Irreducible tensor product modules over the $W$-algebra $W(2,2)$

In this paper, we construct a class of non-weight modules $\mathrm{\Omega }(\lambda ,\alpha ,h)\otimes \mathrm{\text{Ind}}(V)$ over the $W$-algebra $W(2,2)$ by taking tensor products of modules $\mathrm{\Omega }(\lambda ,\alpha ,h)$ with restricted modules $\mathrm{\text{Ind}}(V)$. Then, we give the necessary and sufficient conditions for the irreducibility of such tensor product modules and determine the conditions under which two irreducible tensor product modules are isomorphic. These non-weight modules are different from known non-weight modules. Finally, we transform some tensor product modules into induced modules from modules of certain subalgebras over the $W$-algebra $W(2,2)$. And the conditions for these induced modules to be irreducible are determined by applying the established results.