Quadrature Domains and the Faber Transform

TL;DR

This work develops a unified, transform-based framework to study quadrature domains (QDs) and their weighted variants. Central to the approach is the Faber transform, which pairs the quadrature function $h$ with the Riemann map $\varphi$ in both bounded and unbounded, as well as weighted, settings; this enables explicit direct/inverse relations and constructive classifications, including a complete one-point QD classification. The paper extends classical QD theory to power-weighted (PQD) and log-weighted (LQD) domains, deriving analogous Schwarz-function formalisms, boundary regularity results, and a broad roster of explicit Riemann-map representations (e.g., for cardioids, monomial PQDs, and higher-order cases). Additionally, a potential-theoretic perspective via Frostman’s theory and Hele-Shaw flow reveals dualities between local droplets and QDs, enabling a deeper understanding of existence, uniqueness, and evolutionary behavior. The final section hints at a transcendentally dynamical regime for Schwarz reflections, linking geometric boundaries with anti-holomorphic dynamics and advanced special-function phenomena (e.g., Lambert W) in concrete examples.

Abstract

We present a framework for reconstructing any simply connected, bounded or unbounded, quadrature domain $Ω$ from its quadrature function $h$. Using the Faber transform, we derive formulae directly relating $h$ to the Riemann map for $Ω$. Through this approach, we obtain a complete classification of one point quadrature domains with complex charge. We proceed to develop a theory of weighted quadrature domains with respect to weights of the form $ρ_a(w)=|w|^{2(a-1)}$ when $a > 0$ ("power-weighted" quadrature domains) and the limiting case of when $a=0$ ("log-weighted" quadrature domains). Furthermore, we obtain Faber transform formulae for reconstructing weighted quadrature domains from their respective quadrature functions. Several examples are presented throughout to illustrate this approach both in the simply connected setting and in the presence of rotational symmetry.

Figures (21)

• Figure 1: $\Omega\in\text{QD}\left(w^{-1}+\alpha_1w^{-2}\right)$ (the complement of the shaded region) for $\alpha_1\in\{0,.2,.4,.45,.5\}$.
• Figure 2: $\Omega\in\text{QD}(\alpha w)$ (the complement of the shaded region) for $\alpha \in\{-.7,-.3,0,.3,.7\}$.
• Figure 3: Illustration of the potential-theoretic interpretation of quadrature domains for the cardioid, $\Omega\in\text{QD}\left(\frac{1}{w}+\frac{1}{2w^2}\right)$. $\overline{C^{\Omega}}$ (left), $\overline{h}$ (center), $\overline{C^{\Omega}}-\overline{h}$ (right).
• Figure 4: $\Omega_t\in\text{QD}\left(\frac{1}{w-2}\right)$ (complements of shaded regions) for increasing $t$ up to the critical time (left) and after the critical time (right).
• Figure 5: $\Omega_t\in\text{QD}\left(\frac{\alpha}{w-2}\right)$ (complements of shaded) for $\alpha\in\{-.9,-.7,-.4,-.2\}$ for increasing $t$ up to the critical time.

Theorems & Definitions (100)

• Definition 1.0.1: Bounded quadrature domain
• Definition 1.0.2: Unbounded quadrature domain
• Example 1.1: The ellipse
• Definition 1.0.3: Quadrature domain
• Lemma 1.1
• Theorem 1.3
• Theorem 1.4: AharonovShapiro
• Theorem 1.5
• Theorem 1.6
• Definition 1.6.1: Interior Faber transform