A frequency localized maximum principle and its applications to time fractional equations with rough coefficients
Abstract
We establish a frequency localized maximum principle for parabolic second-order partial differential equations with the Caputo time fractional derivative ∂tα of order α∈(0,1). The leading coefficients are uniformly elliptic, measurable in time, and independent of space. We construct suitable equivalent norms on frequency localized subspaces of L∞(ℝd), for which a frequency localized maximum principle holds. The corresponding result may fail in the usual L∞ norm even for the classical heat equation. As an application, we prove existence, uniqueness, and regularity estimates in Sobolev-Zygmund spaces with Muckenhoupt weights in time for equations with spatially variable coefficients and zero initial data. The coefficients are strongly measurable in time with values in a Hölder-Zygmund space of positive order and uniformly bounded in this space.
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BibTeX
@misc{choi2026frequency,
title = {{A frequency localized maximum principle and its applications to time fractional equations with rough coefficients}},
author = {Jae-Hwan Choi and Junhee Ryu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24476},
eprint = {2610.24476},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24476},
}
RIS
TY - UNPB
TI - A frequency localized maximum principle and its applications to time fractional equations with rough coefficients
AU - Jae-Hwan Choi
AU - Junhee Ryu
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24476
UR - https://arxiv.org/abs/2610.24476
AB - We establish a frequency localized maximum principle for parabolic second-order partial differential equations with the Caputo time fractional derivative ∂_t^α of order α∈(0,1). The leading coefficients are uniformly elliptic, measurable in time, and independent of space. We construct suitable equivalent norms on frequency localized subspaces of L_∞(ℝ^d), for which a frequency localized maximum principle holds. The corresponding result may fail in the usual L_∞ norm even for the classical heat equation. As an application, we prove existence, uniqueness, and regularity estimates in Sobolev-Zygmund spaces with Muckenhoupt weights in time for equations with spatially variable coefficients and zero initial data. The coefficients are strongly measurable in time with values in a Hölder-Zygmund space of positive order and uniformly bounded in this space.
ER -