An application of the regularized Siegel-Weil formula on unitary groups to a theta lifting problem

Let $U(2)$ and $U(2,1)$ be the pair of unitary groups over a global field $F$ and $\pi$ an irreducible cuspidal representation of $U(2)$ which satisfies a certain $L$-function condition. By using a regularized Siegel-Weil formula, we can show that the global theta lifting of $\pi$ in $U(2,1)$ is non-trivial if every local factor $\pi _{\upsilon}$ of $\pi$ has a local theta lifting (Howe lifting) in $U(2,1)(F_{\upsilon})$.