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Gradient self-improvement for mixed local and nonlocal parabolic equations
Abstract
We introduce a new method for studying gradient higher integrability for mixed local and nonlocal parabolic equations. More precisely, for p>2d/(d+2) and s ∈ (0,1), we prove that if the inhomogeneity F ∈ Lp(1+σ)loc for some σ>0, then every weak solution satisfies ∇ u ∈ Lp(1+eps)loc for some eps>0, together with quantitative local estimates. The proof relies on stopping time arguments and an intrinsic Calderón-Zygmund-type covering decomposition in which the inhomogeneity and nonlocal energy contributions are treated by classical and fractional maximal estimates.
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BibTeX
@misc{nakamura2026gradient,
title = {{Gradient self-improvement for mixed local and nonlocal parabolic equations}},
author = {Kenta Nakamura},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13722},
eprint = {2609.13722},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13722},
}
RIS
TY - UNPB
TI - Gradient self-improvement for mixed local and nonlocal parabolic equations
AU - Kenta Nakamura
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13722
UR - https://arxiv.org/abs/2609.13722
AB - We introduce a new method for studying gradient higher integrability for mixed local and nonlocal parabolic equations. More precisely, for p>2d/(d+2) and s ∈ (0,1), we prove that if the inhomogeneity F ∈ L^p(1+σ)_loc for some σ>0, then every weak solution satisfies ∇ u ∈ L^p(1+eps)_loc for some eps>0, together with quantitative local estimates. The proof relies on stopping time arguments and an intrinsic Calderón-Zygmund-type covering decomposition in which the inhomogeneity and nonlocal energy contributions are treated by classical and fractional maximal estimates.
ER -