Parole chiave

**Abstract** In this talk I will be discussing a unique continuation problem for $\overline\partial$: $|\overline\partial u|\leq V|u|$. We will determine under what (sharp) conditions on the potential $V$, so that the solution of the inequality is identically zero whenever it is zero in an (small) open set. As a particular application, we will prove that for any smooth function with compact support $\phi\in C_0^\infty(\mathbb{C}^n)$ it is true that
$$\int_{\phi\not=0}\bigg|\frac{\overline\partial \phi}{\phi}\bigg|^2=\infty,$$
and there is a $\psi\in C_0^\infty(\mathbb{C}^n)$ so that it is true that
$$\int_{\psi\not=0}\bigg|\frac{\overline\partial \psi}{\psi}\bigg|^p<\infty$$
for any $p<2$. (Based on a joint work with Yuan Zhang)