04-93 W.D. Evans, R.T. Lewis

On the Rellich inequality with magnetic potentials (61K, LaTeX) Mar 26, 04

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Abstract. In lectures given in 1953 at New York University, Franz Rellich proved that for all $f\in C_0^{\infty}(R^n \setminus {0})$ and $n\neq 2$ $$ C(n) \int_{R^n} [|f(x)|^2/|x|^4] dx \le \int_{R^n} |\Delta f(x)|^2 dx $$ where the constant $C(n):=n^2(n-4)^2/16$ 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-Wiedl with Hardy-type inequalities in 2-dimensions, the authors show that the inclusion of a magnetic field $B= curl(A)$ of Aharonov-Bohm type yields non-trivial Rellich-type inequalities in $L^2(R^n)$, including the cases of $n=2$ and $n=4$: $$ C(n,\alpha) \int_{R^n} [|f(x)|^2/|x|^{\alpha+4}] dx \le \int_{R^n} [|\Delta_A f(x)|^2/|x|^\alpha] dx, $$ where $\Delta_A = (\nabla-iA)^2$ is the magnetic Laplacian. As in the Laptev-Weidl inequality, the constant $C(n,\alpha)$ depends upon the distance of the magnetic flux $\tilde\Psi$ to the integers. When the flux $\tilde\Psi$ is an integer and $\alpha=0$, the inequalities reduce to Rellich's inequality. Results are also given in $L^p(R^n), 1<p< \infty $ for $n=2$ and $n=3$. These results complement recent work of Davies and Hinz.

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