Localization and summability of multiple Hermite series (Q1363189)

Let \(\mu_k=2|k|+n\) and \(\Phi_k(x)\) be the eigenvalues and orthonormalized eigenfunctions of the operator \(A=-\Delta+x^2\) in \(L^2(\mathbb{R}^n),\) let \[ E^\alpha_\lambda f(y)=\sum_{\mu_k<\lambda} (1-\mu_k/\lambda)^\alpha f_k\Phi_k(y) \] be the corresponding Riesz means of order \(\alpha >0\) of a function \(f\) which is locally in \(L^1(\mathbb{R}^n),n\geq 2\). Localization theorems are given for some classes of functions that are analogous to the classical Szegő results [\textit{G. Szegő}, ``Orthogonal polynomials'' (1959; Zbl 0089.27501)]. The conditions are as follows: (Theorem 1.) \[ \begin{aligned} \int_{\mathbb{R}^n}a(\lambda,x)(1-x^2/\lambda)^{-1/4}|x|^{-(n+1)/2-\alpha}|f(x)|dx &= o(\lambda ^{\alpha/2-(n-1)/4})\\ \int_{\mathbb{R}^n}b(\lambda,x)|f(x)|dx &= o(\lambda ^{\alpha +1/3}), \end{aligned} \] where \(a(\lambda,x)\) is the characteristic function of the set \(\{x\in\mathbb{R}^n:A<x^{2}<\lambda-\lambda^{1/3}\}\) and \(b(\lambda,x)\) is the characteristic function of the set \(\{x\in\mathbb{R}^n:|x^2-\lambda|< \lambda^{1/3+\varepsilon }\}\); (Theorem 2.) \(f\) is differentiable at infinity, \(f(x)=O(|x|^\beta)\) as \(|x|\to\infty\) for \(\alpha >(n-1)/2,\) \(\beta <2\alpha -n+2,\) and \[ \int_{\mathbb{R}^n}a(\lambda,x)(1-x^{2}/\lambda)^{-3/4} |x|^{-(n+1)/2-\alpha -1}|\bigtriangledown f(x)|dx = o(\lambda ^{\alpha /2-(n-1)/4}). \] Theorem 3 gives conditions which ``interpolate'' between the first and third one. Moreover, exactness of the second condition is proved by the example \(f(x)=|x|^\beta\). The localization principle and summability on the Lebesgue set are established as consequences of the given three Theorems.