  We estimate the upper and lower bounds of the Hewitt$\textbf{-}$Stromberg
dimensions. In particular, these results give new proofs of theorems on the
multifractal formalism which is based on the Hewitt$\textbf{-}$Stromberg
measures and yield results even at points $q$ for which the upper and lower
multifractal Hewitt$\textbf{-}$Stromberg dimension functions differ. Finally,
concrete examples of a measure satisfying the above property are developed.
