  The connection between Yang--Mills gauge fields on $4$-dimensional orientable
compact Riemannian manifolds and modified L\'evy Laplacians is studied. A
modified L\'evy Laplacian is obtained from the L\'evy Laplacian by the action
of an infinite dimensional rotation. Under the assumption that the 4-manifold
has a nontrivial restricted holonomy group of the bundle of self-dual 2-forms,
the following is proved. There is a modified L\'evy Laplacian such that a
parallel transport in some vector bundle over the 4-manifold is a solution of
the Laplace equation for this modified L\'evy Laplacian if and only if the
connection corresponding to the parallel transport satisfies the Yang--Mills
self-duality (anti-self-duality) equations. An analogous connection between the
Laplace equation for the L\'evy Laplacian and the Yang--Mills equations was
previously known.
