A multiplier theorem for special Hermite expansions (Q2767423)

Let \(\mathbb{C}^n\) be the \(n\)-dimensional complex Euclidean space and \(m(\nu)= m(|\nu|)\) be a bounded function. The author studies the multiplier operator \(T_m\) defined by NEWLINE\[NEWLINET_m f(z)= (2\pi)^{-n/2} \sum_{\nu\in \mathbb{Z}^n_+} m(\nu) f*\Phi_{\nu\nu}(z),\quad z\in \mathbb{C}^n,NEWLINE\]NEWLINE where NEWLINE\[NEWLINEf* g(z)= \int_{\mathbb{C}^n} f(z- w) g(w) e^{i\text{ Im}\langle z,\overline w\rangle/2} dwNEWLINE\]NEWLINE is the twist convolution, NEWLINE\[NEWLINEf(z)= (2\pi)^{-n/2} \sum_{\nu\in \mathbb{Z}^n_+} f* \Phi_{\nu\nu}(z),NEWLINE\]NEWLINE and \(\{\Phi_{\alpha\beta}\}\) is an orthogonal basis on \(L^2(\mathbb{C}^n)\) generated by the Hermite function. After defining certain differential condition on \(m(\nu)\), with fractional order, the author uses the Littlewood-Paley theory to establish an \(L^p\), \(1< p< \infty\), multiplier theorem for the operator \(T_m\). As applications, some multiplier theorems related to Hermite expansions and Laguerre expansions are also discussed.