A class of weight modules over the twisted Heisenberg-Virasoro algebra and gap-$p$ Virasoro algebras

In this paper, we construct a class of simple weight modules over the twisted Heisenberg-Virasoro algebra and gap-$p$ Virasoro algebras from restricted modules over some positive part subalgebra of the twisted Heisenberg-Virasoro algebra. These modules are new. In particular, when $p=2$, the gap-$p$ Virasoro algebra is the mirror Heisenberg-Virasoro algebra and we obtain many new simple weight modules for the mirror Heisenberg-Virasoro algebra.