A new Leray formula for smooth functions on bounded domains in C^n

By means of a new technique of integral representations in C n given by the authors, we establish a new abstract formula with a vector function W for smooth functions on bounded domains in C n , which is different from the well-known Leray formula. This new formula eliminates the term that contains the parameter A from the classical Leray formula, and especially on some domains the uniform estimates for the $\bar \partial - equation$ are very simple. From the new Leray formula, we can obtain correspondingly many new formulas for smooth functions on many domains in C n , which are different from the classical ones, when we properly select the vector function W.

By means of a new technique of integral representations in Cn given by the authors, we establish a new abstract formula with a vector function W for smooth functions on bounded domains in Cn, which is different from the well-known Leray formula, This new formula eliminates the term that contains the parameter λ from the classical Leray formula, and especially on some domains the uniform estimates for the -equation are very simple. From the new Leray formula, we can obtain correspondingly many new formulas for smooth functions on many domains in Cn, which are different from the classical ones, when we properly select the vector function W.