An alternative proof of $$L^q$$ L q – $$L^r$$ L r estimates of the Oseen semigroup in higher dimensional exterior domains

$$L^q$$ L q – $$L^r$$ L r decay estimates of the Oseen semigroup in n-dimensional exterior domains were well established by Kobayashi and Shibata (Math Ann 310:1–45, 1998) ( $$n=3$$ n = 3 ), Enomoto and Shibata (J Math Fluid Mech 7:339–367, 2005) ( $$n\ge 3$$ n ≥ 3 ) and Maekawa (J Inst Math Jussieu, 2019, https://doi.org/10.1017/s1474748019000355) ( $$n=2$$ n = 2 ). The same result has been recently proved by the present author (Hishida in Math Ann 372:915–949, 2018, Arch Ration Mech Anal 238:215–254, 2020) for a generalized Oseen evolution operator in 3-dimensional exterior domains, where rotation as well as translation of a rigid body is taken into account and, moreover, both translational and angular velocities can be time-dependent. The approach developed there can be considerably simplified if both the non-autonomous character and rotation are absent. As a consequence, an alternative short proof of decay estimates of the Oseen semigroup can be available without relying on analysis of the resolvent and the argument works for $$n\ge 3$$ n ≥ 3 as well. I thus believe that the presentation of the proof would be worth publishing here.