Regularized Periods and the global Gan-Gross-Prasad conjecture : The case of $U(n+2r) \times U(n)

In this paper, we introduce regularized trilinear periods on certain non-reductive groups. It has two direct applications. Firstly, it enables us to define the regularized Bessel periods and the regularized Fourier-Jacobi periods for all classical and metaplectic groups. Secondly, by using the properties of the regularized Fourier-Jacobi periods, we can prove one direction of the full Gan-Gross-Prasad conjecture on skew-hermitian unitary groups.