 define the function $f_0: {\mathbb{R}}^k \to {\mathbb{R}}$ by $$f_0(t_0,\dots,t_k) := \int_{s_0 \geq t_0, \dots, s_k \geq t_k} F_0(s_0,\dots,s_k)\ ds_0 \dots ds_k,$$ thus $f_0$ is smooth, not identically zero and supported on the region $$\l