On Poincaré--Sobolev Levels for Fractional GJMS Operators on Hyperbolic Space

Abstract

We study the monotonicity, attainment, and strict-gap regions of the Poincaré--Sobolev levels associated with fractional GJMS operators $P_s$ on hyperbolic space, where $0<s<n/2$. For noninteger $s$, $P_s$ is not conformally intertwined with the Euclidean fractional Laplacian $(-\Delta {)}^s$, and we therefore also consider an auxiliary family ${\tilde{P}}_s$ for which this intertwining property holds. We establish explicit lower bounds for the Hardy term in the corresponding fractional Hardy--Sobolev--Maz'ya inequalities and derive strict-gap regimes for both families. To prove attainment, we decompose minimizing sequences into bounded-frequency and high-frequency parts. The high-frequency contribution is controlled by the spectral growth of the multipliers, while the bounded-frequency part, after suitable hyperbolic isometries, yields a nonzero weak limit, including at the spectral bottom. The Brezis--Lieb lemma then gives strong convergence and attainment of the infimum. Consequently, the corresponding Brezis--Nirenberg problems admit nontrivial weak solutions for parameters at or below the respective spectral bottoms: for all subcritical exponents, and for the critical exponent under the strict-gap condition. These results extend the corresponding integer-order theory to fractional GJMS operators.