Self-improving properties of weighted norm inequalities on metric measure spaces

This work discusses self-improving properties of the Muckenhoupt condition and weighted norm inequalities for the Hardy-Littlewood maximal function on metric measure spaces with a doubling measure. Our main result provides direct proofs of these properties by applying a Whitney covering argument and a technique inspired by the Calderón-Zygmund decomposition. In particular, this approach does not rely on reverse Hölder inequalities.