The operators obtained by taking conditional expectation of continuous time martingale transforms are studied, both on the circle
T
T
and on
R
n
{{\mathbf {R}}^n}
. 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
L
p
(
R
n
)
,
1
>
p
>
∞
{L^p}({{\mathbf {R}}^n}),\,1 > p > \infty
, with constants that depend only on
p
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
L
p
(
R
n
)
,
1
>
p
>
∞
{L^p}({{\mathbf {R}}^n}),\,1 > p > \infty
, with constants independent of
n
n
.