The kinetic Kato square root problem

Abstract

We solve the kinetic Kato square root problem for the whole-space Kolmogorov operator $$H_A=\partial_t-X\cdot\nabla_Y-\operatorname{div}_X(A\nabla_X)$$ on \(\mathrm L^2\) 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.