Riesz means associated with certain product type convex domain
type: Item
instance of: scholarly article
title: Riesz means associated with certain product type convex domain
zbMATH Open document ID: 1219.42007
author: Sunggeum Hong
author: Joonil Kim
author: Chan Woo Yang
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-05-16T00:00:00Z
review text: Let \(f\in{\mathcal S}(\mathbb R^n)\) and denote by \(\widehat f\) the Fourier transform of \(f\). For \(n\geq 3\) and \(\xi=(\xi_0, \xi_1, \dots, \xi_{n-2}) \in\mathbb R^2\times\mathbb R^{n-2}\), define \(\rho(\xi):=\max\{|\xi_0|, |\xi_1|, \dots, |\xi_{n-2}|\}\). For \(\delta\in(0,\infty)\), define the convolution operator \(T_\varepsilon^\delta\) by \(\widehat{T_\varepsilon^\delta f}(x) :=(1-\frac{\rho(\xi)}{\varepsilon})_+^\delta\widehat f(\xi)\), \(T^\delta:=T_1^\delta\) and the maximal operator \(T_*^\delta\) by \[ T_*^\delta f(x) :=\sup_{\varepsilon>0}|T_\varepsilon^\delta f(x)|,\quad x=(x_0, x_1,\dots, x_{n-2})\in\mathbb R^2\times\mathbb R^{n-2},\;n\geq 3, \] where \(t_+^\delta:=t^\delta\) for \(t\in(0,\infty)\) and 0 otherwise. The authors show that there exists a positive constant \(C\) independent of \(f\) such that for all \(f\in H^p(\mathbb R^n)\), \(\|T_*^\delta f\|_{L^p(\mathbb R^n)} \leq C\|f\|_{H^p(\mathbb R^n)}\) if and only if \(p\in(\frac{2(n-1)}{2n-1}, 1)\) and \(\delta\in(n(\frac{1}{p}-1)+\frac{1}{2}, \infty)\), where \(H^p(\mathbb R^n)\) stands for the classical Hardy spaces. When \(p\in(\frac{2(n-1)}{2n-1}, 1)\) and \(\delta\in(n(\frac{1}{p}-1)+\frac{1}{2}, \infty)\) or \(p=\frac{2(n-1)}{2n-1}\) and \(\delta\in(n(\frac{1}{p}-1)+\frac{1}{2}, \infty)\), the authors show that \(|\{x\in\mathbb R^n:\;T_*^\delta (f)(x)>\alpha\}|\leq C\alpha^{-p}\|f\|_{H^p(\mathbb R^n)}^p\), where the positive constant \(C\) is independent of \(f\) or \(\alpha\). Finally, the authors show that the convolution operator \(T^\delta\) is bounded from \(L^p(\mathbb R^n)\) to \(L^p(\mathbb R^n)\) if and only if \(\delta>\max\{2|\frac{1}{p}-\frac{1}{2}|-\frac{1}{2}, 0\}\) and \(p\in[1,\infty)\). The authors also prove that if \(p=\frac {2(n-1)}{2n-1}\) and \(\delta = \frac {2n-1}{2(n-1)}\) or if \(p=1\) and \(\delta=\frac 12\), then \(T^\delta\) is unbounded from \(H^p(\mathbb R^n)\) to weak-\(L^p(\mathbb R^n)\).
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 42B30
zbMATH DE Number: 5888547
zbMATH Keywords: multiplier
zbMATH Keywords: certain product type
zbMATH Keywords: convex domain
zbMATH Keywords: \(L^p\) bounded
zbMATH Keywords: weak type \((p,p)\)
reviewed by: Dachun Yang
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.02.070
OpenAlex ID: W2136619464
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DOI: 10.1016/J.JMAA.2011.02.070
description: scientific article; zbMATH DE number 5888547