Higher order Riesz transforms associated with Bessel operators
type: Item
instance of: scholarly article
title: Higher order Riesz transforms associated with Bessel operators
zbMATH Open document ID: 1159.42010
author: Jorge J. Betancor
author: Teresa Martínez
author: Lourdes Rodríguez-Mesa
author: Juan C. Fariña
published in: Arkiv för Matematik
publication date: 2008-10-27T00:00:00Z
review text: The authors study Riesz transforms \(R_\mu^{(k)}\) of order \(k\geq 1\) related to the Bessel operator \[ \Delta_\mu := - \frac{\partial^2}{\partial x^2} -\frac{2\mu+1}{x} \frac{\partial}{\partial x}=D^* D, \quad \mu> -\frac{1}{2}, \] where \(D:=\partial/ \partial x\) and \(D^*= -x^{-2\mu-1}D x^{2\mu+1}\) denotes the adjoint operator of \(D\) in \(L^2(x^{2\mu+1}dx)\). The Riesz transform \[ R_\mu^{(k)} f(x) = \omega_k f(x) + \lim_{\varepsilon\to0}\int_{|x-y|>\varepsilon} f(y) R_\mu^{(k)} (x,y) y^{2\mu+1} \, dy, \quad x\in(0,\infty) \] where \(\omega_k\) are constants, is initially defined on the space of smooth functions having compact supporton \((0,\infty)\). Note that the kernel \(R_\mu^{(k)} (x,y)\) is connected with the Poisson kernel associated with the operator \(\Delta_{\mu}\). One of main results of the paper shows that for every \(p\in(1,\infty)\) the above defined operator \(R_\mu^{(k)}\) can be extended as a bounded operator from \(L^p(x^{2\mu+1}dx)\) into \(L^p(x^{2\mu+1}dx)\), and also as a bounded operator from \(L^{1}(x^{2\mu+1}dx)\) into \(L^{1,\infty}(x^{2\mu+1}dx)\). Moreover, weighted inequalities for \(R_\mu^{(k)}, k\geq 1\) are studied and the boundedness of the operator \(R_\mu^{(k)}\) in appropriate weighted spaces is established.
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 42B35
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 5357903
zbMATH Keywords: Riesz transforms
zbMATH Keywords: Laplacian
zbMATH Keywords: boundedness
reviewed by: Victor D. Didenko
MaRDI profile type: Publication
full work available at URL: s11512-008-0078-9
OpenAlex ID: W2014825136
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DOI: 10.1007/S11512-008-0078-9
description: scientific article; zbMATH DE number 5357903