Critical Fractional and Hardy–Lieb–Thirring Inequalities

Definitions and mathematical statement

The supplied record poses two related questions concerning Lieb--Thirring inequalities. Their full analytic formulations are not included in the record.

1. Critical inequalities for powers of the Laplacian. Seek a critical Lieb--Thirring inequality for arbitrary powers of the Laplacian, including noninteger powers. Writing the operator formally as $( -\Delta)^s$, the supplied record does not specify the range of $s$, the underlying space or its dimension, the operator realization, the admissible potentials, the spectral quantity to be bounded, or the meaning of the critical exponent. Thus no particular inequality involving $( -\Delta)^s$ can be stated precisely from the supplied evidence alone.
2. Sharp half-line Hardy--Lieb--Thirring constants. Let $r\in(0,\infty)$ and $0<\theta\leq1$, and define \[ w_\theta(r)=r^{1-\theta}. \] The potential-weight expression specified in the record is \[ V(r)w_\theta(r)=V(r)r^{1-\theta}. \] For every $\theta\in(0,1]$, determine the sharp constant $C_\theta$ in the intended half-line Hardy--Lieb--Thirring inequality with this weight. Here a sharp constant means the least constant for which the fully specified inequality holds for every admissible potential. The supplied record reports that the inequality is known throughout this parameter range, but it does not supply its formula. In particular, it does not identify the Hardy operator or coefficient, its realization and boundary condition at $r=0$, the spectral moment, the admissible class and sign convention for $V$, or the placement and normalization of $C_\theta$. The weight alone therefore does not determine a numerical sharp-constant problem.

The original wording also asks for a proof of the half-line bound while explicitly saying that it is known for $0<\theta\leq1$. Both qualifications are retained: existence in that range is a supplied claim, and determining the sharp constant is the stated further question. Completing either analytic formulation requires source clarification; no missing operator, exponent, or boundary condition is imposed here.

Short English statement

The record asks for critical Lieb--Thirring inequalities for arbitrary, including noninteger, Laplacian powers and for the best constants in a half-line Hardy--Lieb--Thirring bound with weight $V(r)r^{1-\theta}$, for every $0<\theta\leq1$. The exact inequalities and essential operator hypotheses are missing, so this draft preserves the questions and explicitly records the unresolved specifications.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001167$, exact source problem code catalogue entry, title: Critical Fractional and Hardy--Lieb--Thirring Inequalities. Project registry references: data/unsolvedmath/problems.json and catalog/problems.json. These identify the supplied record; no independent file inspection is claimed.
• Low Eigenvalues of Laplace and Schrödinger Operators (2009), Oberwolfach Reports, pp. 355--428, DOI 10.4171/owr/2009/06. Supplied report citation and literature-source link: https://doi.org/10.4171/owr/2009/06. The record attributes the problem to the workshop-report collection and supplies no individual proposer.
• Lieb--Thirring inequalities on the half-line. Supplied literature-source link: https://arxiv.org/abs/math/0611247.
• Hardy--Lieb--Thirring inequalities for fractional operators. Supplied literature-source link: https://arxiv.org/abs/math/0610593.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports that the half-line inequality was proved for all $0<\theta\leq1$, while the requested critical inequality for arbitrary Laplacian powers is established only in important classes such as positive integer powers. This is supplied dated curation, not an independently verified literature assessment or a claim about current status.
• The supplied statement status is corrected_verified; its verification note reports comparison with the official Oberwolfach report and normalization of notation, context, and question form. That verification claim is part of the supplied metadata and has not been independently checked here.