  For multiparameter bilinear paraproduct operators $B$ we prove the estimate
$$
  B: L^p X L^q --> L^r, 1<p,q\le{}\infty. $$ Here, $1/p+1/q=1/r$ and special
attention is paid to the case of $0<r<1$. (Note that the families of
multiparameter paraproducts are much richer than in the one parameter case.)
These estimates are the essential step in the version of the multiparameter
Coifman-Meyer theorem proved by C. Muscalu, J. Pipher, T. Tao, and C. Thiele.
We offer a different proof of these inequalities.
