A sharp three-particle fractional Hardy inequality and an angular Selberg-type identity
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2026
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| author | Mahadevan, Rajesh Olivares, Franco Zuniga, Andres |
| author_facet | Mahadevan, Rajesh Olivares, Franco Zuniga, Andres |
| contents | We establish a sharp three-particle fractional Hardy inequality for the Laplacian of order $s\in(0,1)$ in dimension $d\geq 4-2s$ (Theorem 1.1), involving an explicit intrinsically three-body interaction potential $V_{s,3}$. The inequality holds with the optimal two-particle fractional Hardy constant $C_{fH}(d,s)$, which is shown to be sharp relative to the fixed potential $V_{s,3}$. This potential $V_{s,3}$ strictly dominates the standard pairwise Coulomb-type interaction and captures genuine three-body effects. As a consequence, we derive a nontrivial many-particle fractional Hardy inequality for $N\geq 3$, and, in the regime $N>d+1$, obtain an improved Coulomb-type inequality with a strictly larger constant, agreeing in spirit with the results of Hoffmann-Ostenhof et al. [M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, and J. Tidblom, "Many-particle Hardy inequalities", J. Lond. Math. Soc. 77 (2008), no. 2, 99-115] and Lundholm [D. Lundholm, "Geometric extensions of many-particle Hardy inequalities", J. Phys. A: Math. Theor. 48 (2015), no. 17, 175203]. The proof relies on a fractional ground-state representation method adapted to three-particle interactions, combined with an explicit evaluation of the resulting nonlocal interaction term. This evaluation is achieved through a new singular integral identity of Selberg-type (Theorem 1.2), extending the three-fold formula of Grafakos-Morpurgo [L. Grafakos and C. Morpurgo, "A Selberg integral formula and applications", Pacific J. Math. 191 (1999), no. 1, 85-94] beyond the radial setting. This identity provides the analytic mechanism underlying the emergence of the three-body potential and may be of independent interest in harmonic analysis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_30586 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A sharp three-particle fractional Hardy inequality and an angular Selberg-type identity Mahadevan, Rajesh Olivares, Franco Zuniga, Andres Mathematical Physics Analysis of PDEs Primary 35R11, Secondary 26D10, 47G30, 35P15 We establish a sharp three-particle fractional Hardy inequality for the Laplacian of order $s\in(0,1)$ in dimension $d\geq 4-2s$ (Theorem 1.1), involving an explicit intrinsically three-body interaction potential $V_{s,3}$. The inequality holds with the optimal two-particle fractional Hardy constant $C_{fH}(d,s)$, which is shown to be sharp relative to the fixed potential $V_{s,3}$. This potential $V_{s,3}$ strictly dominates the standard pairwise Coulomb-type interaction and captures genuine three-body effects. As a consequence, we derive a nontrivial many-particle fractional Hardy inequality for $N\geq 3$, and, in the regime $N>d+1$, obtain an improved Coulomb-type inequality with a strictly larger constant, agreeing in spirit with the results of Hoffmann-Ostenhof et al. [M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, and J. Tidblom, "Many-particle Hardy inequalities", J. Lond. Math. Soc. 77 (2008), no. 2, 99-115] and Lundholm [D. Lundholm, "Geometric extensions of many-particle Hardy inequalities", J. Phys. A: Math. Theor. 48 (2015), no. 17, 175203]. The proof relies on a fractional ground-state representation method adapted to three-particle interactions, combined with an explicit evaluation of the resulting nonlocal interaction term. This evaluation is achieved through a new singular integral identity of Selberg-type (Theorem 1.2), extending the three-fold formula of Grafakos-Morpurgo [L. Grafakos and C. Morpurgo, "A Selberg integral formula and applications", Pacific J. Math. 191 (1999), no. 1, 85-94] beyond the radial setting. This identity provides the analytic mechanism underlying the emergence of the three-body potential and may be of independent interest in harmonic analysis. |
| title | A sharp three-particle fractional Hardy inequality and an angular Selberg-type identity |
| topic | Mathematical Physics Analysis of PDEs Primary 35R11, Secondary 26D10, 47G30, 35P15 |
| url | https://arxiv.org/abs/2605.30586 |