[Paper Review] Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality

Rupert L. Frank, Élliott H. Lieb|arXiv (Cornell University)|Apr 27, 2009

Nonlinear Partial Differential Equations15 references4 citations

TL;DR

This paper presents a new proof of the sharp Hardy-Littlewood-Sobolev (HLS) inequality in the diagonal case using inversion positivity under conformal transformations via inversions in spheres, rather than symmetric decreasing rearrangements. It establishes that the minimizers are functions of the form $ f(x) = \alpha(\beta + |x - y|^2)^{-(2N - \lambda)/2} $, and derives the sharp constant $ \mathcal{H}_{N,\lambda,p,p} = \pi^{\lambda/2} \frac{\Gamma((N-\lambda)/2)}{\Gamma(N - \lambda/2)} \left( \frac{\Gamma(N)}{\Gamma(N/2)} \right)^{1 - \lambda/N} $, valid for $ N \geq 3 $, $ \lambda \geq N - 2 $, and $ N = 1,2 $ with $ \lambda < N $.

ABSTRACT

We give a new proof of certain cases of the sharp HLS inequality. Instead of symmetric decreasing rearrangement it uses the reflection positivity of inversions in spheres. In doing this we extend a characterization of the minimizing functions due to Li and Zhu.

Motivation & Objective

• To provide a new proof of the sharp Hardy-Littlewood-Sobolev inequality in the diagonal case without relying on symmetric decreasing rearrangements.
• To extend the characterization of minimizers from continuous functions to finite Borel measures, establishing their absolute continuity and specific functional form.
• To demonstrate that reflection positivity under inversions in spheres—rather than planes—can be used to derive symmetry and extremality properties in variational problems.
• To unify and generalize the geometric approach of Li and Zhu using conformal invariance and inversion positivity in the context of the HLS inequality.

Proposed method

• Utilizes conformal invariance of the HLS functional under inversions in spheres and reflections in hyperplanes, showing $ I_\lambda[f] = I_\lambda[\Theta_B f] $ for ball inversions $ \Theta_B $.
• Applies reflection positivity via inversions in hemi-balls to compare measure values in nested regions, establishing monotonicity and symmetry properties.
• Employs a geometric construction of hemi-balls $ \tilde{B} $ such that $ \Theta_{\tilde{B}}(B') \subset B $, enabling comparison of measure masses across regions.
• Uses radial symmetry and continuity arguments to show that the density $ v $ of the minimizing measure is continuous and symmetric decreasing.
• Applies Taylor expansion and inversion symmetry to derive a differential equation $ r_a^2 \partial_r v(a) + 2N|a|v(a) = 0 $, whose solution yields the explicit form of the minimizer.
• Establishes that the only invariant measures under these conformal transformations are those with densities $ v(a) = C (r_0^2 + |a|^2)^{-N} $, up to constants.

Experimental results

Research questions

• RQ1 Can the sharp HLS inequality be proven without symmetric decreasing rearrangement, using instead conformal invariance and reflection positivity?
• RQ2 What is the precise class of functions that minimize the HLS functional under conformal transformations?
• RQ3 How does inversion positivity in spheres differ from reflection positivity in planes in characterizing extremal functions?
• RQ4 Can the minimizers of the HLS inequality be characterized for general Borel measures, not just continuous functions?
• RQ5 What is the exact form of the sharp constant in the diagonal HLS inequality, and how is it derived via geometric and conformal methods?

Key findings

• The sharp constant in the diagonal HLS inequality is $ \mathcal{H}_{N,\lambda,p,p} = \pi^{\lambda/2} \frac{\Gamma((N-\lambda)/2)}{\Gamma(N - \lambda/2)} \left( \frac{\Gamma(N)}{\Gamma(N/2)} \right)^{1 - \lambda/N} $, valid for $ N \geq 3 $, $ \lambda \geq N - 2 $, and $ N = 1,2 $, $ \lambda < N $.
• Equality in the HLS inequality holds if and only if $ f(x) = \alpha(\beta + |x - y|^2)^{-(2N - \lambda)/2} $ and $ g(x) = \alpha'(\beta + |x - y|^2)^{-(2N - \lambda)/2} $ for $ \alpha, \alpha' \in \mathbb{C} $, $ \beta > 0 $, $ y \in \mathbb{R}^N $.
• The minimizers are shown to be absolutely continuous with respect to Lebesgue measure, with densities of the form $ v(x) = C (r_0^2 + |x|^2)^{-N} $, derived from a radial ODE via inversion symmetry.
• The method establishes continuity and differentiability of the density function $ v $, using inversion-based comparison and radial limits.
• The proof extends to measures, showing that any finite Borel measure invariant under the relevant conformal transformations must be absolutely continuous with the above density.
• The approach provides a new geometric characterization of extremizers, replacing moving planes with moving spheres and reflection positivity through spheres.