AI 中文总结
该研究针对有界约翰域上的解析函数，建立了含距离加权的不同阶导数积分的范数等价性，确定了边界维数条件的适用范围，并通过构造反例证明普通约翰条件无法弱化。
AI 中文摘要
设Ω⊂ℂ是有界约翰域，记δ(z)=dist(z,∂Ω)。对1<p<∞且α>dim_A(∂Ω)-2，我们建立了Ω上解析函数g的积分∫_Ω |g|^p δ^α dA与∫_Ω |g'|^p δ^{α+p} dA之间的范数等价性，其中含一个点赋值项以确定加性常数。导数项的估计是局部的，对所有真平面域均成立；反之，该估计由约翰域上的距离加权庞加莱不等式导出。取α=mp-2可得到m阶与m+1阶导数的对应比较。当m≥2时，边界维数条件自动满足，因此唯一依赖维数的情形是1<p<2时∫_Ω |f'|^p δ^{p-2} dA与∫_Ω |f''|^p δ^{2p-2} dA的比较。我们利用自相似Rohde雪花证明，该限制在拟共形圆盘中是严格的；还对每个s>1构造了一个内尖s-约翰域，其上该比较不成立，表明普通约翰条件一般不能被弱化。
英文摘要
Let $Ω\subset\mathbb C$ be a bounded John domain and set $δ(z)=\text{dist}(z,\partialΩ)$. For $1<p<\infty$ and $α>\text{dim}_A(\partialΩ)-2$, we establish a norm equivalence between $\int_Ω|g|^pδ^α\,dA$ and $\int_Ω|g'|^pδ^{α+p}\,dA$ for analytic functions $g$ on $Ω$, with a point-evaluation term fixing the additive constant. The estimate of the derivative term is local and holds on every proper planar domain, whereas the converse follows from a distance-weighted Poincaré inequality on John domains. Taking $α=mp-2$ yields the corresponding comparison between the $m$-th and $(m+1)$-st derivatives. For $m\ge2$ the boundary-dimension condition is automatic, so the only dimension-sensitive case is the comparison between $\int_Ω|f'|^pδ^{p-2}\,dA$ and $\int_Ω|f''|^pδ^{2p-2}\,dA$ when $1<p<2$. We show that this restriction is sharp within the class of quasidisks by using self-similar Rohde snowflakes. We also construct, for every $s>1$, an inward-cusp $s$-John domain on which the comparison fails, showing that the ordinary John condition cannot in general be weakened.
Comments15 pages, 2 figures
