In the first half of the 20th century, the initial study of so-called the Sobolev type inequalities was motivated by the question- Can one control the size of a function by the size of its gradient in the higher dimensional Euclidean space? Later in the second half, the Sobolev type inequalities found applications in proving some embedding theorems associated with the Sobolev space, and then to study the local behavior of solutions of certain elliptic partial differential equations, such as to prove the Harnack inequalities and the Holder’s continuity. The Poincaré type inequalities were also studied together due to the similar phenomena and applications as of the Sobolev type inequalities.

At the earlier developmental stage, the Sobolev and Poincaré inequalities were established associated to the metric balls, more precisely to the Euclidean balls. The richness of applications of these inequalities in metric spaces motivated mathematicians to investigate such inequalities in various complicated geometrical structures. One of such geometrical structures is the space of homogeneous type, that is, the quasi-metric space equipped with the doubling measure.

The Poincaré and Sobolev type inequalities in the quasi-metric spaces are studied deeply throughout the first two decades of this 21st century. In 2008, G. Tian and X.J. Wang investigated Sobolev inequalities in a space of homogeneous type so-called the Monge-Ampère quasi-metric structure, for the first time, based on the geometry of the Monge-Ampère sections studied by Caffarelli and Gutierrez in 1990s. Later in 2014, D. Maldonado developed Poincaré inequalities under the minimal assumptions in the Monge-Ampère quasi-metric structure.

In this dissertation we first focus on improving the known Poincaré inequalities in the Monge-Ampère quasi-metric structure by weakening the hypotheses, for instance with cheaper

assumptions on the Monge-Ampère measure, and then develop new such inequalities by imposing some stronger conditions on the Monge-Ampère measure. Finally, we present the application of these Poincaré inequalities in establishing the corresponding Sobolev inequalities. The proofs of both Poincaré and Sobolev inequalities developed earlier in the Monge-Ampère quasi-metric structure involve the Green’s functions. We use a completely different approach to establish such inequalities, which is proudly a novelty of our work. Towards the end of this dissertation we study the geometry of the Monge-Ampère sections in the form of the Whitney decomposition.