Riesz transforms associated to Bessel operators

Abstract

For $ν>0$, we consider the Bessel operator $S_ν$ defined on $L^{2}(ℝ^{+}, x^{2ν} dx)$ by $S_{\nu}=-\frac{d^{2}}{\,dx^{2}}-\frac{2\nu}{x}\frac{d}{dx}$. We prove, in a simple way, that the Riesz transform associated to $S_ν$ is bounded on $L^{p}(ℝ^{+}, x^{2ν}dx), 1 < p < ∞$, with a constant only depending on $p$. We also give a weighted version and estimate the constant.