Sharp $L^2$ estimates for the drift heat equation on shrinking Ricci solitons

We prove an $L^2$ estimate for the drift heat equation on a complete gradient shrinking Ricci soliton. This estimate has a time-dependent weight which is Gaussian in its spatial asymptotics. When transferred and scaled to an estimate for the heat equation along the Ricci flow of the soliton, this estimate is uniform up to the singular time.