Mellin transform approach to multidimensional Carlson-type theorems

This paper presents multidimensional extensions of the classical Carlson theorem on zeros of entire functions. We establish new uniqueness theorems for holomorphic functions in several complex variables that vanish on discrete cones of arbitrary rank. Our approach provides a unified framework, grounded in the application of the Mellin transform, which generalizes and connects previously known results by Avanissian-Gay and Sargos-Morimoto. The main result is the proof that a function, holomorphic in a tubular domain over a rational cone and satisfying specific growth conditions, must be identically zero if it vanishes on a discrete cone. The developed method yields sharp exponential type estimates that ensure uniqueness. These findings have potential applications in problems of analytic continuation, interpolation theory, and the analysis of multidimensional difference equations.