Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0708.0269

Sharp k-order Sobolev inequalities in the hyperbolic space ${\Bbb H}^n$

Genqian Liu
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we obtain the sharp $k$-th order Sobolev inequalities in the hyperbolic space ${\H}^n$ for all $k=1,2,3,\cdots$. This gives an answer to an open question raised by Aubin in [5, p.$\;$176-177] for $W^{k,2}({\H}^n)$ with $k>1$. In addition, we prove that the associated Sobolev constants are optimal.

Explore related subjects

math.APmath.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Genqian Liu. 2013-09-30. Sharp k-order Sobolev inequalities in the hyperbolic space ${\Bbb H}^n$. https://arxiv.org/abs/0708.0269

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Smooth solutions to the chord log-Minkowski problem

In integral geometry generalized with Aleksandrov's variational theory, Lutwak-Xi-Yang-Zhang [Comm. Pure Appl. Math. 77 (2024)] recently opened the door to researching the cone-chord measures and their log-Minkowski problem stemming from the chord integrals, named as the chord log-Minkowski problem. In the smooth category, the solvability of the chord log-Minkowski problem amounts to dealing with a nonlocal {M}onge-{A}mpère equation involving a Riesz potential. In this paper, to study the chord log-Minkowski problem, we first present some novel results on the boundary regularity of the Riesz potential. Based on these results, we obtain the regularity and existence for the chord log-Minkowski problem from the perspective of a nonlocal {M}onge-{A}mpère equation and a nonlocal Gauss curvature flow equation.

math.AP↗
arXiv2026

A proof of Onsager's Conjecture for the SQG equation

We construct solutions to the SQG equation that fail to conserve the Hamiltonian while having the maximal allowable regularity for this property to hold. This result solves the generalized Onsager conjecture on the threshold regularity for Hamiltonian conservation for SQG. We also prove nonuniqueness for arbitrary smooth data and an h-principle result.

math.AP↗