Sharp maximal function inequalities and boundedness for Toeplitz type operator related to general fractional integral operators
type: Item
instance of: scholarly article
title: Sharp maximal function inequalities and boundedness for Toeplitz type operator related to general fractional integral operators
zbMATH Open document ID: 1277.47044
author: Lanzhe Liu
published in: Banach Journal of Mathematical Analysis
publication date: 2013-03-20T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bjma/1358864555
review text: The author considers some operators modelled by Toeplitz operators and integral operators with general kernels and proves some inequalities concerning estimates between maximal functions associated to them. Namely, for a kernel family \(F_t(x,y)\), \(t\geq 0\), \((x,y)\in \mathbb R^n \times \mathbb R^n\), a given Banach space \(H\) (of functions defined on \([0,\infty)\)), he deals with the operators \(F_t(f)(x)=\int_{\mathbb R^n} F_t(x,y)f(y)\,dy\) and \(T(f)(x)= \|F_t(f)(x)\|_H\), where the norm is taken in the \(t\)-variable and \(T\) is assumed to be bounded on \(L^2(\mathbb R^n)\). Given a locally integrable function \(b\) and inspired by the commutator \([b,T]=bT(f)-T(bf)\), he considers combinations \(F^b_t(f)(x)=\sum_{k=1}^m F_t^{k,1}(M_b I_\alpha F^{k,2}_t (f))+ F_t^{k,3}(I_\alpha M_b F^{k,4}_t (f)) \), where \(F^{k,1}(f)\) stands for \(F_t(f)\) or \(\pm I\), \(\|F_t^{k,2}(f)\|_H\), \(\|F_t^{k,3}(f)\|_H\) and \(\|F_t^{k,4}(f)\|_H\) are bounded operators on \(L^p(\mathbb R^n)\), \(I_\alpha\) is the fractional integral and \(M_b\) is the multiplication by \(b\). Denoting \(T^b(f)= \|F^b_t (f)(x)\|\), it is shown, under certain conditions on the kernel functions \(F_t\) (Calderón-Zygmund type conditions) and \(b\in \mathrm{Lip}_\beta\), the boundedness of \(T\) and \(T^b\) in spaces such as \(L^p\) or generalized Morrey spaces. The author presents some applications to Littlewood-Paley, Marcinkiewicz and maximal Bochner-Riesz operators, where the space \(H\) is selected in a convenient way in each case. Reviewer's remark. Very similar results with the same introduction, analogous proof and applications have been given in [\textit{S. Guo}, \textit{C.-X. Huang} and \textit{L.-Z. Liu}, An. Univ. Craiova, Ser. Mat. Inf. 39, No. 1, 35--47 (2012; Zbl 1274.42033), \url{http://inf.ucv.ro/~ami/index.php/ami/article/view/442/402}], with the main difference that the fractional integral operator is not considered there.
reviewed by: Oscar Blasco
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 6147126
zbMATH Keywords: Toeplitz type operator
zbMATH Keywords: sharp maximal function
zbMATH Keywords: Morrey space
zbMATH Keywords: \(BMO\)
zbMATH Keywords: Lipschitz function
MaRDI profile type: Publication
OpenAlex ID: W2026761715
DOI: 10.15352/BJMA/1358864555
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description: scientific article; zbMATH DE number 6147126