 0493 W.D. Evans, R.T. Lewis
 On the Rellich inequality with magnetic potentials
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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 $C(n):=n^2(n4)^2/16$ 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 LaptevWiedl with
Hardytype inequalities in 2dimensions, the authors show that the inclusion of a magnetic field $B= curl(A)$ of AharonovBohm type yields nontrivial Rellichtype 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 $\Delta_A = (\nablaiA)^2$ is the magnetic Laplacian. As in the
LaptevWeidl 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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