Hypercontractivity Inequality on Epsilon-Product Spaces

Explore the concept of hypercontractivity inequality on ε-product spaces in this comprehensive lecture by Siqi Liu from UC Berkeley. Delve into the properties of the noise operator T_ρ over L^q spaces and understand the implications of (p,q)-hypercontractivity inequalities. Learn how hypercontractivity characterizes the smoothing effect of the noise operator on functions in various spaces, including Boolean hypercubes, Boolean slices, Grassmann schemes, and symmetric groups. Discover the applications of these results in small-set expansion theorems, level-k inequalities, and the KKL theorem. Examine the (2,4)-hypercontractivity for general ε-product spaces, defined by distributions μ over random variables with bounded correlation. Understand how this definition generalizes previous results and captures new spaces like spectral high-dimensional expanders. Gain insights from the joint work of Siqi Liu, Tom Gur, and Noam Lifshitz in this advanced exploration of hypercontractivity and its applications in mathematics and theoretical computer science.