On Poincar\'e-Sobolev level involving fractional GJMS operators on hyperbolic space

TL;DR

This paper analyzes the Poincaré–Sobolev levels associated with fractional GJMS operators on hyperbolic space, establishing new bounds, existence results, and characterizations, especially addressing the challenges posed by non-integer orders.

Contribution

It introduces a conformally related operator ilde{ ext{P}}_s, derives explicit Hardy bounds, and characterizes Poincaré–Sobolev levels, advancing understanding of fractional operators on hyperbolic space.

Findings

Established new lower bounds for Hardy terms in fractional Hardy–Sobolev–Maz'ya inequalities.

Proved existence of solutions to the Brezis–Nirenberg problem on hyperbolic space for fractional GJMS operators.

Provided a complete characterization of Poincaré–Sobolev levels for these operators.

Abstract

This paper is devoted to a qualitative analysis of the Poincar\'e--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 \((-\Delta)^s\); this link is crucial for importing Euclidean theory. We therefore introduce \(\widetilde{\mathcal{P}}_s\) (\(s>0\)), which is conformally related to \((-\Delta)^s\). Our purpose in the paper is to analyze the monotonicity, attainability, and strict-gap regions of the Poincar\'e--Sobolev levels associated with \(\mathcal{P}_s\) and \(\widetilde{\mathcal{P}}_s\). First, we reinterpret the Brezis--Nirenberg problem through the lens of Poincar\'e--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 Poincar\'e--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 Poincar\'e--Sobolev levels \(H_{n,s}\) and \(\widetilde H_{n,s}\).