Hermite and special Hermite expansions revisited
The author studies the Bochner-Riesz means for the Hermite and special Hermite expansions, i.e., the eigenfunctions expansions of the Hermite operator \(H=-\Delta +|x|^2\) on \({\mathbb{R}}^n\) and of the special Hermite operator \[ L=-\Delta + |z|^2/4 - i \sum_{j=1}^n \left(x_j \frac{\partial}{\partial y_j}-y_j\frac{\partial} {\partial x_j}\right) \] on \({\mathbb{C}}^n.\) He gives several reasons to consider local estimates instead of global ones. The main result of the paper is the so-called restriction theorem for the spectral projection operator (more exactly, its local version). Namely, for any compact subset \(B\) of \({\mathbb{R}}^n\) and for \(1\leq p\leq 2(n+1)/(n+3)\) the estimate \[ \|P_k\Phi _B f\|_2 \leq C_B k^{\delta(p)/2-1/4} \|f \|_p \] is proved for any \(f\in L^p({\mathbb{R}}^n)\) where \(P_kf\) are the projections \[ P_k f=\sum_{|\alpha |=k} (f,\Phi _{\alpha }) \Phi _{\alpha } \] for the Hermite expansions, \(C_B\) depends only on \(B,\) and \(\delta (p)\) is the critical index \(\delta (p) =\max \{n|1/p-1/2|-1/2,0\}.\) An analogous assertion is given for the special Hermite expansions.
scientific article; zbMATH DE number 1425142