  We study the consistency of four-point functions of half-BPS chiral primary
operators of weight p in four-dimensional N=4 superconformal field theories.
The resulting conformal bootstrap equations impose non-trivial bounds for the
scaling dimension of unprotected local operators transforming in various
representations of the R-symmetry group. These bounds generalize recent bounds
for operators in the singlet representation, arising from consistency of the
four-point function of the stress-energy tensor multiplet.
