On Poincaré-Sobolev level involving fractional GJMS operators on hyperbolic space

This paper is devoted to a qualitative analysis of the Poincaré–Sobolev level associated with the fractional GJMS operators ${\mathcal{P}}_s$ $(s\in (0,\frac{n}{2})\setminus \mathbb{N})$ on the hyperbolic space ${\mathbb{H}}^n$. In contrast to the integer-order case, when $s\mathrm{\notin }\mathbb{N}$ 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 $(-\mathrm{\Delta }{)}^s$; this link is crucial for importing Euclidean theory. We therefore introduce ${\tilde{\mathcal{P}}}_s$ ($s>0$), which is conformally related to $(-\mathrm{\Delta }{)}^s$. 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 ${\tilde{\mathcal{P}}}_s$. 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 $k\in (0,\frac{n}{2})$. We then establish new, explicit lower bounds for the Hardy term in fractional Hardy–Sobolev–Maz’ya inequalities involving both ${\mathcal{P}}_s$ and ${\tilde{\mathcal{P}}}_s$. 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 ${\tilde{H}}_{n,s}$.