Function Spaces on Quantum Tori and Their Applications

Explore function spaces on quantum tori and their applications in this 53-minute conference talk by Xiao Xiong at BIMSA. Delve into the definitions of Sobolev, Besov, and Triebel-Lizorkin spaces on quantum tori, and discover how functional analysis methods are adapted to these spaces. Examine fundamental properties, including the lifting theorem, embedding inequalities, and Littlewood-Paley type characterizations for Besov and Triebel-Lizorkin spaces. Learn about concrete characterizations using Poisson and heat semigroups, as well as differences. Conclude with two practical applications of function space theory: pseudo-differential theory and Connes' noncommutative geometry.