Skip to content
DraconiticResearch
Search Sources API
203,203 studies31 sourcesUpdated 15 min ago
PreprintarXiv licencearXiv

Optimal AB-program for Poincaré--Sobolev and logarithmic Sobolev inequalities on hyperbolic spaces

  • Anh Do
  • Debdip Ganguly
  • Nguyen Lam
  • Guozhen Lu
Open at sourcePDFDOICite

Abstract

In this paper, we prove that we can subtract the full spectral gap ((N−1)/p)p of the p-Laplacian and keep the sharp Sobolev constant S(N,p) of ℝN at the same time in a large family of p. For p=2, this was proved by Benguria, Frank and Loss in dimension three, and it was shown by Hebey to be false in dimensions N≥4. For p≠2, it was proved by Nguyen for N≥4 and 2N/(N−1)≤ p<N. By employing, among other tools, a suitable conjugation that maps the radial p-Dirichlet integral on ℍN onto its Euclidean counterpart on ℝN while preserving the Lp norm, together with a one-dimensional tangent line inequality, we establish such sharp Poincaré--Sobolev inequalities for N=3 and 2≤ p<3, for N=4 and (1+√(13))/2≤ p<4, and for N≥5 and min{3,(1+√(4N−3))/2}≤ p<N. In particular, in all cases we also obtain an explicit remainder term. Moreover, since the change of function keeps the Lp norm, we prove the sharp logarithmic Sobolev inequalities with the full spectral gap and the sharp Euclidean constant for N=3 and 2≤ p≤3, and for N≥4 on an explicit interval of p. For p≥3, we also use the change of variable t=sinhρ and the Hardy inequality, which give a second remainder term. At the endpoint p=N, this remainder gives a Hardy--Trudinger--Moser inequality on ℍN under a weaker constraint than the one of Liang, Lu, Wang and Yang, in every dimension N≥3. Finally, we show that for 1<p<(1+√(4N−3))/2, which includes all 1<p<2, the full spectral gap can still be subtracted with some positive constant in front of the Sobolev term, and that this constant is smaller than S(N,p)p when p<min{2,√N}. Together with our other results, this shows that the full spectral gap can be subtracted for every 1<p<N.

Related

  1. Poincaré inequalities, logarithmic growth and doubling
    Joaquim Martin
    arXivPreprintarXiv licenceFull textAnalysis & PDEsProbabilityarXiv
  2. Rellich-Poincaré inequalities and improved Rellich inequalities
    Paolo Caldiroli, Roberta Musina
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  3. Caffarelli-Kohn-Nirenberg and Weighted Gaussian Poincaré Inequalities: a complete characterization of sharp L2 stability and Lp extensions
    Anh Xuan, Nguyen Lam, Guozhen Lu +1
    arXiv (Cornell University)PreprintLicence not statedFull text11 citedGeneral mathematicsOpenAlex
  4. Parabolic Poincaré inequalities and maximal function estimates for systems of partial differential equations
    Wontae Kim, Juha Kinnunen
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  5. Poincaré Beta Balls: Radial Laws, Hyperbolic Shell Metric Transforms, and Critical-Limit Regimes
    Shigeaki Yokota
    arXivPreprintCC BY 4.0Full textGeometry & topologyProbabilityarXiv

Cite

BibTeXRIS
BibTeX
@misc{do2026optimal,
  title = {{Optimal AB-program for Poincaré--Sobolev and logarithmic Sobolev inequalities on hyperbolic spaces}},
  author = {Anh Do and Debdip Ganguly and Nguyen Lam and Guozhen Lu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01161},
  eprint = {2611.01161},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01161},
}
RIS
TY  - UNPB
TI  - Optimal AB-program for Poincaré--Sobolev and logarithmic Sobolev inequalities on hyperbolic spaces
AU  - Anh Do
AU  - Debdip Ganguly
AU  - Nguyen Lam
AU  - Guozhen Lu
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01161
UR  - https://arxiv.org/abs/2611.01161
AB  - In this paper, we prove that we can subtract the full spectral gap ((N−1)/p)^p of the p-Laplacian and keep the sharp Sobolev constant S(N,p) of ℝ^N at the same time in a large family of p. For p=2, this was proved by Benguria, Frank and Loss in dimension three, and it was shown by Hebey to be false in dimensions N≥4. For p≠2, it was proved by Nguyen for N≥4 and 2N/(N−1)≤ p<N. By employing, among other tools, a suitable conjugation that maps the radial p-Dirichlet integral on ℍ^N onto its Euclidean counterpart on ℝ^N while preserving the L^p norm, together with a one-dimensional tangent line inequality, we establish such sharp Poincaré--Sobolev inequalities for N=3 and 2≤ p<3, for N=4 and (1+√(13))/2≤ p<4, and for N≥5 and min{3,(1+√(4N−3))/2}≤ p<N. In particular, in all cases we also obtain an explicit remainder term. Moreover, since the change of function keeps the L^p norm, we prove the sharp logarithmic Sobolev inequalities with the full spectral gap and the sharp Euclidean constant for N=3 and 2≤ p≤3, and for N≥4 on an explicit interval of p. For p≥3, we also use the change of variable t=sinhρ and the Hardy inequality, which give a second remainder term. At the endpoint p=N, this remainder gives a Hardy--Trudinger--Moser inequality on ℍ^N under a weaker constraint than the one of Liang, Lu, Wang and Yang, in every dimension N≥3. Finally, we show that for 1<p<(1+√(4N−3))/2, which includes all 1<p<2, the full spectral gap can still be subtracted with some positive constant in front of the Sobolev term, and that this constant is smaller than S(N,p)^p when p<min{2,√N}. Together with our other results, this shows that the full spectral gap can be subtracted for every 1<p<N.
ER  -