On the Rellich inequality with magnetic potentials

In lectures given in 1953 at New York University, Franz Rellich proved that for all f∈C₀^∞(Rⁿ \{0}) and n≠2where the constant C(n):=n²(n−4)²/16 is sharp. For n=2 extra conditions were required for f, and for n=4, C(4)=0, producing a trivial inequality. Influenced by recent work of Laptev-Weidl on Hardy-type inequalities in R², the authors show that for n≥2, the inclusion of a magnetic field B=curl(A) of Aharonov-Bohm type yields non-trivial Rellich-type inequalities of the formwhere Δ_A=(∇−iA)² is the magnetic Laplacian. As in the Laptev-Weidl inequality, the constant C(n,α) depends upon the distance of the magnetic flux to the integers Z. When the flux is an integer and α=0, the inequalities reduce to Rellich’s inequality.The first author gratefully acknowledges the hospitality and support of the Mathematics Department at UAB where much of this work was done.