Lieb-Thirring Inequalities for Fourth-Order Operators in Low Dimensions

TL;DR

This paper establishes Lieb-Thirring inequalities for fourth-order differential operators in low dimensions, providing sharp constants and deriving related Sobolev-type inequalities, advancing understanding of spectral bounds for higher-order operators.

Contribution

It develops Lieb-Thirring inequalities for fourth-order operators on the half-line, including sharp constants and conditions, and derives associated Sobolev-type inequalities for specific dimensions.

Findings

Proved Lieb-Thirring inequalities for fourth-order operators in dimensions 1 and 3.

Established sharp constants in Hardy-Rellich inequalities.

Derived Sobolev-type inequalities as corollaries.

Abstract

This paper considers 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 tr((-Delta)^2 - C^{HR}_{d,2} / (|x|^4) - V(x))^{-\gamma} < C_\gamma \int_{R^d} V(x)_+^{\gamma + d/4} dx for gamma \geq 1 - d/4, where C^{HR}_{d,2} is the sharp constant in the Hardy-Rellich inequality and where C_\gamma > 0 is independent of V, is proved for dimensions d = 1,3. As a corollary of this inequality a Sobolev-type inequality is obtained.