Clunie--Sheil-Small系数差猜想的反例：通过拟共形映射
A counterexample to the Clunie--Sheil-Small coefficient-difference conjecture via quasiconformal maps
- Shantou University（汕头大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
通过构造调和K-拟共形映射，否证了Clunie--Sheil-Small系数差猜想，并揭示了其Hardy空间性质与共形情形的差异，同时给出了Beltrami系数在Sobolev空间时的精确范围。
AI中文摘要：
我们否证了Clunie--Sheil-Small猜想 $\bigl||a_n|-|b_n|\bigr|\nle n$，该猜想针对单位圆盘 $\nmathbb{D}$ 中归一化的单叶调和函数 $f=h+\noverline g$，其中 $a_n$ 和 $b_n$ 分别是 $h$ 和 $g$ 的Taylor系数。对于每个 $K>1$，我们构造了一个调和 $K$-拟共形映射，使得当 $n\n\to\ninfty$ 时，$|a_n|-|b_n|\nasymp n^{1+\nvarepsilon_K}$，其中 $\nvarepsilon_K>0$。然而，该映射满足猜想中的两个个体系数界。接下来，我们证明同一映射属于调和Hardy空间 $h^p$ 当且仅当 $0<p<1/(2+\nvarepsilon_K)$。因此，调和拟共形映射不保留完整的共形Hardy范围 $p<1/2$。有趣的是，当解析伸张在固定双曲半径的圆盘上的振荡在接近边界时趋于零，后者范围得以恢复。这尤其适用于伸张具有有限Dirichlet能量的情况。对于一般的 $K$-拟共形映射，我们施加额外假设：其Beltrami系数属于Sobolev空间 $W^{1,s}(\nmathbb{D})$，其中 $s\nge1$。当 $1\nle s<2$ 时，每个这样的映射在 $0<p<1/(2K)$ 范围内具有有界的 $p$-积分均值；当 $s\nge2$ 时，在 $0<p<1/2$ 范围内。这两个范围都是精确的。对于 $s\nge2$，每个映射还具有与映射到同一像的共形映射相同的临界Hardy指数。
英文摘要：
We disprove the Clunie--Sheil-Small conjecture $\bigl||a_n|-|b_n|\bigr|\le n$ for normalized univalent harmonic functions $f=h+\overline g$ in the unit disk $\mathbb{D}$, where $a_n$ and $b_n$ are the Taylor coefficients of $h$ and $g$, respectively. For every $K>1$, we construct a harmonic $K$-quasiconformal map which has $|a_n|-|b_n|\asymp n^{1+\varepsilon_K}$ as $n\to\infty$, with $\varepsilon_K>0$. Nevertheless, this map satisfies both individual coefficient bounds in the conjecture. Next, we show that the same map belongs to the harmonic Hardy space $h^p$ if and only if $0<p<1/(2+\varepsilon_K)$. Thus, harmonic quasiconformal maps do not retain the full conformal Hardy range $p<1/2$. Interestingly, the latter range is recovered when the oscillation of the analytic dilatation on disks of fixed hyperbolic radius tends to zero near the boundary. This holds, in particular, when the dilatation has finite Dirichlet energy. For general $K$-quasiconformal maps, we impose the additional assumption that their Beltrami coefficients belong to the Sobolev space $W^{1,s}(\mathbb{D})$ for some $s\ge1$. Every such map has bounded $p$-integral means for $0<p<1/(2K)$ when $1\le s<2$, and for $0<p<1/2$ when $s\ge2$. Both these ranges are sharp. For $s\ge2$, each map also has the same critical Hardy exponent as a conformal map onto the same image.
