Pointwise estimates of the size of characters of compact Lie groups

Pointwise bounds for characters of representations of the classical, compac t, connected, simple Lie groups are obtained which allow us to study the si ngularity of central measures. For example, we find the minimal integer k s uch that any continuous orbital measure convolved with itself k times belon gs to L-2. We also prove that if k = rank G then mu(2k) is an element of L- 1 for all central, continuous measures mu. This improves upon the known cla ssical result which required the exponent to be the dimension of the group G.