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arXiv · 2305.06797

Optimal Sobolev inequalities in the hyperbolic space

Zdeněk Mihula
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Abstract

We find the optimal function norm on the left-hand side of the $m$th order Sobolev type inequality $\|u\|_{Y(\mathbb{H}^n)} \leq C \|\nabla_g^m u\|_{X(\mathbb{H}^n)}$ in the $n$-dimensional hyperbolic space $\mathbb{H}^n$, $1\leq m < n$. The optimal function norm in the inequality among all rearrangement-invariant function norms is completely characterized. A variety of concrete examples of optimal function norms is provided. The examples include delicate limiting cases, and especially when $m\geq3$, seem to provide new, improved inequalities in these limiting cases.

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Zdeněk Mihula. 2023-05-11. Optimal Sobolev inequalities in the hyperbolic space. https://doi.org/10.1142/s0219199726500227

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