Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 10 min ago
PreprintarXiv licencearXiv

Hardy-Sobolev-Maz'ya inequalities for 1<p<2

  • Avas Banerjee
  • Michał Kijaczko
  • Vivek Sahu
Open at sourcePDFDOICite

Abstract

We establish Hardy--Sobolev--Maz'ya inequalities in the subquadratic range 1<p<2 for singularities supported on affine subspaces and for arbitrary domains. We first prove a Hardy--Sobolev--Maz'ya inequality for singularities supported on affine subspaces of arbitrary codimension 1≤ k<N, showing that the deficit in the sharp Hardy inequality controls a suitable weighted Sobolev norm. Combining the same convexity estimate with the one-dimensional approach of Frank and Loss in [9], we also establish Hardy--Sobolev--Maz'ya inequalities on arbitrary domains with the mean distance to the boundary and thereby solve a problem posed in [9]. These results complete the previously unresolved subquadratic range 1<p<2 and, together with the known results for p≥ 2, provide a unified Hardy--Sobolev--Maz'ya theory for all 1<p<N associated with plane singularities and general domains.

Related

  1. On improved Hardy-type inequalities in Log-Weighted Sobolev Spaces
    Sourav Ghosh
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  2. On the Stability of Critical Points of the Hardy-Littlewood-Sobolev Inequality
    Kuan Liu, Qian Zhang, Wenming Zou
    Potential AnalysisPeer-reviewedLicence not statedFull text10 citedGeneral mathematicsOpenAlex
  3. Diagonal operators on Janson-Sobolev and Janson-Sobolev-Hardy spaces
    Krystian Kazaniecki, Richard Lechner
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  4. A Sharp Mass Threshold for Fractional Choquard Equations with Lower Hardy-Littlewood-Sobolev and L2-Critical Terms
    Yongpeng Chen, Zhipeng Yang
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  5. Uniqueness of Extremizers and Convergence of the Euler-Lagrange Iteration for the Critical Hardy-Littlewood-Sobolev Inequality
    Hanwen Lu
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{banerjee2026hardysobolevmazya,
  title = {{Hardy-Sobolev-Maz'ya inequalities for $\boldsymbol1<p<2$}},
  author = {Avas Banerjee and Michał Kijaczko and Vivek Sahu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22540},
  eprint = {2610.22540},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22540},
}
RIS
TY  - UNPB
TI  - Hardy-Sobolev-Maz'ya inequalities for 1<p<2
AU  - Avas Banerjee
AU  - Michał Kijaczko
AU  - Vivek Sahu
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.22540
UR  - https://arxiv.org/abs/2610.22540
AB  - We establish Hardy--Sobolev--Maz'ya inequalities in the subquadratic range 1<p<2 for singularities supported on affine subspaces and for arbitrary domains. We first prove a Hardy--Sobolev--Maz'ya inequality for singularities supported on affine subspaces of arbitrary codimension 1≤ k<N, showing that the deficit in the sharp Hardy inequality controls a suitable weighted Sobolev norm. Combining the same convexity estimate with the one-dimensional approach of Frank and Loss in [9], we also establish Hardy--Sobolev--Maz'ya inequalities on arbitrary domains with the mean distance to the boundary and thereby solve a problem posed in [9]. These results complete the previously unresolved subquadratic range 1<p<2 and, together with the known results for p≥ 2, provide a unified Hardy--Sobolev--Maz'ya theory for all 1<p<N associated with plane singularities and general domains.
ER  -