\(L^1\) estimates for maximal functions and Riesz transform on real rank 1 semisimple Lie groups
type: Item
instance of: scholarly article
title: \(L^1\) estimates for maximal functions and Riesz transform on real rank 1 semisimple Lie groups
zbMATH Open document ID: 0911.22006
author: Takeshi Kawazoe
published in: Journal of Functional Analysis
publication date: 1999-05-04T00:00:00Z
review text: Let \(G\) be a real rank one semisimple Lie group and \(K\) a maximal compact subgroup of \(G\). Let \(f\in L^1(G//K)\) and denote by \(F_f\) the Abel transform of \(f\) and let \(F^1_f(x)= e^{\rho (x)}F(x)\), where \(\rho\) is the half sum of the positive roots of the Lie algebra of \(G\). For a real function \(F\) one introduces two Fourier multipliers \({\mathcal C}_+'\) and \({\mathcal Q}_s (s\geq 0)\): \[ \begin{matrix} ({\mathcal C}^1_+F)^\sim (\lambda)= c(-\lambda- i\rho)^{-1} F^\sim (\lambda), \\ ({\mathcal Q}_s F)^\sim (\lambda)= \bigl(1+| \lambda |^{-1}\bigr)^s F^\sim (\lambda), \end{matrix}\quad \lambda\in R. \] One defines a subspace \(H^1_s(G//K)\) of \(L^1(G//K)\) (which is analogous to the Hardy space \(H^1(R))\) as \[ H^1_s(G//K)= \bigl\{f\in L^1_{\text{loc}}(G//K); \quad {\mathcal Q}_s {\mathcal C}^1_+F^1_f \in H^1(R) \bigr\}. \] By a detailed analysis of the Harish-Chandra expansion and the Gangolli expansion of the zonal spherical functions of \(G\), the author obtains some basic properties of \(f\) satisfying \({\mathcal C}^1_+ F^1_f \in H^1(R)\) and some criteria by which one can judge whether a radial maximal operator from \(H^1_s (G//K)\) to \(L^1(G//K)\) is bounded. Then, he applies these criteria to radial maximal operators for suitable dilations, to the heat and Poisson maximal operators. He also studies the Riesz transform. Finally he shows that these operators are in \((L^p, L^p)\) for \(p>1\) and that they are of weak type (1,1) and from \(H^1_s (G//K)\) to \(L^1 (G//K)\) for suitable values of \(s\geq 0\).
reviewed by: Fuliu Zhu
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 1211636
zbMATH Keywords: semisimple Lie group
zbMATH Keywords: Abel transform
zbMATH Keywords: Fourier multipliers
zbMATH Keywords: zonal spherical functions
zbMATH Keywords: radial maximal operator
zbMATH Keywords: heat and Poisson maximal operators
zbMATH Keywords: Riesz transform
MaRDI profile type: Publication
full work available at URL: jfan.1998.3256
OpenAlex ID: W2077220455
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DOI: 10.1006/JFAN.1998.3256
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description: scientific article; zbMATH DE number 1211636