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The use of certain critical-exponent Sobolev norms is an important feature ofmethods employed by Taubes to solve the anti-self-dual and similar non-linearelliptic partial differential equations. Indeed, the estimates one can obtainusing these critical-exponent norms appear to be the best possible when oneneeds to bound the norm of a Green's operator for a Laplacian, depending on aconnection varying in a non-compact family, in terms of minimal data such asthe first positive eigenvalue of the Laplacian or the L^2 norm of the curvatureof the connection. Following Taubes, we describe a collection ofcritical-exponent Sobolev norms and general Green's operator estimatesdepending only on first positive eigenvalues or the L^2 norm of theconnection's curvature. Such estimates are particularly useful in the gluingconstruction of solutions to non-linear partial differential equationsdepending on a degenerating parameter, such as the approximate, referencesolution in the anti-self-dual or PU(2) monopole equations. We apply them hereto prove an optimal slice theorem for the quotient space of connections. Theresult is optimal in the sense that if a point [A] in the quotient space isknown to be just L^2_1-close enough to a reference point [A_0], then theconnection 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 thecovariant Laplacian defined by A_0 and the L^2 norm of the curvature of A_0. Inthis paper we shall for simplicity only consider connections overfour-dimensional manifolds, but the methods and results can adapted tomanifolds of arbitrary dimension to prove slice theorems which apply when thereference connection is allowed to degenerate.Comment: LaTeX 2e, 43 pages. Estimates in section 5.2 strengthene

Critical-exponent Sobolev norms and the Slice Theorem for the quotient space of connections