  Using a codimension-$1$ algebraic cycle obtained from the Poincar\'e line
bundle, Beauville defined the Fourier transform on the Chow groups of an
abelian variety $A$ and showed that the Fourier transform induces a
decomposition of the Chow ring $CH^*(A)$. By using a codimension-$2$ algebraic
cycle representing the Beauville--Bogomolov class, we give evidence for the
existence of a similar decomposition for the Chow ring of hyperK\"ahler
varieties deformation equivalent to the Hilbert scheme of length-$2$ subschemes
on a K3 surface. We indeed establish the existence of such a decomposition for
the Hilbert scheme of length-$2$ subschemes on a K3 surface and for the variety
of lines on a very general cubic fourfold.
