Our primary objective in this project is to extend results from spaces with doubling volume properties to those with exponential volume growth. Building on successful works on multipliers, our initial focus is exploring the implications of using symbols of broader generality in pseudo-differential operators on Harmonic NA groups. The aim is to identify conditions under which these operators become L^p-bounded, expanding on findings from the multiplier case. This research addresses a gap in pseudo-differential calculus for Harmonic NA groups, involving defining symbol classes, establishing kernel representations, and exploring algebraic structures in this context. Furthermore, we intend to formulate a theory of weighted estimates that integrates the geometric properties of Gromov hyperbolic space and adopts the combinatorial approach proposed by Naor and Tao. More broadly, our aim is to characterize weight class within the context of Harmonic manifolds exhibiting purely exponential volume growth. This exploration will unveil several directions in the field of harmonic analysis, marking an uncharted territory for weighted estimates in the context of spaces with exponential volume growth.