Keller maps and holonomic $D$-modules

Abstract: Let $\kk$ be a field of characteristic $p$ and $R=\kk[x_1,\dots,x_n]$. Let $D$ denote the ring of $\kk$-linear differential operators on $R$. Let $\varphi:R\to R$ be a $\kk$-algebra endomorphism whose Jacobian is a unit in $R$. We show that there is a unique $\kk$-algebra endomorphism $Φ:D\to D$ such that $Φ|_R=\varphi$ and that the restriction of scalar via $Φ$ preserves holonomicity. We also construct an example of a $\kk$-algebra endomorphism of $D$ that does not preserve holonomicity.