  In this paper, we establish the Fefferman--Stein type inequality for area
integral and non-tangential maximal function on the Shilov boundary studied by
Nagel and Stein in 2004. The technique here is inspired by Fefferman--Stein
(1972) and Merryfield (1985) but we bypass the use of Fourier or group
structure as these were not available on the polynomial domains of finite type.
Direct applications include the maximal function characterisation of product
Hardy space and the weak type endpoint estimate for product Calder\'on--Zygmund
operators (such as the Cauchy--Szeg\H{o} projection) on the Shilov boundary.
