A Mixed Radial--Angular Littlewood--Paley Criterion for the 3D Navier--Stokes Equations

We construct a linear decomposition of vector fields on R3 that refines the standard homogeneous Littlewood–Paley dyadic frequency decomposition by a further splitting of each dyadic shell into spherical-harmonic angular sectors. The decomposition is orthogonal in L2 and satisfies a Bernstein-type inequality with a constant independent of the angular degree ℓ, a consequence of the L2-contractivity of the spectral projections on the sphere. By composing this decomposition with the Kozono–Taniuchi BMO refinement of the Beale–Kato–Majda criterion, we obtain a conditional regularity criterion for the 3D incompressible Navier–Stokes equations expressed solely in terms of the mixed radial–angular spectrum of the vorticity. The nonlinear transfer tensor is isolated and expressed as a product of angular Gaunt coefficients and radial coupling constants. We prove that the balanced Gaunt coefficient decays only as O(ℓ−1/2), which is insufficient for the required summability. Moreover, we show that while the radial coupling is geometrically local for non‑resonant frequency interactions, the resonant regime is not controlled by routine paraproduct arguments. The remaining analytical challenge is encapsulated in a summability hypothesis on these coefficients, whose verification would imply unconditional regularity; the work's contribution is the reduction and the precise localization of the two obstructions.