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Potential estimates and improved Sobolev regularity for nonlocal fully nonlinear elliptic equations

  • Yao Zhang
  • Lingwei Ma
  • Zhenqiu Zhang
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Abstract

This paper investigates pointwise and oscillatory potential estimates for viscosity solutions and their gradients to a class of nonlocal fully nonlinear elliptic equations. The key ingredient of our argument is to establish precise pointwise bounds for a novel fractional sharp maximal function defined in terms of affine functions. As an immediate consequence, we prove higher order Sobolev regularity results of Calderón-Zygmund type for such equations without imposing any global boundedness assumptions on viscosity solutions. The potential theory developed herein extends the work of Daskalopoulos-Kuusi-Mingione (Commun. Partial Differ. Equ. 39: 574-590, 2014) for fully nonlinear second order elliptic equations to the fractional fully nonlinear setting.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026potential,
  title = {{Potential estimates and improved Sobolev regularity for nonlocal fully nonlinear elliptic equations}},
  author = {Yao Zhang and Lingwei Ma and Zhenqiu Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31082},
  eprint = {2610.31082},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31082},
}
RIS
TY  - UNPB
TI  - Potential estimates and improved Sobolev regularity for nonlocal fully nonlinear elliptic equations
AU  - Yao Zhang
AU  - Lingwei Ma
AU  - Zhenqiu Zhang
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31082
UR  - https://arxiv.org/abs/2610.31082
AB  - This paper investigates pointwise and oscillatory potential estimates for viscosity solutions and their gradients to a class of nonlocal fully nonlinear elliptic equations. The key ingredient of our argument is to establish precise pointwise bounds for a novel fractional sharp maximal function defined in terms of affine functions. As an immediate consequence, we prove higher order Sobolev regularity results of Calderón-Zygmund type for such equations without imposing any global boundedness assumptions on viscosity solutions. The potential theory developed herein extends the work of Daskalopoulos-Kuusi-Mingione (Commun. Partial Differ. Equ. 39: 574-590, 2014) for fully nonlinear second order elliptic equations to the fractional fully nonlinear setting.
ER  -