Sharp Weighted Hardy-Type Inequalities for Homogeneous Magnetic Potentials

Abstract

We prove a sharp magnetic improvement of the family of weighted inequalities on the upper half-space $\mathbb{R}^{N+1}_+$ that interpolate between the Hardy inequality and the trace Hardy inequality. The magnetic potentials are of Aharonov-Bohm type, $\mathbf{A}(x)=|x|^{-1}\boldsymbol{a}(x/|x|)$, where the tangential field $\boldsymbol{a}$ may be singular on a closed subset of the sphere. We compute the sharp constant explicitly: it is given by the constant of the corresponding nonmagnetic inequality, evaluated in a virtual dimension that depends on the first eigenvalue of the magnetic Laplacian on the sphere. In particular, the magnetic trace Hardy constant strictly exceeds its Euclidean value whenever this eigenvalue is positive. The proof rests on a ground state representation of the deficit, which also shows that the sharp constant is not attained in the energy space. As a consequence, we obtain the sharp Hardy inequality for the fractional powers $(-Δ_{\mathbf{A}})^σ$, $0<σ<1$, of the magnetic Laplacian: the constant is the same on smooth functions supported away from the singular set and on the form domain of $(-Δ_{\mathbf{A}})^σ$, and it is not attained. Examples include the planar Aharonov-Bohm potential with nonintegral flux and, in dimension three, the Aharonov-Bohm potential and the Dirac monopole.