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The Cauchy problem for the Maxwell-Klein-Gordon equations in Lorenz gauge in$n$ space dimensions ($n \ge 2$) is locally well-posed for low regularity data,in two and three space dimensions even for data without finite energy. Theresult relies on the null structure for the main bilinear terms which was shownto be not only present in Coulomb gauge but also in Lorenz gauge by Selberg andTesfahun, who proved global well-posedness for finite energy data in threespace dimensions. This null structure is combined with product estimates forwave-Sobolev spaces given systematically by d'Ancona, Foschi and Selberg.

Low regularity local well-posedness for the (N+1)-dimensional Maxwell-Klein-Gordon equations in Lorenz gauge