Abstract
In lectures given in 1953 at New York University, Franz Rellich proved that for all f∈C0∞(Rn \{0}) and n≠2
where 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 form
where 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.
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The first author gratefully acknowledges the hospitality and support of the Mathematics Department at UAB where much of this work was done.
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Evans, W., Lewis, R. On the Rellich inequality with magnetic potentials. Math. Z. 251, 267–284 (2005). https://doi.org/10.1007/s00209-005-0798-5
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DOI: https://doi.org/10.1007/s00209-005-0798-5