|Sep 18, 2018
Abstract: Decomposing a function is widely used tool to deal with many situations. Among them, one important tool is decomposing some specific functions(Fourier supported in a neighborhood of some hyper-surfaces) with pieces of small supports in the Fourier side, rather than the physical side. After the decomposition, we need to control our original function with decomposed pieces. Precisely, we can estimate the L^p norm of the function with the square mean of decomposed pieces by using Cauchy-Schwarz inequality. However, Cauchy-Schwarz inequality makes large coefficient in front of the square mean. The conjecture is how small this coefficient can be. We will prove the l^2 decoupling conjecture for compact hyper-surfaces with positive definite second fundamental form. Although decoupling conjecture is a weaker version of the square function estimate problem, there are various applications such as discrete restriction phenomena, Strichartz estimates for torus, and some number theory problems. We will see how the decoupling conjecture used in these situations.