Almost everywhere summability on nilmanifolds
HTML articles powered by AMS MathViewer
- by Andrzej Hulanicki and Joe W. Jenkins
- Trans. Amer. Math. Soc. 278 (1983), 703-715
- DOI: https://doi.org/10.1090/S0002-9947-1983-0701519-0
- PDF | 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 $\Sigma (\varphi ,{e_j}){e_j},\varphi \in {L^p}(G/\Gamma ), 1 \leqslant p < \infty$.References
- Ronald R. Coifman and Guido Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin-New York, 1971 (French). Étude de certaines intégrales singulières. MR 0499948
- Jacques Dixmier, Opérateurs de rang fini dans les représentations unitaires, Inst. Hautes Études Sci. Publ. Math. 6 (1960), 13–25 (French). MR 136684
- Roe W. Goodman, Nilpotent Lie groups: structure and applications to analysis, Lecture Notes in Mathematics, Vol. 562, Springer-Verlag, Berlin-New York, 1976. MR 0442149
- A. Hulanicki, Subalgebra of $L_{1}(G)$ associated with Laplacian on a Lie group, Colloq. Math. 31 (1974), 259–287. MR 372536, DOI 10.4064/cm-31-2-259-287
- A. Hulanicki, Commutative subalgebra of $L^{1}(G)$ associated with a subelliptic operator on a Lie group G, Bull. Amer. Math. Soc. 81 (1975), 121–124. MR 358229, DOI 10.1090/S0002-9904-1975-13664-0
- J. W. Jenkins, Dilations and gauges on nilpotent Lie groups, Colloq. Math. 41 (1979), no. 1, 95–101. MR 550634, DOI 10.4064/cm-41-1-95-101
- Giancarlo Mauceri, Riesz means for the eigenfunction expansions for a class of hypoelliptic differential operators, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 4, v, 115–140 (English, with French summary). MR 644345
- Benjamin Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans. Amer. Math. Soc. 139 (1969), 231–242. MR 249917, DOI 10.1090/S0002-9947-1969-0249917-9
- Eileen L. Poiani, Mean Cesàro summability of Laguerre and Hermite series, Trans. Amer. Math. Soc. 173 (1972), 1–31. MR 310537, DOI 10.1090/S0002-9947-1972-0310537-9
- Leonard F. Richardson, $N$-step nilpotent Lie groups with flat Kirillov orbits, Colloq. Math. 52 (1987), no. 2, 285–287. MR 893545, DOI 10.4064/cm-52-2-285-287
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Elias M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory. , Annals of Mathematics Studies, No. 63, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0252961
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- Michael E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, No. 34, Princeton University Press, Princeton, N.J., 1981. MR 618463
Bibliographic Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 278 (1983), 703-715
- MSC: Primary 22E30; Secondary 43A55, 43A85
- DOI: https://doi.org/10.1090/S0002-9947-1983-0701519-0
- MathSciNet review: 701519