Certain square functions on product spaces

We study certain singular integral operators and related square functions o n product spaces R-n x R-m (n greater than or equal to 2, m greater than or equal to 2), whose integral kernels are obtained from kernels which are ho mogeneous in each factor R-n and R-m and locally in L-q (q > 1) away from R -n x {0} and {0} x R-m by means of polynomial distortions in the radial var iables. We prove that, if the kernels satisfy the mean zero condition (1.1) , the singular integral operators and square functions are bounded on L-p ( R-n x R-m), p epsilon (1, infinity), and the bounds are independent of the coefficients of the polynomials.