On Poincaré-Sobolev level involving fractional GJMS operators on hyperbolic space
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866908849506615296 |
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| author | Chen, Huyuan Chen, Rui |
| author_facet | Chen, Huyuan Chen, Rui |
| contents | This paper is devoted to a qualitative analysis of the Poincaré--Sobolev level associated with the fractional GJMS operators \(\mathcal{P}_s\) \(\bigl(s\in(0,\tfrac n2)\setminus\mathbb N\bigr)\) on the hyperbolic space \(\mathbb H^n\). In contrast to the integer-order case, when \(s\notin\mathbb N\) the operator \(\mathcal{P}_s\) does not enjoy the conformal covariance that allows one, in the upper half-space or ball model, to relate it to the Euclidean fractional Laplacian \((-Δ)^s\); this link is crucial for importing Euclidean theory. We therefore introduce \(\widetilde{\mathcal{P}}_s\) (\(s>0\)), which is conformally related to \((-Δ)^s\). Our purpose in the paper is to analyze the monotonicity, attainability, and strict-gap regions of the Poincaré--Sobolev levels associated with \(\mathcal{P}_s\) and \(\widetilde{\mathcal{P}}_s\). First, we reinterpret the Brezis--Nirenberg problem through the lens of Poincaré--Sobolev levels, connecting earlier results for the Euclidean Laplacian and for operators \(\mathcal{P}_k\) on \(\mathbb H^n\) with integer \(k\in(0,\tfrac n2)\). We then establish new, explicit lower bounds for the Hardy term in fractional Hardy--Sobolev--Maz'ya inequalities involving both \(\mathcal{P}_s\) and \(\widetilde{\mathcal{P}}_s\). By applying the concentration--compactness principle together with a detailed analysis of the strict-gap regions for the Poincaré--Sobolev levels, we prove the existence of solutions to the Brezis--Nirenberg problem on \(\mathbb H^n\) for both operators. Finally, combining the Hardy lower bounds with criteria for attainability, we obtain a complete characterization of the Poincaré--Sobolev levels \(H_{n,s}\) and \(\widetilde H_{n,s}\). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03299 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Poincaré-Sobolev level involving fractional GJMS operators on hyperbolic space Chen, Huyuan Chen, Rui Analysis of PDEs 35R11, 58J40, 35J30, 53C21 This paper is devoted to a qualitative analysis of the Poincaré--Sobolev level associated with the fractional GJMS operators \(\mathcal{P}_s\) \(\bigl(s\in(0,\tfrac n2)\setminus\mathbb N\bigr)\) on the hyperbolic space \(\mathbb H^n\). In contrast to the integer-order case, when \(s\notin\mathbb N\) the operator \(\mathcal{P}_s\) does not enjoy the conformal covariance that allows one, in the upper half-space or ball model, to relate it to the Euclidean fractional Laplacian \((-Δ)^s\); this link is crucial for importing Euclidean theory. We therefore introduce \(\widetilde{\mathcal{P}}_s\) (\(s>0\)), which is conformally related to \((-Δ)^s\). Our purpose in the paper is to analyze the monotonicity, attainability, and strict-gap regions of the Poincaré--Sobolev levels associated with \(\mathcal{P}_s\) and \(\widetilde{\mathcal{P}}_s\). First, we reinterpret the Brezis--Nirenberg problem through the lens of Poincaré--Sobolev levels, connecting earlier results for the Euclidean Laplacian and for operators \(\mathcal{P}_k\) on \(\mathbb H^n\) with integer \(k\in(0,\tfrac n2)\). We then establish new, explicit lower bounds for the Hardy term in fractional Hardy--Sobolev--Maz'ya inequalities involving both \(\mathcal{P}_s\) and \(\widetilde{\mathcal{P}}_s\). By applying the concentration--compactness principle together with a detailed analysis of the strict-gap regions for the Poincaré--Sobolev levels, we prove the existence of solutions to the Brezis--Nirenberg problem on \(\mathbb H^n\) for both operators. Finally, combining the Hardy lower bounds with criteria for attainability, we obtain a complete characterization of the Poincaré--Sobolev levels \(H_{n,s}\) and \(\widetilde H_{n,s}\). |
| title | On Poincaré-Sobolev level involving fractional GJMS operators on hyperbolic space |
| topic | Analysis of PDEs 35R11, 58J40, 35J30, 53C21 |
| url | https://arxiv.org/abs/2602.03299 |