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, Krejčiř\'ık, 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 $(\frac{2-p}{p}{)}^p$, 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 $$\left[\right(\frac{2-p}{p}{)}^2+(\frac{\sin (\pi \text{dist}(\beta ,\mathbb{Z}))}{\pi }{)}^2{]}^{p/2},1<p<2.$$ 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.

arXiv comment12 pages, comments welcome