Local derivations on the Lie algebra $W(2,2)$

The present paper is devoted to studying local derivations on the Lie algebra $W(2,2)$ which has some outer derivations. Using some linear algebra methods in \cite{CZZ} and a key construction for $W(2,2)$ we prove that every local derivation on $W(2,2)$ is a derivation. As an application, we determine all local derivations on the deformed ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_3$ algebra.