Littlewood-Paley and pseudo-differential operators on Herz-type spaces over Vilenkin groups
type: Item
instance of: scholarly article
title: Littlewood-Paley and pseudo-differential operators on Herz-type spaces over Vilenkin groups
zbMATH Open document ID: 1069.43002
author: Yueping Zhu
publication date: 2005-06-07T00:00:00Z
review text: The author studies Littlewood-Paley and pseudo-differential operators on Herz-type spaces over Vilenkin groups. Suppose that \(G\) is a Vilenkin group with character group \(\Gamma\), the Littlewood-Paley operator on \(G\) is defined by \[ Sf(x)= \Biggl\{\sum_{n\in\mathbb{Z}} |(f^\wedge\chi_{\Gamma_{n+1}\setminus\Gamma_n})^\vee(x)|^2 \Biggr\}^{{1\over 2}},\quad x\in G, \] and the pseudo-differential operator on \(G\) is defined by \[ T_\sigma f(x)= \int_\Gamma\sigma(\gamma) f^\wedge(\gamma) \gamma(x)\, d\gamma \] with \(\sigma\in S^m_p\). Moreover, the Herz-type space on \(G\) is \(\dot K^{\alpha,p}_q(\omega_1,\omega_2; G)\) with norm \[ \| f\|_{\dot K^{\alpha,p}_q(\omega_1,\omega_2)}= \Biggl\{\sum^{+ \infty}_{l=-\infty} |\omega_1(G_1)|^{\alpha p}\| f\chi_{G_l\setminus G_{l+1}}\|^p_{L^q_{\omega_2(G)}}\Biggr\}^{{1\over p}}, \] and the Herz-type Hardy space on \(G\) is \[ H\dot K^{\alpha,p}_q(\omega_1, \omega_2; G)= \{f\in S'(G): f^*\in \dot K^{\alpha, p}_q(\omega_1, \omega_2; G)\} \] with \(f^*(x)= \sup_{n\in\mathbb{Z}}|f^*\Delta_n(x)|\), \(\| f\|_{H\dot K^{\alpha,p}_q(\omega_1, \omega_2; G)}= \| f^*\|_{\dot K^{\alpha,p}_q(\omega_1, \omega_2; G)}\). The main results are: 1. The operator \(S\) is bounded from \(H\dot K^{\alpha,p}_q(\omega_1, \omega_2; G)\) to \(\dot K^{\alpha, p}_q(\omega_1, \omega_2; G)\) for \(1< q< \infty\), \(0< p<\infty\), \(1-{1\over q}\leq \alpha< \infty\) where \(\omega_1,\omega_2\in A_1\) are certain nonnegative weights; 2. the operator \(T_\sigma\) is bounded on \(H\dot K^{\alpha,p}_q(\omega_1,\omega_2; G)\) for \(1< q<\infty\), \(0< p< 1\) and \(\alpha> \max(1-{1\over q},{1\over p}- 1)\) with the symbol \(\sigma\in S^m_p\), \(m\leq \alpha\), \(\rho\geq 1\).
Mathematics Subject Classification ID: 43A22
Mathematics Subject Classification ID: 43A70
zbMATH DE Number: 2174190
zbMATH Keywords: Vilenkin group
zbMATH Keywords: Herz-type Hardy space
zbMATH Keywords: Littlewood-Paley function
reviewed by: Su Weiyi
MaRDI profile type: Publication
full work available at URL: s10114-004-0384-5
OpenAlex ID: W1990199890
cites work: Some new Hardy spaces on locally compact Vilenkin groups
cites work: Herz-type Hardy spaces on Vilenkin groups and their applications
cites work: Fourier Analysis on Local Fields. (MN-15)
cites work: Littlewood-Paley and multiplier theorems on weighted spaces over locally compact Vilenkin groups
cites work: Q3154313
cites work: Q4489533
cites work: Q4244354
DOI: 10.1007/S10114-004-0384-5
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 2174190