Research Problems in Function Theory — Problem 3.30

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\Gamma\subset\mathbb C\cong\mathbb R^2$ be a smooth Jordan curve, meaning an embedded smooth closed curve bounding a bounded interior region. Let $n(z)$ be its inward unit normal and $ds_z$ arc-length measure. For distinct $z,z'\in\Gamma$, define the double-layer kernel \[K(z,z')=\frac{n(z)\cdot(z'-z)}{|z'-z|^2},\] which equals $\cos\phi/|z-z'|$ with the source's inward-normal convention. Use its continuous diagonal extension, which exists for a smooth curve. Define \[(T_\Gamma f)(z')=\int_\Gamma f(z)K(z,z')\,ds_z\qquad(f\in C(\Gamma)).\] Its rank is the dimension of its range in $C(\Gamma)$; a circle gives a constant kernel and rank one.

Statement recorded in the supplied source.

Is there any smooth Jordan curve $\Gamma$ that is not a circle for which $\operatorname{rank}T_\Gamma<\infty$? Equivalently, are circles the only smooth Jordan curves giving finite rank? [S2]

Short English statement

Are circles the only smooth simple closed curves whose double-layer boundary operator has finite rank?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303030 (AMR-022-3030). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory, arXiv:1809.07200, 2018. Problem 3.30, its complete Update 3.30, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.