  This paper discusses some regularity of almost periodic solutions of the
Poisson's equation $-\Delta u = f$ in $\mathbb{R}^n$, where $f$ is an almost
periodic function. It has been proved by Sibuya [Almost periodic solutions of
Poisson's equation. Proc. Amer. Math. Soc., 28:195--198, 1971.] that if $u$ is
a bounded continuous function and solves the Poisson's equation in the
distribution sense, then $u$ is an almost periodic function. In this work, we
relax the assumption of the usual boundedness into boundedness in the sense of
distribution which we refer to as a bounded generalized function. The set of
bounded generalized functions are wider than the set of usual bounded
functions. Then, upon assuming that $u$ is a bounded generalized function and
solves the Poisson's equation in the distribution sense, we prove that this
solution is bounded in the usual sense, continuous and almost periodic.
Moreover, we show that the first partial derivatives of the solution $\partial
u/ \partial x_i$, $i=1, \ldots, n$, are also continuous, bounded, and almost
periodic functions. The technique is based on extending a representation
formula using Green's function for Poisson's equation for solutions in the
distribution sense. Some useful properties of distributions are also shown that
can be used to study other elliptic problems.
