On the equisummability of Hermite and Fourier expansions (Q5933524)

Let \(1\leq p\leq \infty\) and \(f\in L^p({\mathbb R}^n)\). The Bochner-Riesz means associated to the Fourier transform of \(f\) are defined by \[ S^\delta_t(f)(x)=(2\pi)^{-{n\over 2}}\int_{|y|<t}e^{ixy}\left(1-{|y|^2\over t^2}\right)^\delta \widehat{f}(y) dy. \] The Bochner-Riesz means associated to the Hermite expansion of \(f\) are defined by \[ (SH)^\delta_t(f)(x)=\sum_{0\leq <t} \left(1-{2k+n\over t}\right)^\delta P_k(f)(x) \] where \(P_k\) is the orthogonal projection of \(L^2({\mathbb R}^n)\) onto the \(k\)th eigenspace spanned by the \(n\)-dimensional Hermite functions \(\Phi_\alpha\) which are eigenfunctions of the Hermite operator \(H=-\Delta +|x|^2\) with eigenvalue \(2|\alpha|+n\) satisfying \(|\alpha|=k.\) Here I use the symbol \( (SH)^\delta_t\) instead of \(S^\delta_R\) in the original paper to distinguish it with \(S^\delta_t\). The paper investigates the equivalence between these two classes of operators in the uniform boundedness from \(L^p\) to \(L^p\). One of the theorems is as follows. Theorem 2.2. \(E\sigma_N^\delta E\) are uniformly bounded on \(L^p({\mathbb R}^n)\) if and only if \(S_t^\delta\) are uniformly bounded, provided \(\delta \geq \max \{0, {n\over 2}-1\}\), where \(E\) stands for the operator \(E(f)(x)=e^{-{1\over 2}|x|^2}f(x)\) and \(\sigma^\delta_N\) denotes Cesàro operators. In this theorem the authors take the Cesàro operators \(\sigma^\delta_N\) in the place of the operators \( (SH)^\delta_t\), since it is well known that they are equivalent in the uniformly boundedness on \(L^p({\mathbb R}^n)\). In the proof the transplantation theorem in [\textit{C. E. Kenig, R. J. Stanton} and \textit{P. A. Tomas}, J. Funct. Anal. 46, 28-44 (1982; Zbl 0506.47014)] is employed.