The $L^1$-realization of an elliptic operator with unbounded coefficients

We establish an $L^1$-theory for a class of elliptic operators with unbounded coefficients, extending the semigroup generation results of Boutiah et al. (J. Differential Equations 264 (2018), 2184–2204) and Canale et al. (Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), 581–601) to the endpoint $p=1$. The $L^1$-realization generates a positive analytic $C_0$-semigroup that is compact for every positive time. We characterize the generator domain through weighted estimates and prove consistency with the corresponding $L^p$-semigroups for $1<p<\mathrm{\infty }$.