Almost Price's law in Schwarzschild and decay estimates in Kerr for Maxwell field

Abstract: We consider in this work the asymptotics of a Maxwell field in Schwarzschild and Kerr spacetimes. In any subextremal Kerr spacetime, we show energy and pointwise decay estimates for all components under an assumption of a basic energy and Morawetz estimate for spin $\pm 1$ components. If restricted to slowly rotating Kerr, we utilize the basic energy and Morawetz estimates proven in an earlier work to further improve these decay estimates such that the total power of decay for all components of Maxwell field is $-7/2$. In the end, depending on if the Newman--Penrose constant vanishes or not, we prove almost sharp Price's law decay $\tau^{-5+}$ (or $\tau^{-4+}$) for Maxwell field and $\tau^{-\ell -4+}$ (or $\tau^{-\ell -3+}$) for any $\ell$ mode of the field towards a static solution on a Schwarzschild background. All estimates are uniform in the exterior of the black hole.