Fourth-order operators with unbounded coefficients in $L^1$ spaces

We prove that operators of the form $A=-a(x{)}^2{\mathrm{\Delta }}^2$, with suitable growth conditions on the coefficient $a(x)$, generate analytic semigroups in $L^1({\mathbb{R}}^N)$. In particular, we deduce generation results for the operator $A:=-(1+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{\alpha }{\mathrm{\Delta }}^2$, $0\le \alpha \le 2$. Moreover, we characterise the maximal domain of $A$ in $L^1({\mathbb{R}}^N)$.