In this paper, we introduce a new space that generalizes the \(\phi \) -Hilfer space with the \(\xi (\cdot )\) -Laplacian operator, denoted \((\phi ,{\xi }(\cdot ))\) -HFDS. We refer to this new space as the \(\phi \) -fractional space with anisotropic \(\overrightarrow{\xi }(\cdot )\) -Laplacian operator, abbreviated as \((\phi ,\overrightarrow{\xi }(\cdot ))\) -HFDAS. We prove that \((\phi ,\overrightarrow{\xi }(\cdot ))\) -HFDAS is a separable, and reflexive Banach space. Furthermore, we extend some well-known properties and embedding results of the \((\phi ,\xi (\cdot ))\) -HFDS space to \((\phi ,\overrightarrow{\xi }(\cdot ))\) -HFDAS. Moreover, we illustrate an application of \((\phi ,\overrightarrow{\xi }(\cdot ))\) -HFDAS by solving a differential equation via variational methods.
