  An explicit Bellman function is used to prove a bilinear embedding theorem
for operators associated with general multi-dimensional orthogonal expansions
on product spaces. This is then applied to obtain $L^p,$ $1<p<\infty,$
boundedness of appropriate vectorial Riesz transforms, in particular in the
case of Jacobi polynomials. Our estimates for the $L^p$ norms of these Riesz
transforms are both dimension-free and linear in $\max(p,p/(p-1)).$ The
approach we present allows us to avoid the use of both differential forms and
general spectral multipliers.
