Further evaluation of Wahl vanishing theorems for surface singularities in characteristic $p$

Let $(\mathrm{Spec }R, \frak m)$ be a rational double point defined over an algebraically closed field $k$ of characteristic $p\geq 0$. We evaluate further the dimensions of the local cohomology groups which were treated by Wahl in 1975 as vanishing theorem C (resp. D) under the assumption that $p$ is a very good prime (resp. good prime) with respect to $(\mathrm{Spec }R, \frak m)$. We use Artin's classification of rational double points and completely determine the dimensions $\dim_k H_E^1(S_X)$, $\dim_k H_E^1(S_X\otimes \mathcal O_X(E))$, supplementing Wahl's theorems. In the proof we construct derivations concretely which do not lift to the minimal resolution $X\to \mathrm{Spec }R$, as well as non-trivial equisingular families which inject into a versal deformation of the rational double point $(\mathrm{Spec }R, \mathfrak m)$.