On Hermite pseudo-multipliers (Q473179)

The authors define Hermite pseudo-mutipliers on the Euclidean space \( \mathbb{R}^{n}\) and discuss their link with Hermite expansions. Earlier \textit{J. Epperson} [Proc. Am. Math. Soc. 124, No. 7, 2061--2068 (1996; Zbl 0889.42012)] studied the \(L^{p}(\mathbb{R})\) boundedness of Hermite pseudo-mutipliers which are bounded on \(L^{2}(\mathbb{ R)}\). The authors prove a variation of a theorem of \textit{G. Mauceri} [Funct. Anal. 39, 408--429 (1980; Zbl 0458.42008)] about the \(L^{p}(\mathbb{R}^{n})\) boundedness of operators \(M\) which are known to be bounded on \(L^{2}(\mathbb{R}^{n})\). As an application of this, they prove results about the \(L^{p}(\mathbb{R} ^{n})\) boundedness of Hermite pseudo-multipliers. These results are analogues of Epperson's result mentioned above.