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An integral representation formula for      ( 0 , q ) (0,q)       forms is constructed on a strictly pseudoconvex domain      D D       in      C n \mathbf {C}^n       by using only the local geometry of the boundary of      D D      . By combining this representation with elementary results about compact operators in Banach spaces, one obtains the solution of the Levi problem and, more importantly, an integral solution operator for      ∂  ¯  \bar {\partial }       on      D D      . The construction does not need any a priori knowledge of the solvability of      ∂  ¯  \bar {\partial }       and thus allows us to establish fundamental global results by a direct and elementary method.

An elementary integral solution operator for the Cauchy-Riemann equations on pseudoconvex domains in 𝐶ⁿ