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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Almost everywhere summability on nilmanifolds


Authors: Andrzej Hulanicki and Joe W. Jenkins
Journal: Trans. Amer. Math. Soc. 278 (1983), 703-715
MSC: Primary 22E30; Secondary 43A55, 43A85
MathSciNet review: 701519
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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}\vert{T_{K_t}}\varphi (x)\vert$ (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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DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0701519-0
PII: S 0002-9947(1983)0701519-0
Article copyright: © Copyright 1983 American Mathematical Society