AI 中文总结
研究二维阿哈罗诺夫 - 玻姆势\(L^p\) - 哈代不等式中常数\(\lambda_\beta(p)\)相关问题，利用扭曲角常数的无紧致性双边界给出显式常数及新不等式，还讨论了\(p\geq2\)时的情况。
AI 中文摘要
对于二维阿哈罗诺夫 - 玻姆势\(A_\beta\)，通量\(\beta\notin\mathbb{Z}\)且\(1<p<2\)，Cazacu等人通过紧致性论证证明其\(L^p\) - 哈代不等式中的常数\(\lambda_\beta(p)\)严格超过自由常数\((\frac{2 - p}{p})^p\)，并提出构造性证明及\(\lambda_\beta(p)\)与依赖于\(\text{dist}(\beta,\mathbb{Z})\)的量的可比性问题。本文利用扭曲角常数的无紧致性双边界回答了这两个问题，给出了显式哈代常数\([(\frac{2 - p}{p})^2+(\frac{\sin(\pi\text{dist}(\beta,\mathbb{Z}))}{\pi})^2]^{p/2}\)。此外，当\(p\geq2\)时，阿哈罗诺夫 - 玻姆场产生具有通常齐次权重\(|x|^{-p}\)的\(L^p\) - 哈代不等式。本文方法还为复AB势提供了具有显式常数的新\(L^p\) - 哈代不等式。
英文摘要
For the two-dimensional Aharonov-Bohm potential $A_β$ with flux $β\notin\mathbb{Z}$ and $1<p<2$, Cazacu, Krejčiř\'ık, Lam and Laptev proved by a compactness argument that their constant $λ_β(p)$ in the $L^p$-Hardy inequality strictly exceeds the free constant $\big(\tfrac{2-p}{p}\big)^p$, and asked for a constructive proof with explicit estimates and for comparability of $λ_β(p)$ with a quantity depending on $\text{dist}(β,\mathbb{Z})$. We answer both questions by using a compactness-free two-sided bound for the twisted angular constant. Our explicit Hardy constant is $$\big[\big(\tfrac{2-p}{p}\big)^{2}+\big(\tfrac{\sin(π\text{dist}(β,\mathbb{Z}))}π\big)^{2}\big]^{p/2},\quad 1<p<2.$$ As a byproduct we observe that when $p\ge 2$ the Aharonov--Bohm field produces an $L^p$-Hardy inequality with the usual homogeneous weight $|x|^{-p}$. Our approach also provides new $L^p$-Hardy inequalities with explicit constants for the complex AB potentials.
Comments12 pages, comments welcome
