This paper especially focuses on a general estimate, called  $(f-\mathcal{M}{)}^k$, for the $\overline{\partial }$-Neumann problem

$$(f-\mathcal{M}{)}^k\Vert f(\Lambda )\mathcal{M}u{\Vert }^2\le c(\Vert \overline{\partial }u{\Vert }^2+\Vert {\overline{\partial }}^{\ast }u{\Vert }^2+\Vert u{\Vert }^2)+C_{\mathcal{M}}\Vert u{\Vert }_{-1}^2$$

for any $u\in C_c^{\mathrm{\infty }}(U\cap \overline{\mathrm{\Omega }}{)}^k\cap \text{Dom}({\overline{\partial }}^{\ast })$, where f(Λ) is the tangential pseudodifferential operator with symbol $f((1+|\xi {|}^2{)}^{\frac{1}{2}})$,  $\mathcal{M}$ is a multiplier, and U is a neighborhood of a given boundary point $z_0$. Here the domain Ω is q-pseudoconvex or q-pseudoconcave at $z_0$. We want to point out that under a suitable choice of f and $\mathcal{M}$, $(f-\mathcal{M}{)}^k$ is the subelliptic, superlogarithmic, compactness and so on. Generalizing the Property (P) by Catlin (1984), we define Property $(f-\mathcal{M}-P{)}^k$. The result we obtain in here is: Property $(f-\mathcal{M}-P{)}^k$ yields the $(f-\mathcal{M}{)}^k$ estimate. The paper also aims at exhibiting some relevant classes of domains which enjoy Property $(f-\mathcal{M}-P{)}^k$.