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

TL;DR

This paper introduces and analyzes the real, complex, and analytic Neumann–Poincaré configuration constants for convex planar domains and proves c_R(Ω) = c_C(Ω) =: c(Ω) while showing a(Ω) < 1 for all such Ω. It develops a Three-measures theorem as the central tool, derives a sharp bound a(W) governing polynomial functional calculi, and applies these results to obtain refined spectral-constant bounds for operators in terms of their numerical ranges, including explicit ellipse-based constants. The work also provides explicit calculations for ellipses, integral curvature-based estimates, and near-extremal quadrilateral examples, linking geometric boundary properties to operator-theoretic constants. These results yield improved universal bounds in functional calculus and deepen the connection between boundary potential theory and operator theory, with practical implications for estimating ||p(T)|| in terms of the numerical range or a domain containing it.

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.

Figures (7)

• Figure 1: Example domain $\Omega$ with corner of angle $\theta_{\zeta'}$ at $\zeta'$, and a circle of radius $R_{\zeta, \sigma}$ with center $m$, tangent to $\partial \Omega$ at $\sigma$ and passing through $\zeta$.
• Figure 2: A triangular image of a complex-valued function $g$ contained in a disk of radius $1$, with three points on the boundary of a disk.
• Figure 3: The initial disk $\mathbb{D}_K$ is the dashed circle, and we assume that $\partial \mathbb{D}_K \cap K$ is contained in the black thick arc. Then $K$ will be contained in the grey disk which is obtained from $\mathbb{D}_K$ by first translating $\mathbb{D}_K$ in the direction of the positive real axis, and then slightly shrinking the translated disk. This contradicts the minimality of $\mathbb{D}_K$.
• Figure 4: The thick arc $J$ between $a$ and $b$ is the smallest containing the compact set $K$. It follows that the shorter arc between the antipodal points $\tilde{a}$ an $\tilde{b}$ must contain points of $K$.
• Figure 5: The unit disk in dark grey with the strip $S_\delta$ removed. The dotted circle containing the dark grey area has a radius slightly smaller than $1$.

Theorems & Definitions (53)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 4
• Corollary 5
• Theorem 6
• Theorem 7
• Proposition 8
• Proposition 9
• Lemma 10