In the present work, we revisit several structural properties of almost automorphic and compact almost automorphic functions from the Euclidean space \({\mathbb {R}}^m\) ( \(m \ge 1\) ) with values in a Banach space \({\mathbb {X}}\) . When \({\mathbb {X}}\) is a Banach algebra, it is proven that the spaces formed by these functions are also Banach algebras. As applications, first we prove regularity of almost automorphic solution of Poisson’s equation; that is, we prove that bounded continuous functions with weak (distributional) almost automorphic Laplacian are compact almost automorphic; then, we prove the compact almost automorphy in space variable of classical solutions to heat equation with almost automorphic initial datum.
