Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions

We show that in dimensions $n\ge 6$ that one has global regularity for the Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical Sobolev norm ${\dot{H}}^{n\mathrm{/}2-1}\times {\dot{H}}^{n\mathrm{/}2-2}$ of the initial data is sufficiently small. These results are analogous to those recently obtained for the high-dimensional wave map equation but unlike the wave map equation, the Coulomb gauge non-linearity cannot be iterated away directly. We shall use a different approach, proving Strichartz estimates for the covariant wave equation. This in turn will be achieved by use of Littlewood-Paley multipliers, and a global parametrix for the covariant wave equation constructed using a truncated, microlocalized Cronstrom gauge.