The Weyl calculus with respect to the Gaussian measure and restricted Lp-Lq boundedness of the Ornstein-Uhlenbeck semigroup in complex time

In this paper, we introduce a Weyl functional calculus a↦a(Q,P) for the position and momentum operators Q and P associated with the Ornstein-Uhlenbeck operator L=−Δ+x⋅∇, and give a simple criterion for restricted Lp-Lq boundedness of operators in this functional calculus. The analysis of this non-commutative functional calculus is simpler than...

Shedding new light on Gaussian harmonic analysis

This dissertation consists out of two rather disjoint parts. One part concerns some results on Gaussian harmonic analysis and the other on an optimization problem in optics. In the first part we study the Ornstein–Uhlenbeck process with respect to the Gaussian measure. We focus on two areas. One is on “Gaussian” analogues of classical results in...