Embedding of infinitely differentiable functions into the space of hyperfunctions via Čech–Dolbeault cohomology and its inverse map

Honda, Izawa and Suwa define Čech–Dolbeault representation of hyperfunctions and an embedding of distributions to the space of hyperfunctions. With this embedding, we can regard $C^{\infty}$ functions as hyperfunctions in the framework of Čech–Dolbeault cohomology. This article aims to characterize a Čech–Dolbeault representative which corresponds to the image of the embedding of a $C^{\infty}$ function, and also to construct the inverse map of the embedding of $C^{\infty}$ functions.