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 L p(γ d), 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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Notes
- 1.
Formally speaking, it should be denoted by m(−L) because of (2.7); for simplicity we just write it as m(L).
- 2.
Alternatively, we could define m(L) on the set of polynomials in d-variables, \({\mathcal P}(\mathbb R^d),\) as they have finite Hermite expansion \(f = \sum _{k=0}^\infty {\mathbf {J}}_k f = \sum _{k=0}^\infty \sum _{|\alpha |= k} \langle f, {\mathbf {h}}_{\nu } \rangle _{\gamma _d} {\mathbf {h}}_{\nu }.\)
- 3.
In fact, nowadays this theorem is known as Stein’s universal multiplier theorem.
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Urbina-Romero, W. (2019). Spectral Multiplier Operators with Respect to the Gaussian Measure. In: Gaussian Harmonic Analysis. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-05597-4_6
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