  A strong local form of the ``4/3-law'' in turbulent flow has been proved
recently by Duchon and Robert for a triple moment of velocity increments
averaged over both a bounded spacetime region and separation vector directions,
and for energy dissipation averaged over the same spacetime region. Under
precisely stated hypotheses, the two are proved to be proportional, by a
constant 4/3, and to appear as a nonnegative defect measure in the local energy
balance of singular (distributional) solutions of the incompressible Euler
equations. Here we prove that the energy defect measure can be represented also
by a triple moment of purely longitudinal velocity increments and by a mixed
moment with one longitudinal and two tranverse velocity increments. Thus, we
prove that the traditional 4/5- and 4/15-laws of Kolmogorov hold in the same
local sense as demonstrated for the 4/3-law by Duchon-Robert.
