H^{s,p} estimates for weak solutions of fractional nonlocal equations in divergence form
Abstract
This paper establishes interior Hs,p regularity for weak solutions to fractional nonlocal equations in divergence form with general kernels. Under the $(δ, R)$-vanishing condition on the coefficients, we prove that for p>2, if the source term f∈ Lploc, then the solution u∈ Hs,ploc. In contrast to Byun and Kim \cite{BK2025}, our result holds in the borderline case for f without requiring differentiability assumptions. Our approach simplifies previous methods by using the stability properties of the fractional gradient to derive the good-inequality directly, thereby avoiding reliance on maximal functions. The proof integrates Vitali's covering lemma, Hardy-Littlewood maximal theory, self-improving estimates, and compactness arguments. This work extends classical W1,p regularity to fractional nonlocal settings under minimal kernel assumptions.
Related
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BibTeX
@misc{li2026estimates,
title = {{H^s,p estimates for weak solutions of fractional nonlocal equations in divergence form}},
author = {Zhenjie Li and Lihe Wang and Chunqin Zhou},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07606},
eprint = {2611.07606},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07606},
}
RIS
TY - UNPB
TI - H^{s,p} estimates for weak solutions of fractional nonlocal equations in divergence form
AU - Zhenjie Li
AU - Lihe Wang
AU - Chunqin Zhou
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07606
UR - https://arxiv.org/abs/2611.07606
AB - This paper establishes interior H^s,p regularity for weak solutions to fractional nonlocal equations in divergence form with general kernels. Under the $(δ, R)$-vanishing condition on the coefficients, we prove that for p>2, if the source term f∈ L^p_loc, then the solution u∈ H^s,p_loc. In contrast to Byun and Kim \cite{BK2025}, our result holds in the borderline case for f without requiring differentiability assumptions. Our approach simplifies previous methods by using the stability properties of the fractional gradient to derive the good-inequality directly, thereby avoiding reliance on maximal functions. The proof integrates Vitali's covering lemma, Hardy-Littlewood maximal theory, self-improving estimates, and compactness arguments. This work extends classical W^1,p regularity to fractional nonlocal settings under minimal kernel assumptions.
ER -