Three Revolutions In The Kernel Are Worse Than One

An example is constructed of a purely unrectifiable measure mu for which the singular integral associated to the kernel K(z) = (z) over bar /z(2) is bounded in L-2(mu). The singular integral fails to exist in the sense of principal value mu-almost everywhere. This is in sharp contrast with the results known for the kernel 1/z (the Cauchy transform).