Functional calculus for the Laguerre operator
In this paper, the author studies the \(L^p\) boundedness of spectral multipliers associated to the multidimensional Laguerre operator \({\mathcal L}_\alpha\) on the Euclidean space. It is known that the analysis of the Laguerre operator can be interpreted as the analysis of the Ornstein-Uhlenbeck operator acting on polyradial functions. Using this relationship, the author can reduce the multiplier problem to the study of the boundedness of certain auxiliary operators which involve the purely imaginary powers of \({\mathcal L}_\alpha\). The main result is that if \(M\) is a bounded holomorphic function in the sector \(S= \{z\in\mathbb{C}:|\arg z|<\arcsin|2/p- 1|\}\), satisfying a suitable Hörmander type condition on the boundary, then the spectral operator \(M({\mathcal L}_\alpha)\) is bounded on \(L^q\) (with respect to the Laguerre measure) for a range of values \(q\) which depends on \(p\). The author also shows that holomorphy in the sector \(S\) is a necessary condition for the boundedness of the spectral multiplier whose norm is invariant under dilations.
\(L^p\) boundedness of spectral multipliers
scientific article; zbMATH DE number 2156069
scientific article