Research Problems in Function Theory — Problem 3.34

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 $0<\alpha\le\pi$, interpreting $\alpha$ as the full opening angle of a circular cone. Let $\Omega\subset\mathbb R^n$ be a bounded connected open set such that for every $y\in\partial\Omega$ there are a unit vector $e_y$ and height $h_y>0$ with \[\{y+te:0<t<h_y,\ |e|=1,\ e\cdot e_y>\cos(\alpha/2)\}\subset\Omega. \] The height is allowed to depend on the boundary point; no Lipschitz boundary or uniform positive lower height is imposed. 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.

Does there exist an exponent $p=p_n(\alpha)>0$, independent of the particular domain and function, such that every positive superharmonic $V$ on every such $\Omega$ belongs to $L^p(\Omega)$? If so, characterize the possible exponents in terms of the cone angle (with dimension fixed). [S2]

Short English statement

Does having an interior truncated cone of a fixed angle at each boundary point force a common positive integrability exponent for all positive superharmonic functions, and how does that exponent depend on the angle?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303034 (AMR-022-3034). 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 3.34, its complete Update 3.34, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.