The Kato square root property for fractional elliptic operators in divergence form
Abstract
We establish the Kato square root property for fractional elliptic operators L = −divα(μ∇α·) of order 2α∈ (0,2) on ℝd, built from the Riesz fractional gradient and divergence, with bounded, measurable and uniformly elliptic complex coefficients μ. More precisely, we show that the domain of L1/2 coincides with the distributional fractional Sobolev space Sα,2(ℝd) and that |L1/2 u|L2 ≃ |∇αu|L2, with constants that are locally uniform in α. As a consequence, we recover the Kato square root property for elliptic operators −div(μ∇·) in the limit α→ 1, and we obtain continuity of the square root with respect to the parameter α. The proof follows the second-order approach to the Kato problem. Its two new ingredients are nonlocal off-diagonal estimates of polynomial order for the resolvents of L, and a construction of T(b)-test functions that circumvents the nonlocality of ∇α by factorizing it through Riesz potentials.
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BibTeX
@misc{haardt2026kato,
title = {{The Kato square root property for fractional elliptic operators in divergence form}},
author = {Luca Haardt},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06391},
eprint = {2611.06391},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06391},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Kato square root property for fractional elliptic operators in divergence form
AU - Luca Haardt
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06391
UR - https://arxiv.org/abs/2611.06391
AB - We establish the Kato square root property for fractional elliptic operators L = −div^α(μ∇^α·) of order 2α∈ (0,2) on ℝ^d, built from the Riesz fractional gradient and divergence, with bounded, measurable and uniformly elliptic complex coefficients μ. More precisely, we show that the domain of L^1/2 coincides with the distributional fractional Sobolev space S^α,2(ℝ^d) and that |L^1/2 u|_L^2 ≃ |∇^αu|_L^2, with constants that are locally uniform in α. As a consequence, we recover the Kato square root property for elliptic operators −div(μ∇·) in the limit α→ 1, and we obtain continuity of the square root with respect to the parameter α. The proof follows the second-order approach to the Kato problem. Its two new ingredients are nonlocal off-diagonal estimates of polynomial order for the resolvents of L, and a construction of T(b)-test functions that circumvents the nonlocality of ∇^α by factorizing it through Riesz potentials.
ER -