PreprintCC BY 4.0arXiv
Harnack-type estimates for nonlocal double phase functionals
Abstract
We establish Harnack-type inequalities for local minimizers of nonlocal double phase functionals, involving an explicit two-radius positive-part tail and a negative far-field tail. The key difficulty is the mismatch between the natural (p,q)-growth scales in the available modular supremum estimate and the low integrability exponent provided by the weak Harnack inequality. To overcome this obstacle, we establish a concentric modular estimate and derive from it a local upper estimate valid for every integrability exponent. To the best of our knowledge, our results provide the first Harnack-type theory for such nonlocal double phase functionals under the natural structural conditions.
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BibTeX
@misc{fang2026harnacktype,
title = {{Harnack-type estimates for nonlocal double phase functionals}},
author = {Yuzhou Fang and Chao Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27957},
eprint = {2609.27957},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27957},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Harnack-type estimates for nonlocal double phase functionals
AU - Yuzhou Fang
AU - Chao Zhang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27957
UR - https://arxiv.org/abs/2609.27957
AB - We establish Harnack-type inequalities for local minimizers of nonlocal double phase functionals, involving an explicit two-radius positive-part tail and a negative far-field tail. The key difficulty is the mismatch between the natural (p,q)-growth scales in the available modular supremum estimate and the low integrability exponent provided by the weak Harnack inequality. To overcome this obstacle, we establish a concentric modular estimate and derive from it a local upper estimate valid for every integrability exponent. To the best of our knowledge, our results provide the first Harnack-type theory for such nonlocal double phase functionals under the natural structural conditions.
ER -