A measure on a metric space is said to be doubling if the measure of any ball, of whatever size, is comparable to the measure of the ball of same center and half radius. This is one of the fundamental properties of the Lebesgue measure. If a measure is doubling only on balls of uniformly bounded radii, it is called locally doubling. In this proposal, we focus on three central topics in harmonic analysis, whose environment is a smooth manifold endowed with a locally doubling measure which depends on a positive function, usually referred to as "weight".  These topics are closely related to certain differential operators on the manifold, called weighted Laplacians, which play a similar role to that of the Laplacian in the Euclidean setting. 1) Function spaces on Lie groups. Among these, we shall study embedding theorems and algebra properties of Sobolev, Besov and Triebel-Lizorkin spaces, and their role in the study of nonlinear PDEs associated with invariant operators on the groups. 2) Hardy spaces on weighted manifolds. In particular, we shall consider the Euclidean space endowed either with the Gauss measure or with the measure whose density is the reciprocal of a Gaussian, and study some related singular integrals. 3) Spectral properties of weighted sub-Laplacians on stratified groups. We shall investigate some necessary and/or sufficient conditions to the discreteness of their spectra, by studying some associated Schrödinger-type operators.