A weak second term identity of the regularized Siegel-Weil formula for unitary groups

Following W. T. Gan and S. Takeda, we obtain a weak second term identity of the regularized Siegel-Weil formula for the unitary dual pair $(U(n,n),U(V))$ , where $V$ is a split hermitian space of dimension $2r$ with $r+1\le n \le 2r-1$ . As an application, we obtain a Rallis inner product formula for theta lifts from $U(W)$ to $U(V)$ for a skew-hermitian space $W$ of dimension $n$ .