Higher order Riesz transforms associated with Bessel operators |
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Authors: | Jorge J Betancor Juan C Fariña Teresa Martinez Lourdes Rodríguez-Mesa |
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Institution: | (1) Departamento de Análisis Matemático, Universidad de la Laguna, Campus de Anchieta, Avda. Astrofísico Francisco Sánchez, s/n, ES-38271 La Laguna (Sta. Cruz de Tenerife), Spain;(2) Departamento de Matemáticas, Faculdad de Ciencias, Universidad Autónoma de Madrid, ES-28049 Madrid, Spain |
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Abstract: | In this paper we investigate Riesz transforms R
μ
(k) of order k≥1 related to the Bessel operator Δμ
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 dλ(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 of weights for the doubling measure dλ. These weighted results extend the ones obtained by Andersen and Kerman. |
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Keywords: | |
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