Double-layer potentials, configuration constants and applications to numerical ranges

Abstract

Given a compact convex planar domain $Ω$ with non-empty interior, the classical Neumann's configuration constant $c_{\mathbb{R}}(Ω)$ is the norm of the Neumann-Poincaré operator $K_Ω$ acting on the space of continuous real-valued functions on the boundary $\partial Ω$, modulo constants. We investigate the related operator norm $c_{\mathbb{C}}(Ω)$ of $K_Ω$ on the corresponding space of complex-valued functions, and the norm $a(Ω)$ on the subspace of analytic functions. This change requires introduction of techniques much different from the ones used in the classical setting. We prove the equality $c_{\mathbb{R}}(Ω) = c_{\mathbb{C}}(Ω)$, the analytic Neumann-type inequality $a(Ω) < 1$, and provide various estimates for these quantities expressed in terms of the geometry of $Ω$. We apply our results to estimates for the holomorphic functional calculus of operators on Hilbert space of the type $\|p(T)\| \leq K \sup_{z \in Ω} |p(z)|$, where $p$ is a polynomial and $Ω$ is a domain containing the numerical range of the operator $T$. Among other results, we show that the well-known Crouzeix-Palencia bound $K \leq 1 + \sqrt{2}$ can be improved to $K \leq 1 + \sqrt{1 + a(Ω)}$. In the case that $Ω$ is an ellipse, this leads to an estimate of $K$ in terms of the eccentricity of the ellipse.