Theorem 1 (p. 497): short paths to the boundary on which u > eps/4

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Source. Theorem 1, p. 497, with Remarks 1 and 2 of section II on p. 499 and the proof in section III, pp. 500--501, of Jang-Mei Wu, Length of paths for subharmonic functions, J. London Math. Soc. (2) 32 (1985), 497--505, doi:10.1112/jlms/s2-32.3.497, the edition named on the source card.

Read depth. Claims checked: the statement and Remarks 1 and 2 were read clause by clause on the page images of the print; the proof was not checked. Nothing here is independently reviewed.

Statement

Let $D$ be a simply connected domain in $\mathbb C$ and $a\in D$. Let $u$ be subharmonic in $D$ with $0\le u\le1$ and $u(a)=\varepsilon>0$. Then there are paths $\gamma$ and $\Gamma$ from $a$ to points $b$ and $B$ of $\partial D$ respectively, with $\gamma\setminus\{b\}\subseteq D$ and $\Gamma\setminus\{B\}\subseteq D$, such that $u>\tfrac14\varepsilon$ on $\gamma\setminus\{b\}$ and on $\Gamma\setminus\{B\}$, and

$$L(\gamma)\le c_1\Bigl(1+\log\frac1\varepsilon\Bigr)\operatorname{diam}D, \qquad (1.1)$$ $$L(\Gamma)\le c_2\Bigl(1+\log\frac1\varepsilon\Bigr)\varepsilon^{-2}\, d(a,\partial D), \qquad (1.2)$$

where $L$ denotes length and $c_1$, $c_2$ are absolute constants.

The theorem refines Theorem A of the paper (p. 497), which it attributes to Lewis, Rossi and Weitsman as a stronger version of Hall's lemma: for $D$ simply connected with $0\in D$ and $u$ subharmonic with $0\le u\le1$ and $u(0)=\varepsilon>0$, there is a path $\gamma$ from $0$ to a point $b\in\partial D$ with $\gamma\setminus\{b\}\subseteq D$, $u>0$ on $\gamma\setminus\{b\}$ and $L(\gamma)\le C\varepsilon^{-c}d(0,\partial D)$. Lewis, Rossi and Weitsman asked how small the exponent $c$ can be; Theorem 1 replaces $\varepsilon^{-c}$ by $(1+\log(1/\varepsilon))\varepsilon^{-2}$.

Sharpness (p. 499)

• Remark 1: in (1.2), $\varepsilon^{-2}$ cannot be replaced by $\varepsilon^{-c}$ for any $c<2$, by an example in the slit disc $\Delta(0,2)\setminus\{x:0\le x\le2\}$, with $a=-2/n$ and $u$ vanishing on two unit segments issuing from $-1/n$. So $c\ge2$ in Theorem A. The paper says it does not know whether the factor $1+\log(1/\varepsilon)$ is necessary in (1.2).
• Remark 2: in (1.1), $1+\log(1/\varepsilon)$ cannot be replaced by $(1+\log(1/\varepsilon))^{1/2}$ or anything smaller, by an example in the unit square with a family of segments removed.

Proof, as a pointer

Section III, pp. 500--501. The path $\gamma$ is built in at most $1+\log\tfrac12\varepsilon/\log(1-\delta_0)$ steps by repeated use of Theorem D (p. 500), a preliminary form of Theorem A from the paper of Lewis, Rossi and Weitsman, on nested components of sublevel sets of $\max\{0,u-\tfrac12\varepsilon\}$, which gives (1.1). For $\Gamma$ the paper localizes to the component of $D\cap\Delta(a,R)$ containing $a$, $R=40\varepsilon^{-2}d(a,\partial D)$, subtracts a harmonic measure bounded through the Beurling projection theorem (the Milloux problem, in Ahlfors's Conformal invariants), and applies the first part to the result. The proof was not checked here.

Dependencies

Theorem D, quoted from J. Lewis, J. Rossi and A. Weitsman, On the growth of subharmonic functions along paths, Ark. Mat. 22 (1984), 109--119; the Beurling projection theorem.

Bears on

The paper names no Erdős problem. Theorem 1 is the step that the proof of Theorem 2 applies in components of sublevel sets of $u$ (p. 502).