Beijing Institute of Mathematical Sciences and Applications

This course presents semigroup methods in harmonic analysis, with an emphasis on Markov semigroups and their links to classical real-variable techniques. We begin with the heat and Poisson semigroups on $\mathbb{R}^d$ (kernels, subordination, and basic estimates), then introduce Markov semigroups on probability spaces $(E,\mathcal{B},\mu)$ through a range of examples. After review of Euclidean Calder\'on--Zygmund theory---in particular the Riesz transforms and their $L^p$ and weak-type bounds---we turn to Gaussian harmonic analysis for the Ornstein--Uhlenbeck operator, where Lebesgue measure is replaced by the Gaussian measure and the Mehler kernel plays the role of the heat kernel.