Unique continuation property for $\bar{\partial }$ with $L^{2n}$ potentials

Abstract

We establish the sharp unique continuation property for the Cauchy--Riemann operator $\bar{\partial }$ in higher dimensions. Let $0\inΩ\subset \C^n$ be a domain. Suppose $u \in W_{\mathrm{loc}}^{1,2}(Ω; \C^N)$ satisfies the differential inequality \[ |\bar{\partial} u| \leq V |u| \quad \text{a.e.~on } Ω \] with $V\in L_{\mathrm{loc}}^{2n}(\Omega )$. If $u$ is $L^2$ flat at the origin, then $u\equiv 0$ on $\Omega$.

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