  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 $r+1 \leq n \leq
2r-1$. 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 $n$.
