This monograph contains a study of the global Cauchy problem for the Yang-Mills equations on \((6+1)\) and higher dimensional Minkowski space, when the initial data sets are small in the critical gauge covariant Sobolev space \(\dot{H}_A^{(n-4)/{2}}\). Regularity is obtained through a certain "microlocal geometric renormalization" of the equations which is implemented via a family of approximate null Crönstrom gauge transformations. The argument is then reduced to controlling some degenerate elliptic equations in high index and non-isotropic \(L^p\) spaces, and also proving some bilinear estimates in specially constructed square-function spaces.

Table of Contents

• Introduction
• Some gauge-theoretic preliminaries
• Reduction to the "main a-priori estimate"
• Some analytic preliminaries
• Proof of the main a-priori estimate
• Reduction to approximate half-wave operators
• Construction of the half-wave operators
• Fixed time \(L^2\) estimates for the parametrix
• The dispersive estimate
• Decomposable function spaces and some applications
• Completion of the proof
• Bibliography