Research Problems in Function Theory — Problem 3.31

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 an unbounded domain $D\subset\mathbb R^n$, $n\ge2$, and write $x=(x_1,\ldots,x_n)$. A harmonic majorant of a nonnegative function $v$ is a finite harmonic function $H$ on $D$ with $H\ge v$. Assume $|x|$ has a harmonic majorant. Let $\Psi$ be the least harmonic majorant of $|x_1|$, and write $\log^+t=\max\{0,\log t\}$ for $t>0$, with $\log^+0=0$. Consider the conditions \[ (A)\quad\exists C,R>0\ \forall x\in D,\ |x|\ge R:\ \Psi(x)\le C|x|, \] \[ (B)\quad |x_1|\log^+|x_1|\text{ has a harmonic majorant on }D. \] The source records that $(A)$ implies $(B)$ under the standing assumption. In the planar comparison assume $0\in D$ and let $F:\mathbb D\to D\subset\mathbb C$ be a holomorphic universal covering map normalized by $F(0)=0$. It lies in $H^1$, meaning $\sup_{0<r<1}(2\pi)^{-1}\int_0^{2\pi}|F(re^{i\theta})|\,d\theta<\infty$. For $w\ne F(0)$, set \[N(1,w,F)=\sum_{F(z)=w}\log\frac1{|z|},\] where solutions in $\mathbb D$ are counted with multiplicity. The value at $w=0$ is immaterial to the following Lebesgue integral. Put \[ (J)\quad\int_{-\infty}^{\infty}N(1,iv,F)\log^+|v|\,dv<\infty. \] The source relates $(B)$ to $\operatorname{Re}F\in L\log L$ on the unit circle, where this means integrability of $|\operatorname{Re}F^*|\log^+|\operatorname{Re}F^*|$ for the almost-everywhere radial boundary values $F^*$.

Statement recorded in the supplied source.

(a) In dimension two, determine the relationship between $(A)$ and the counting-function condition $(J)$. (b) In every dimension $n\ge2$, decide whether $(A)$ is the correct condition needed for $(B)$, under the standing existence of a harmonic majorant of $|x|$. In particular, does there exist a domain where $|x|$ has a harmonic majorant but $|x_1|\log^+|x_1|$ does not? [S2]

Short English statement

Relate linear growth of the least harmonic majorant to the planar Nevanlinna integral, and decide whether the logarithmic majorant can fail when the norm itself has a harmonic majorant.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2303031 (AMR-022-3031). 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 3.31, its complete Update 3.31, and the relevant chapter notation. https://arxiv.org/pdf/1809.07200.