In this paper, we study the global boundary regularity of the $$\bar{\partial }$$ ∂ ¯ - equation on an annulus domain $$\Omega $$ Ω between two strictly $$q$$ q -convex domains with smooth boundaries in $$\mathbb{C }^n$$ C n for some bidegree. To this finish, we first show that the $$\bar{\partial }$$ ∂ ¯ -operator has closed range on $$L^{2}_{r, s}(\Omega )$$ L r , s 2 ( Ω ) and the $$\bar{\partial }$$ ∂ ¯ -Neumann operator exists and is compact on $$L^{2}_{r,s}(\Omega )$$ L r , s 2 ( Ω ) for all $$r\ge 0$$ r ≥ 0 , $$q\le s\le n-q- 1$$ q ≤ s ≤ n − q − 1 . We also prove that the $$\bar{\partial }$$ ∂ ¯ -Neumann operator and the Bergman projection operator are continuous on the Sobolev space $$W^{k}_{r,s}(\Omega )$$ W r , s k ( Ω ) , $$k\ge 0$$ k ≥ 0 , $$r\ge 0$$ r ≥ 0 , and $$q\le s\le n-q-1$$ q ≤ s ≤ n − q − 1 . Consequently, the $$L^{2}$$ L 2 -existence theorem for the $$\bar{\partial }$$ ∂ ¯ -equation on such domain is established. As an application, we obtain a global solution for the $$\bar{\partial }$$ ∂ ¯ equation with Hölder and $$L^p$$ L p -estimates on strictly $$q$$ q -concave domain with smooth $$\mathcal C ^2$$ C 2 boundary in $$\mathbb{C }^n$$ C n , by using the local solutions and applying the pushing out method of Kerzman (Commun Pure Appl Math 24:301–380, 1971).