Abstract
In this chapter, we study spectral multiplier operators for Hermite polynomial expansions. First, we consider Meyer’s multiplier theorem, which is one of the most basic and most useful results for Hermite expansions. Then, we consider spectral multipliers of Laplace transform type. In both cases, we prove their boundedness in Lp(γd), for 1 < p < ∞. For the case of spectral multipliers of Laplace transform type, we also study the boundedness in the case p = 1. Finally, we discuss the fact that the Ornstein–Uhlenbeck operator has a bounded holomorphic functional calculus.
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