Twisted convolution and Riesz means (Q1283054)

Using some estimates of Askey and Wainger for Laguerre functions, the authors improve a result in [\textit{S. Thangavelu}, Ark. Mat. 29, No. 2, 307-321 (1991; Zbl 0765.42009)]. Consider the twisted Laplacian on \(\mathbb{R}^{2n}\), \(n\geq 1\), \[ -\Delta_x- \Delta_y+ 1/4(| x|^2+| y|^2)- i \sum^n_{j=1} \Biggl(x_j{\partial\over\partial y_j}- y_j{\partial\over\partial x_j}\Biggr),\quad (x,y)\in \mathbb{R}^n\times \mathbb{R}^n. \] The spectrum of this operator is discrete and consists of points \(2k+n\), \(k=0\), \(1,\dots\)\ . Let \(Q_k\) denote the spectral projection operator on the eigenspace that corresponds to the \(k\)th eigenvalue \(2k+n\). The Riesz means of order \(\delta\) are defined by \[ T^\delta_R f= \sum^\infty_{k=0} (1-(2k+ n)/R)^\delta_+ Q_kf. \] The authors prove that \(T^\delta_R\) is bounded on \(L^p(\mathbb{C}^n,dz)\) uniformly for \(R>0\), provided \(1\leq p<2(2n+1)/(2n+ 3)\) and \(\delta> 2n(1/p- 1/2)-1/2\). To get this theorem, the authors reduce the problem to an \(L^p-L^2\) restriction type estimate for \(Q_k\), that is analogous to the well-known restriction theorem of Fefferman-Stein-Tomas.