Improved L${}^p$-Poincaré inequalities on the hyperbolic space

Abstract

We investigate the possibility of improving the $p$-Poincaré inequality $\Vert {\nabla }_{{\mathbb{H}}^N}u{\Vert }_p\ge {\Lambda }_p\Vert u{\Vert }_p$ on the hyperbolic space, where $p>2$ and ${\Lambda }_p:=[(N-1)/p{]}^p$ is the best constant for which such inequality holds. We prove several different, and independent, improved inequalities, one of which is a Poincaré-Hardy inequality, namely an improvement of the best $p$-Poincaré inequality in terms of the Hardy weight $r^{-p}$, $r$ being geodesic distance from a given pole. Certain Hardy-Maz'ya-type inequalities in the Euclidean half-space are also obtained.

File conformal to the printed one. Appeared in Nonlinear Analysis

References in corpus (2)

• Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem
• Improved higher order Poincaré inequalities on the hyperbolic space via Hardy-type remainder terms

Cited by in corpus (6)

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• Sharp constant for Poincaré-type inequalities in the hyperbolic space
• On some strong Poincaré inequalities on Riemannian models and their improvements
• On Higher order Poincaré Inequalities with radial derivatives and Hardy improvements on the hyperbolic space
• Improved Poincaré-Hardy inequalities on certain subspaces of the Sobolev space
• The sharp Poincaré--Sobolev type inequalities in the hyperbolic spaces ${\mathbb{H}}^n$