Skip to content
DraconiticResearch
Search Sources API
203,203 studies31 sourcesUpdated 36 min ago
PreprintarXiv licencearXiv

Asymptotic Formulas for Negative Sobolev Norms and Applications

  • Huaiqian Li
Open at sourcePDFDOICite

Abstract

This paper establishes an asymptotic formula for negative Sobolev norms as the fractional order tends to zero. In the Euclidean setting, under a mild boundedness condition, the product of the order and the norm raised to the power p converges to a dimension-independent constant multiple of the corresponding Lp norm. The proof relies on heat-kernel regularization, weak compactness, and an Abelian--Tauberian argument. The result is further extended to a general measure space equipped with a family of bounded and continuous operators that covering nonlinear and non-semigroup settings. We also present several applications in analysis and probability, including an absolute-continuity criterion for measures, a random-distribution regularity result, a construction of square-integrable local times for fractional Brownian motion, and limiting formulas for truncated maximal operators and martingales.

Related

  1. On the equivalence of Sobolev norms in infinite dimensions
    Zhouzhe Wang
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  2. Optimal interpolation formulas in the Sobolev space of periodic functions
    Kholmat Shadimetov, Khurshidjon Khayatov
    Results in Applied MathematicsPeer-reviewedCC BYFull textGeneral mathematicsDOAJ
  3. Error bounds in Sobolev norms for approximations with norm constrained ReLU neural networks
    Xianjun Li, Yunfei Yang
    arXivPreprintarXiv licenceFull textStatisticsarXiv
  4. Posterior contraction rates in Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families
    Emanuele Dolera, Stefano Favaro, Matteo Giordano
    arXivPreprintarXiv licenceFull textStatisticsarXiv
  5. Random Projections for Spectral Algorithms: Optimal Rates in Sobolev Norms
    Caixing Wang, Yue Wang
    arXivPreprintarXiv licenceFull textStatisticsarXiv

Cite

BibTeXRIS
BibTeX
@misc{li2026asymptotic,
  title = {{Asymptotic Formulas for Negative Sobolev Norms and Applications}},
  author = {Huaiqian Li},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26002},
  eprint = {2609.26002},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26002},
}
RIS
TY  - UNPB
TI  - Asymptotic Formulas for Negative Sobolev Norms and Applications
AU  - Huaiqian Li
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26002
UR  - https://arxiv.org/abs/2609.26002
AB  - This paper establishes an asymptotic formula for negative Sobolev norms as the fractional order tends to zero. In the Euclidean setting, under a mild boundedness condition, the product of the order and the norm raised to the power p converges to a dimension-independent constant multiple of the corresponding L^p norm. The proof relies on heat-kernel regularization, weak compactness, and an Abelian--Tauberian argument. The result is further extended to a general measure space equipped with a family of bounded and continuous operators that covering nonlinear and non-semigroup settings. We also present several applications in analysis and probability, including an absolute-continuity criterion for measures, a random-distribution regularity result, a construction of square-integrable local times for fractional Brownian motion, and limiting formulas for truncated maximal operators and martingales.
ER  -