Martingale Transforms and Related Singular

The operators obtained by taking conditional expectation of continuous time martingale transforms are studied, both on the circle T and on R". Using a Burkholder-Gundy inequality for vector-valued martingales, it is shown that the vector formed by any number of these operators is bounded on LP(R"), 1 < p < oo, with constants that depend only on p and the norms of the matrices involved. As a corollary we obtain a recent result of Stein on the boundedness of the Riesz transforms on LP(K'), 1 < p < oo, with constants independent of n. 0. Introduction. For / e Lp(R"), 1 < p < oo, we define the Riesz transforms by Rjf(x)=hm^V2Tt^l)j yJ^ldy jJK .-0 I 2 )Jw>e \y\" + l for j = 1, 2,..., n. These operators are the basic singular integrals in R" and it is well known (see [14]) that if we set