Sato's hyperfunctions are known to be represented as the boundary values of harmonic functions as well as those of holomorphic functions. The author obtains a bijective Poisson mapping \[ P\colon\mathcal S^*{}'(\mathbb R^n)ongrightarrow\mathcal S^*{}'(S^*\mathbb R^n)\cap H(S^*\mathbb R^n) \] where $H(S^*\mathbb R^n)$ is a kind of Hardy subspace of $\mathcal B(S^*\mathbb R^n)$. Moreover, the author has an isomorphism between Sobolev spaces \[ P\colon W^s(\mathbb R^n)ongrightarrow W^{s+(n-1)/4}(S^*\mathbb R^n)\cap H(S^*\mathbb R^n). \] There are some similar results in case of other functions.