  Let $(A,\mathfrak{m})$ be a complete equicharacteristic Noetherian domain of
dimension $d + 1 \geq 2$. Assume $k = A/\mathfrak{m}$ 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 $J \neq 0$. Let $ f \in J$ with $f
\neq 0$. Set $R = A_f$. Then $R$ is a regular domain of dimension $d$. We show
$R$ contains naturally a field $\ell \cong k((X))$. Let $\mathfrak{g}$ be the
set of $\ell$-linear derivations of $R$ and let $D(R)$ be the subring of
$Hom_\ell(R,R)$ generated by $\mathfrak{g}$ and the multiplication operators
defined by elements in the ring $R$. We show that $D(R)$, the ring of
$\ell$-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.
