On the ring of differential operators of certain regular domains

Let $(A,\mathfrak{m})$ be a complete equicharacteristic Noetherian domain of dimension $d+1\ge 2$. Assume $k=A\mathrm{/}\mathfrak{m}$ has characteristic zero and that $A$ is not a regular local ring. Let $Sing(A)$ the singular locus of $A$ be defined by an ideal $J$ in $A$. Note $J\mathrm{\ne }0$. Let $f\in J$ with $f \neq 0$\mathrm{.}Set$R = A_f$\mathrm{.}Then$R$isaregulardomainofdimension$d$\mathrm{.}Weshow$R$containsnaturallyafield$\ell \cong k((X))$\mathrm{.}Let$\mathfrak{g}$ be the set of $\mathrm{\ell }$-linear derivations of $R$ and let $D(R)$ be the subring of $Ho{m}_{\mathrm{\ell }}(R,R)$ generated by $\mathfrak{g}$ and the multiplication operators defined by elements in the ring $R$. We show that $D(R)$, the ring of $\mathrm{\ell }$-linear differential operators on $R$, is a left, right Noetherian ring of global dimension $d$. This enables us to prove Lyubeznik’s conjecture on $R$ modulo a conjecture on roots of Bernstein-Sato polynomials over power series rings.