Decoupling Estimates for the Paraboloid and the Truncated Cone

Decoupling theory has become a cornerstone of modern harmonic analysis. This thesis centers on the detailed proof of the decoupling theorem for the paraboloid, following the groundbreaking work of Bourgain and Demeter. Additionally, we provide a proof of a stronger inequality, the square function estimate for the truncated cone, as established by Guth, Wang, and Zhang; finally, we will discuss various applications of decoupling theory, highlighting the versatility and the importance of this result.