Critical Hardy–Lieb–Thirring Inequalities for Fourth-Order Operators in Low Dimensions

Abstract

This paper considers Hardy–Lieb–Thirring inequalities for higher order differential operators. A result for general fourth-order operators on the half-line is developed, and the trace inequality

$$\mathrm{tr}\left( (-\Delta)^2 - C^{\mathrm{HR}}_{d,2}\frac{1}{|x|^4} - V(x) \right)_-^{\gamma}\leq C_\gamma\int\limits_{\mathbb{R}^d} V(x)_+^{\gamma + \frac{d}{4}}\,\mathrm{d}x, \quad \gamma \geq 1 - \frac d 4,$$

where ${C^{\mathrm{HR}}_{d,2}}$ is the sharp constant in the Hardy–Rellich inequality and where C _γ >  0 is independent of V, is proved for dimensions d =  1, 3. As a corollary of this inequality, a Sobolev-type inequality is obtained.