  For a commutative algebra $A$ over $\mathbb{C}$,denote
$\mathfrak{g}=\text{Der}(A)$. A module over the smash product $A\#
U(\mathfrak{g})$ is called a jet $\mathfrak{g}$-module, where $U(\mathfrak{g})$
is the universal enveloping algebra of $\mathfrak{g}$.In the present paper, we
study jet modules in the case of $A=\mathbb{C}[t_1^{\pm 1},t_2]$.We show that
$A\#U(\mathfrak{g})\cong\mathcal{D}\otimes U(L)$, where $\mathcal{D}$ is the
Weyl algebra $\mathbb{C}[t_1^{\pm 1},t_2, \frac{\partial}{\partial
t_1},\frac{\partial}{\partial t_2}]$, and $L$ is a Lie subalgebra of $A\#
U(\mathfrak{g})$ called the jet Lie algebra corresponding to
$\mathfrak{g}$.Using a Lie algebra isomorphism $\theta:L \rightarrow
\mathfrak{m}_{1,0}\Delta$, where $\mathfrak{m}_{1,0}\Delta$ is the subalgebra
of vector fields vanishing at the point $(1,0)$, we show that any irreducible
finite dimensional $L$-module is isomorphic to an irreducible
$\mathfrak{gl}_2$-module. As an application, we give tensor product
realizations of irreducible jet modules over $\mathfrak{g}$ with uniformly
bounded weight spaces.
