[Paper Review] The sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half-space

Rafael D. Benguria, Rupert L. Frank+1 more|ArXiv.org|May 25, 2007

Nonlinear Partial Differential Equations5 references3 citations

TL;DR

This paper establishes that the sharp constant in the Hardy-Sobolev-Maz'ya inequality on the three-dimensional upper half-space is equal to the classical Sobolev constant $ S_3 = 3(\pi/2)^{4/3} $, using a duality argument that connects the problem to a sharp Hardy-Littlewood-Sobolev inequality. The result shows the inequality is always strict for non-zero functions, resolving a long-standing open problem in dimension three where the sharp constant was previously unknown and distinct from the $ n \geq 4 $ case.

ABSTRACT

It is shown that the sharp constant in the Hardy-Sobolev-Maz'ya inequality on the three dimensional upper half space is given by the Sobolev constant. This is achieved by a duality argument relating the problem to a Hardy-Littlewood-Sobolev type inequality whose sharp constant is determined as well.

Motivation & Objective

• To determine the sharp constant in the Hardy-Sobolev-Maz'ya inequality for the three-dimensional upper half-space $ \mathbb{H}^3 $, where the constant had remained unknown despite progress in higher dimensions.
• To resolve the discrepancy between the sharp constant in the inequality and the classical Sobolev constant in dimension three.
• To establish that the inequality is always strict for non-zero functions, implying no non-trivial minimizers exist in this case.

Proposed method

• A duality argument is employed to relate the Hardy-Sobolev-Maz'ya inequality on $ \mathbb{H}^3 $ to a Hardy-Littlewood-Sobolev-type inequality on the unit ball via conformal transformations.
• The problem is transformed using stereographic projection to the hyperboloid $ \mathbb{P}^3 $, where the functional becomes invariant under Möbius transformations.
• The sharp constant in the resulting inequality is determined by leveraging Lieb’s sharp constant for the Hardy-Littlewood-Sobolev inequality in the context of the conformally invariant operator $ (-\Delta - \frac{1}{4y^2})^{-\alpha/2} $.
• The analysis relies on asymptotic estimates of a kernel function $ F(A) $, derived from a generalized hypergeometric-type integral, to control the operator norm and establish sharpness.
• The proof uses the method of competing symmetries and correction terms to Fatou’s lemma to show existence of maximizers in related variational problems.
• The key equation is the duality identity $ |(f,g)|^2 \leq C \cdot (f, Q^{\alpha/2}f) \|g\|_p^2 $, which leads to the sharp constant when $ n=3 $, $ \alpha=2 $.

Experimental results

Research questions

• RQ1 What is the sharp constant in the Hardy-Sobolev-Maz'ya inequality for the three-dimensional upper half-space $ \mathbb{H}^3 $?
• RQ2 Does the sharp constant in this case coincide with the classical Sobolev constant $ S_3 $?
• RQ3 Why does the sharp constant fail to be attained in dimension three, unlike in higher dimensions ($ n \geq 4 $)?
• RQ4 How does the conformal invariance of the problem help in determining the sharp constant?
• RQ5 Can the duality approach between the Hardy-Sobolev-Maz'ya and Hardy-Littlewood-Sobolev inequalities be used to derive sharp constants in other dimensions?

Key findings

• The sharp constant in the Hardy-Sobolev-Maz'ya inequality on $ \mathbb{H}^3 $ is exactly the classical Sobolev constant $ S_3 = 3(\pi/2)^{4/3} $, resolving the open problem in dimension three.
• The inequality is always strict for non-zero functions, meaning no non-trivial minimizer exists, in contrast to the $ n \geq 4 $ case where optimizers exist.
• The sharp constant in the associated Hardy-Littlewood-Sobolev inequality for the operator $ (-\Delta - \frac{1}{4y^2})^{-1} $ is determined and shown to be $ 2^{-2} \pi^{-3/2} \frac{\Gamma(\frac{3-2}{2})}{\Gamma(1)} C(3,2) $, with $ C(3,2) $ given explicitly.
• The kernel function $ F(A) $, arising from the integral representation of the Green's function, is shown to be monotone increasing for $ \beta > 1/2 $, with a finite limit as $ A \to \infty $, which is crucial for bounding the operator norm.
• The sharp constant in the $ L^p $-boundedness of $ (-\Delta - \frac{1}{4y^2})^{-\alpha/2} $ is derived for $ \alpha = 2 $, $ n=3 $, and $ p = 6 $, confirming the sharpness of the inequality.
• The result confirms that the sharp constant for convex domains is conjectured to be attained in the half-space case, and this paper verifies it explicitly for $ n=3 $.