  The use of certain critical-exponent Sobolev norms is an important feature of
methods employed by Taubes to solve the anti-self-dual and similar non-linear
elliptic partial differential equations. Indeed, the estimates one can obtain
using these critical-exponent norms appear to be the best possible when one
needs to bound the norm of a Green's operator for a Laplacian, depending on a
connection varying in a non-compact family, in terms of minimal data such as
the first positive eigenvalue of the Laplacian or the L^2 norm of the curvature
of the connection. Following Taubes, we describe a collection of
critical-exponent Sobolev norms and general Green's operator estimates
depending only on first positive eigenvalues or the L^2 norm of the
connection's curvature. Such estimates are particularly useful in the gluing
construction of solutions to non-linear partial differential equations
depending on a degenerating parameter, such as the approximate, reference
solution in the anti-self-dual or PU(2) monopole equations. We apply them here
to prove an optimal slice theorem for the quotient space of connections. The
result is optimal in the sense that if a point [A] in the quotient space is
known to be just L^2_1-close enough to a reference point [A_0], then the
connection A can be placed in Coulomb gauge relative to the connection A_0,
with all constants depending at most on the first positive eigenvalue of the
covariant Laplacian defined by A_0 and the L^2 norm of the curvature of A_0. In
this paper we shall for simplicity only consider connections over
four-dimensional manifolds, but the methods and results can adapted to
manifolds of arbitrary dimension to prove slice theorems which apply when the
reference connection is allowed to degenerate.
