  This paper is a contribution to a program to see symmetry breaking in a
weakly interacting many Boson system on a three dimensional lattice at low
temperature. It is part of an analysis of the "small field" approximation to
the "parabolic flow" which exhibits the formation of a "Mexican hat" potential
well. Bounds on the fluctuation integral covariance, as well as on some other
linear operators, are an important ingredient in our renormalization group step
analysis. These bounds are proven here.
