Analysis of Log-Weighted Quadrature Domains
This work provides a theoretical foundation for quadrature domains with logarithmic singularities, relevant to complex analysis and potential theory, but is incremental as it extends existing frameworks to a specific singular case.
This paper introduces log-weighted quadrature domains (LQDs) with a singular weight, revealing non-unique quadrature data and establishing a generalized Schwarz function characterization. It shows that simply connected LQDs correspond to Riemann maps whose outer factor is the exponential of a rational function, extending classical formulae via the Faber transform.
This paper studies plane domains satisfying a quadrature identity with respect to the singular weight $ρ_0(w)=|w|^{-2}$. These are referred to as log-weighted quadrature domains (LQDs). The logarithmic singularity at $w=0$ leads to phenomena not present in the classical theory: in particular, when the domain contains the origin, the associated quadrature data are no longer unique, but are determined only up to a point charge at $0$. A generalized Schwarz function characterization of LQDs is established together with a natural formulation of the inverse problem in the singular setting. In the simply connected case, it is shown that a domain is an LQD if and only if the outer factor of its Riemann map extends to the exponential of a rational function. This characterization yields explicit formulae relating the quadrature function and the Riemann map via the Faber transform, thereby extending earlier formulae from the non-singular theory. Several basic classes of LQDs are also covered, and explicit examples are computed.