The kinetic Kato square root problem
Abstract
We solve the kinetic Kato square root problem for the whole-space Kolmogorov operator HA=∂t−X·∇Y−divX(A∇X) on L2 with Lebesgue measure, where A is a bounded measurable complex uniformly accretive matrix depending on all variables. We define its realization through a sesquilinear form and identify the square-root domains of the resulting maximal accretive operator and its adjoint with the intrinsic Sobolev space defined by one velocity derivative and half a derivative along the transport flow. The homogeneous norm equivalences depend only on the dimension and the coefficient bounds. The energy also controls a position derivative of order one third. The proof uses second-order resolvents and localized affine testing. A regularized transport symmetrizer gives hidden coercivity, while comparison with a positive Rockland operator supplies intrinsic smoothing. Commutator and dyadic approximation estimates remove the smoothing before a Carleson argument is applied. We also obtain uniform shifted square-root estimates and bounded velocity, transport and position Riesz transforms.
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BibTeX
@misc{nystrom2026kinetic,
title = {{The kinetic Kato square root problem}},
author = {Kaj Nyström},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03104},
eprint = {2611.03104},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03104},
}
RIS
TY - UNPB
TI - The kinetic Kato square root problem
AU - Kaj Nyström
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03104
UR - https://arxiv.org/abs/2611.03104
AB - We solve the kinetic Kato square root problem for the whole-space Kolmogorov operator H_A=∂_t−X·∇_Y−div_X(A∇_X) on L^2 with Lebesgue measure, where A is a bounded measurable complex uniformly accretive matrix depending on all variables. We define its realization through a sesquilinear form and identify the square-root domains of the resulting maximal accretive operator and its adjoint with the intrinsic Sobolev space defined by one velocity derivative and half a derivative along the transport flow. The homogeneous norm equivalences depend only on the dimension and the coefficient bounds. The energy also controls a position derivative of order one third. The proof uses second-order resolvents and localized affine testing. A regularized transport symmetrizer gives hidden coercivity, while comparison with a positive Rockland operator supplies intrinsic smoothing. Commutator and dyadic approximation estimates remove the smoothing before a Carleson argument is applied. We also obtain uniform shifted square-root estimates and bounded velocity, transport and position Riesz transforms.
ER -