Beijing Institute of Mathematical Sciences and Applications

In this course we discuss analysis for the Gaussian measure $\gamma_d$ on $\mathbb{R}^d$ from a semigroup and harmonic–analytic point of view. We begin with the geometry of $\gamma_d$ (global non-doubling and the local-doubling calculus via admissible balls) and the Ornstein–Uhlenbeck operator $L$, introduced through Mehler’s kernel and the associated semigroup. On the spectral side we use Hermite polynomials and the Wiener–It\^o chaos decomposition of $L^2(\gamma_d)$, and we build the basic function spaces $L^p(\gamma_d)$, together with Sobolev and Hardy spaces adapted to $\gamma_d$. Central tools include hypercontractivity and logarithmic Sobolev inequalities, maximal functions, spectral multipliers and the holomorphic functional calculus for $L$, and the behavior of Gaussian Riesz transforms. Throughout, the lack of translation invariance forces techniques different from the Euclidean case; proofs are organized around semigroup methods, covering arguments adapted to $\gamma_d$, and Littlewood–Paley theory in the Gaussian setting. The course prepares participants for modern Gaussian harmonic analysis and for applications in stochastic analysis.