Research Problems in Function Theory — Problem 3.32

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.

Fix $n\ge2$ and an interior-cone half-angle $0<\alpha\le\pi/2$. A bounded Lipschitz domain is a bounded connected open set whose boundary is locally the graph of a Lipschitz function. Require that it admits interior cones of half-angle at least $\alpha$: there is a height $h>0$ such that for each $y\in\partial\Omega$ some unit direction $e_y$ has \[\{y+te:0<t<h,\ |e|=1,\ e\cdot e_y>\cos\alpha\}\subset\Omega. \] The directions may vary with $y$. A positive superharmonic function $V$ is a lower semicontinuous function $V:\Omega\to(0,\infty]$, not identically infinite, satisfying the super-mean inequality on every ball compactly contained in $\Omega$. For $p>0$, the statement $V\in L^p(\Omega)$ means $\int_\Omega V(x)^p\,dx<\infty$.

Statement recorded in the supplied source.

Determine, in terms of $n$ and the cone angle $\alpha$, all exponents $p>0$ for which every positive superharmonic function on every such domain belongs to $L^p(\Omega)$. Determine the sharp range, including endpoint behaviour. [S2]

Short English statement

Find the sharp integrability exponents for positive superharmonic functions on bounded Lipschitz domains with a prescribed interior-cone angle.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303032 (AMR-022-3032). 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 3.32, its complete Update 3.32, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.