Unique continuation for a gradient inequality with Lⁿ potential

We establish a unique continuation property for solutions of the differential inequality |∇ u|≤ V|u|, where V is locally Lⁿ integrable on a domain in Rⁿ. A stronger uniqueness result is obtained if in addition the solutions are locally Lipschitz. One application is a finite order vanishing property in the L² sense for the exponential of W1,n functions. We further discuss related results for the Cauchy-Riemann operator ∂ and characterize the vanishing order for smooth extension of holomorphic functions across the boundary.