Higher order Riesz transforms associated with Bessel operators

Abstract

In this paper we investigate Riesz transforms Rμ(k) of order k≥1 related to the Bessel operator Δμf(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, Rμ(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)=x2μ+1 dx in (0,∞). We also characterize the class of weights ω on (0,∞) for which Rμ(k) maps Lp(ω) into itself and L1(ω) into L1,∞(ω) boundedly. This class of weights is wider than the Muckenhoupt class $\mathcal{A}_{p}^\mu$ of weights for the doubling measure dλ. These weighted results extend the ones obtained by Andersen and Kerman.