  We study the effect of satellite operations on the Upsilon invariant of
Ozsvath-Stipsicz-Szabo. We obtain results concerning when a knot and its
satellites are independent; for example, we show that the set
$\{D_{2^i,1}\}_{i=1}^\infty$ is a basis for an infinite rank summand of the
group of smooth concordance classes of topologically slice knots, for D the
positive clasped untwisted Whitehead double of any knot with positive
tau-invariant, e.g. the right-handed trefoil. We also prove that the image of
the Mazur satellite operator on the smooth knot concordance group contains an
infinite rank subgroup of topologically slice knots.
