Almost everywhere summability on nilmanifolds

Hulanicki, Andrzej; Jenkins, Joe W.

doi:10.1090/S0002-9947-1983-0701519-0

Almost everywhere summability on nilmanifolds
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by Andrzej Hulanicki and Joe W. Jenkins PDF

Trans. Amer. Math. Soc. 278 (1983), 703-715 Request permission

Abstract:

Let $G$ be a stratified, nilpotent Lie group and let $L$ be a homogeneous sublaplacian on $G$. Let $E(\lambda )$ denote the spectral resolution of $L$ on ${L^2}(G)$. Given a function $K$ on $\mathbf {R}^+$, define the operator ${T_K}$ on ${L^2}(G)$ by ${T_k}f = \int _0^\infty {K(\lambda )\;dE(\lambda ) f}$. Sufficient conditions on $K$ to imply that ${T_K}$ is bounded on ${L^1}(G)$ and the maximal operator $K^{\ast } \varphi (x) = \sup _{t > 0}|{T_{K_t}}\varphi (x)|$ (where ${K_t}(\lambda ) = K(t\lambda )$) is of weak type $(1,1)$ are given. Picking a basis ${e_0},{e_1},\ldots$ of ${L^2}(G/\Gamma )$ ($\Gamma$ being a discrete cocompact subgroup of $G$) consisting of eigenfunctions of $L$, we obtain almost everywhere and norm convergence of various summability methods of $\Sigma (\varphi ,{e_j}){e_j},\varphi \in {L^p}(G/\Gamma ), 1 \leqslant p < \infty$.

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