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May 27, 2026Open Access

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

Key Points

This research aims to derive higher-order commutator estimates for the pressure Hessian in incompressible fluid dynamics.
Utilized a localized Littlewood-Paley decomposition.
Employed the Calderón-Zygmund framework to establish sharp bounds.
Focused on fractional Sobolev spaces H(R) for s > 5/2.
Proved sharp bounds for the commutator [R R , u · ∇].
Provided geometric control over localized oscillations in the pressure field.
Avoided structural smallness assumptions on the velocity field.

Abstract

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.

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