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

$${(f-\mathcal M)^{k}} \qquad \| f({\varLambda })\mathcal M u\|^{2}\le c(\|\bar \partial u\|^{2}+\|\bar \partial ^{*}u\|^{2}+\|u\|^{2})+C_{\mathcal M}\|u\|^{2}_{-1} $$

for any $u\in C^{\infty }_{c}(U\cap \bar {\Omega })^{k}\cap \text {Dom}(\bar {\partial }^{*}) $, 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} $.