Growth and nonvanishing of restricted Siegel modular forms arising as Saito-Kurokawa lifts

We study the analytic behavior of the restriction of a Siegel modular form to $\Bbb{H}\times\Bbb{H}$ in the case that the Siegel form is a Saito-Kurokawa lift. A formula of Ichino links this behavior to a family of ${\rm GL}_3 \times {\rm GL}_2$ $L$-functions.