Multivariate spectral multipliers for systems of Ornstein-Uhlenbeck operators (Q2839269)

Let \(L = (L_1, L_2, \cdots L_d)\) be self-adjoint operators on \(L^2(X, \nu)\) which commute strongly. That is, the spectral projections of \(L_j~j =1,2, \cdots d\) commute pairwise. Then, if \(m\) is a function defined on the joint spectrum of \(L,\) one can consider the joint spectral multiplier \(m(L)\) and study its boundedness on \(L^p\) spaces. The paper under review proves a fairly general theorem on the holomorphic extension property of a system of non-negative operators that strongly commute and a Marcinkiewicz type multiplier theorem for the system \(\mathcal{L} = ( \mathcal{L}_1, \cdots \mathcal{L}_d)\) where NEWLINE\[NEWLINE\mathcal{L}_n = -\frac{1}{2} \frac{\partial^2}{\partial x_n^2} + x_n \frac{\partial}{\partial x_n},NEWLINE\]NEWLINE a system of one-dimensional Ornstein-Uhlenbeck operators.NEWLINENEWLINEGiven a non-negative system \(L\) as above and \(m_n,~ n =1, 2, \cdots d,\) bounded continuous functions on \([0, \infty)\) which are spectral multipliers for \(L_n,~ n =1, 2, \cdots d,\) on a fixed \(L^p\)-space (for \(p > 1\)), which satisfies \(\sup_{t > 0}\| m_n(tL_n)\| < \infty, \) assume that each \(m_n\) extends to a bounded holomorphic function to an open set \(E_n^p \subset \mathbb C\) and NEWLINE\[NEWLINE\|m_n\|_{H^\infty(E_n^p)} \leq \sup_{t>0} \|m_n(tL_n)\|.NEWLINE\]NEWLINE Then, every joint \(L^p\)-spectral multiplier \(m\) which satisfies NEWLINE\[NEWLINE\sup_{t_1, t_2, \cdots t_d > 0} \|m(t_1L_1, \cdots t_d L_d) \| < \infty,NEWLINE\]NEWLINE extends to a bounded holomorphic function of several variables in \(E^p = E_1^p \times \cdots \times E_d^p\) and NEWLINE\[NEWLINE \|m\|_{H^\infty(E^p)} \leq \sup_{t_1, t_2, \cdots t_d > 0} \|m(t_1L_1, \cdots t_d L_d) \|.NEWLINE\]NEWLINENEWLINENEWLINEThe Marcinkiewicz multiplier theorem proved in the paper ensures that the class of \(p\)-uniform joint spectral multipliers is rich for the system \((\mathcal{L}_1, \mathcal{L}_2, \cdots \mathcal{L}_d)\).