Explicit constants in $L^p$-Hardy inequalities for Aharonov-Bohm potentials

Abstract

For the two-dimensional Aharonov-Bohm potential $A_\beta$ with flux $\beta\notin\mathbb{Z}$ and $1<p<2$, Cazacu, Krej\v{c}i\v{r}\'{\i}k, Lam and Laptev proved by a compactness argument that their constant $\lambda_\beta(p)$ 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 $\lambda_\beta(p)$ with a quantity depending on $\text{dist}(\beta,\mathbb{Z})$. We answer both questions by using a compactness-free two-sided bound for the twisted angular constant. Our explicit Hardy constant is $$\big[\big(\tfrac{2-p}{p}\big)^{2}+\big(\tfrac{\sin(\pi\text{dist}(\beta,\mathbb{Z}))}{\pi}\big)^{2}\big]^{p/2},\quad 1<p<2.$$ 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.