A Hardy–Sobolev inequality for the twisted Laplacian

Abstract

We prove a strong optimal Hardy–Sobolev inequality for the twisted Laplacian on ℂn. The twisted Laplacian is the magnetic Laplacian for a system of n particles in the plane, corresponding to the constant magnetic field. The inequality we obtain is strong optimal in the sense that the weight cannot be improved. We also show that our result extends to a one-parameter family of weighted Sobolev spaces.