Quantitative Unique Continuation in the Critical Drift Class

This paper establishes quantitative unique continuation and doubling estimates for parabolic operators with critical drift and potential in the scale-invariant Lorentz classb∈Lt∞Lx3b L^ₜ L³ₓb∈Lt∞Lx3, V∈Lt∞Lx3/2, 1V L^ₜ L^3/2, 1ₓV∈Lt∞Lx3/2, 1. Using a sharp Carleman estimate combined with Lorentz–Sobolev absorption, three-cylinder inequalities, and a good-time selection argument, we derive explicit fixed-time doubling bounds and coefficient-dependent propagation of smallness estimates for subsolutions of ∂tf−νΔf+b⋅∇f+Vf≤0. ₜ f - f + b f + V f 0. ∂tf−νΔf+b⋅∇f+Vf≤0. The results provide scale-critical control of amplification rates and yield an explicit upper doubling exponent depending only on a quantitative coefficient functional KucKₔ₂Kuc. The framework is designed to serve as a black-box interface for geometric and rigidity arguments, in particular for applications to vorticity equations arising in the three-dimensional Navier–Stokes system. All constants, dependencies, and interface outputs are made explicit, and the paper exports two standalone theorems: a fixed-time quantitative doubling inequality and a good-time measure bound for the critical coefficient class.