On Calderon-Zygmund Commutator Estimates for the Incompressible Pressure Hessian in Fractional Sobolev Spaces

In this paper, we establish novel higher-order commutator estimates for the pressure Hessian associated with three-dimensional incompressible fluid flows. By employing a localized Littlewood-Paley decomposition and utilizing the classic Calder´on-Zygmund framework, we prove sharp bounds for the commutator R R , u · ∇ in fractional Sobolev spaces H (R ) for s > 5/2. These estimates provide an intrinsic geometric control over the localized oscillations of the pressure field without enforcing structural smallness assumptions on the velocity field.