# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc:1701.333134

Arkiv for Matematik, 46, (2), 219-250, 2006.11
In this paper we investigate Riesz transforms$R$_{μ}^{($k$)}of order$k$≥1 related to the Bessel operator Δ_{μ}$f$($x$)=-$f$”($x$)-((2μ+1)/$x$)$f$’($x$) and extend the results of Muckenhoupt and Stein for the conjugate Hankel transform (a Riesz transform of order one). We obtain that for every$k$≥1,$R$_{μ}^{($k$)}is a principal value operator of strong type ($p$,$p$),$p$∈(1,∞), and weak type (1,1) with respect to the measure$d$λ($x$)=$x$^{2μ+1}$dx$in (0,∞). We also characterize the class of weights ω on (0,∞) for which$R$_{μ}^{($k$)}maps$L$^{$p$}(ω) into itself and$L$^{1}(ω) into$L$^{1,∞}(ω) boundedly. This class of weights is wider than the Muckenhoupt class $\mathcal{A}_{p}^\mu$ of weights for the doubling measure$d$λ. These weighted results extend the ones obtained by Andersen and Kerman.
@inproceedings{jorge2006higher,
title={Higher order Riesz transforms associated with Bessel operators},
author={Jorge J. Betancor, Juan C. Fariña, Teresa Martinez, and Lourdes Rodríguez-Mesa},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203623312130943},
booktitle={Arkiv for Matematik},
volume={46},
number={2},
pages={219-250},
year={2006},
}

Jorge J. Betancor, Juan C. Fariña, Teresa Martinez, and Lourdes Rodríguez-Mesa. Higher order Riesz transforms associated with Bessel operators. 2006. Vol. 46. In Arkiv for Matematik. pp.219-250. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203623312130943.