Critical Lorentz Bounds for Hankel Multipliers with Slow Phases

For compactly supported radial symbols with a logarithmic slow phase near a sphere, we characterize the bounds L^{p,q}_rad → L^{p,r}_rad at p = 2d/(d + 1) and p = 2d/(d − 1) for every 0 < q,r ≤ ∞. We assume Φ is C², a = Φ′ > 0, a(v) → 0, and |a′(v)| ≲ a(v)/v; neither monotonicity of the speed nor divergence of the phase is required. The criteria involve a power supremum of the speed in the Banach ranges and the phase-sensitive amplitude a cos Φ at subunit indices. Finite limiting phases can therefore change boundedness at a fixed speed, and the two critical exponents have different equality cases. Both radial strong endpoints are bounded exactly when sup_v va(v) < ∞. The proofs combine quantitative kernel asymptotics, resonant radial inputs, and weighted Hardy estimates. We distinguish endpoint boundedness from kernel Lorentz integrability and a classical fractional-derivative necessary condition. A smooth phase with narrow speed spikes shows that positivity and decay of the speed alone do not preserve the pointwise criterion, even when the planar Fourier kernel is integrable.