equivalence:

• If $u|_{\partial\mathbb{D}}$ is Hölder-$\alpha$, so is $v|_{\partial\mathbb{D}}$ with a constant depending on $K$ and $\alpha$; the same exponent is preserved (Das et al., 5 Jun 2025).

4. Conjugate-Function and Endpoint Phenomena

Harmonic quasiregular mappings inherit a detailed conjugate-function theory, extending classical results (Riesz, Kolmogorov, Zygmund) to the nonlinear setting:

• For $f = u + iv$ $K$-quasiregular in $\mathbb{D}$ with $u \in h^p$, the mapping $v \in h^p$ for $p > 1$ (Riesz theorem) and $v \in h^q$ for all $q < 1$ if $u \in h^1$ (Kolmogorov theorem), with explicit best constants (Kalaj, 2023, Chen et al., 22 Sep 2025).
• At $p=1$, Zygmund-type results are optimal: $u$ in the Zygmund class $\mathcal{Z}$ (i.e., $u \log^+ |u|$ integrable) is necessary and sufficient for $v\in h^1$, and this is quantitatively sharp even in the quasiregular context (Das et al., 3 Jan 2025, Kalaj, 3 Jan 2025).
• Logarithmic growth and O(log(1/(1-r))) rates are demonstrated for $0Das et al., 5 Jun 2025, Chen et al., 22 Sep 2025).

The subharmonicity regime for $|f|^q$ in higher dimensions has been completely characterized; the sharp range for subharmonicity depends on both the dimension $n$ and the distortion constant $K$ (Kalaj et al., 2011).

5. Growth, Distortion, and Coefficient Theorems

Sharp distortion, growth, and coefficient estimates are established for harmonic quasiregular and quasiconformal mappings:

• Coefficient bounds: For $(K,K')$-elliptic harmonic mappings, $|a_n| + |b_n| \le [K\Lambda + \sqrt{K'}]/n$ where $A f(z) \leq \Lambda$; this improves all previous results (Allu et al., 2022).
• Landau–Bloch type theorems: Explicit univalence radii and image discs are provided under normalization conditions, both for $K$-quasiregular and more general $(K,K')$-elliptic maps, with sharpness (Allu et al., 2022).
• Growth of integral means and of Taylor coefficients for normalized univalent mappings in both the $K$-quasiregular and $(K,K')$-quasiregular settings is now resolved, confirming the Das–Kaliraj conjecture in these subfamilies (Chen, 22 Sep 2025).

Table: Function Space Stability for Harmonic Quasiregular Mappings

Property	Analytic $f$	Harmonic $K$-QR $f$	Sharpness/Achievers
$u\in h^p \implies v\in h^p$	$1<p<\infty$	$1<p<\infty$ ($v(0)=0$)	Equality for affine models
$u\in a^p \implies v\in a^p$	$0<p<\infty$	$0<p<\infty$	Achieved by pure rotations/stretches
$u\in h^1$	$v\in h^q,\ \forall q<1$	$v\in h^q,\ \forall q<1$	Best possible, Zygmund class necessary
$u\in \mathcal{Z}$	$v\in h^1$	$v\in h^1$	No weaker gauge than $\log^+$ possible

6. Bohr Phenomena and Extremal Problems

Bohr-type inequalities and quasi-subordination phenomena have been extended to sense-preserving harmonic $K$-quasiregular mappings (Liu et al., 2020). Sharp radii (Bohr radii) for coefficient sums, both for harmonic mappings and their derivatives, are achieved under quasi-subordination or majorization conditions, with the optimality demonstrated by extremal rotationally symmetric mappings.

These results settle previously open conjectures concerning Bohr's phenomenon for harmonic and analytic mappings in the disk, allowing for optimal treatment of both subordination and majorization structures.

7. Extensions and Geometric Applications

The scope of harmonic quasiregular mapping theory encompasses:

• Higher dimensions: Riesz-type and Zygmund-type inequalities, subharmonicity ranges, and radial growth theorems for $\mathbb{R}^n$.
• Geometric function theory: Precise links to A-harmonic equations, generalized Robin, Hilbert, and Riemann–Hilbert-type problems in multiply connected and non-smooth domains (Gutlyanskii et al., 2015).
• Harmonic extension and rigidity phenomena: Every non-constant quasiregular self-map of the $n$-sphere admits a harmonic extension to the hyperbolic space $\mathbb{H}^{n+1}$, resolving a central conjecture and linking the theory to the geometry of high-dimensional negatively curved spaces (Pankka et al., 2017).
• Operator-theoretic characterizations: Norm estimates for Bergman and Hardy norms of the partial derivatives, sharp $L^p$ and $H^p$ bounds, and detailed spectral theory for related elliptic differential operators (Zhu, 2020).

These advances have implications in elliptic PDE theory, low-regularity geometric analysis, and the construction of quasiconformal and quasiregular structures in geometric topology and dynamics.

• (Gutlyanskii et al., 2015) Gutlyanskii, Ryazanov, Yefimushkin, "On Hilbert, Riemann, Neumann and Poincare problems for plane quasiregular mappings"
• (Das et al., 25 Jul 2025) Das & Rasila, "On harmonic quasiregular mappings in Bergman spaces"
• (Chen et al., 2023) Chen & Huang, "Riesz type theorems for $κ$-pluriharmonic mappings..."
• (Das et al., 3 Jan 2025) Das, Huang, Rasila, "Zygmund's theorem for harmonic quasiregular mappings"
• (Das et al., 5 Jun 2025) Das & Rasila, "Note on real and imaginary parts of harmonic quasiregular mappings"
• (Chen et al., 22 Sep 2025) Chen & Kalaj, "Conjugate type properties of harmonic $(K,K')$-quasiregular mappings"
• (Sun et al., 3 Jan 2026) Das, Kalaj, Rasila, "Characterizations of harmonic quasiregular mappings in function spaces"
• (Chen, 22 Sep 2025) Chen, "Growth type theorems of harmonic $(K,K')$-quasiregular mappings"
• (Allu et al., 2022) Allu & Kumar, "Landau-Bloch type theorem for elliptic and $K$-quasiregular harmonic mappings"
• (Pankka et al., 2017) Pankka & Souto, "Harmonic extensions of quasiregular maps"
• (Zhu, 2020) Zhu, "Norm estimates of the partial derivatives for harmonic mappings and harmonic quasiregular mappings"
• (Kalaj et al., 2011) Kalaj & Manojlović, "Subharmonicity of the modulus of quasiregular harmonic mappings"
• (Kalaj, 2023) Kalaj, "Riesz and Kolmogorov inequality for harmonic quasiregular mappings"
• (Kalaj, 3 Jan 2025) Kalaj, "Zygmund theorem for harmonic quasiregular mappings"
• (Liu et al., 2020) Das & Ponnusamy, "Bohr's phenomenon for the classes of Quasi-subordination and $K$-quasiregular harmonic mappings"

These references collectively provide a comprehensive, technically detailed, and functionally sharp account of the modern theory of harmonic quasiregular mappings.