ON HOEFFDING DECOMPOSITION IN Lp

We give a new proof of a result by J. Bourgain which says that if (Omega, F, P) is a product of probability spaces then V-d-the orthonormal in L-2(Omega, F, P) projection on the space spanned by those X is an element of L-2(Omega, F, P) which depend on most of d-variables is a bounded operator in L-p(Omega, F, P) for 1 < p < infinity. We prove that for X is an element of L-p(Omega, F, P) E|V-d(X)|(P) <= C-p,C-d E|X|(P) with C-p,C-d = (C (p) over cap /ln (p) over cap)(dp) where (p) over cap = max{p, p/p-1} and c is an universal constant.