Hardy spaces, Campanato spaces and higher order Riesz transforms associated with Bessel operators

Abstract

Let $\nu = (\nu_1, \ldots, \nu_n) \in (-1/2, \infty)^n$, with $n \ge 1$, and let $\Delta_\nu$ be the multivariate Bessel operator defined by \[ \Delta_{\nu} = -\sum_{j=1}^n\left( \frac{\partial^2}{\partial x_j^2} - \frac{\nu_j^2 - 1/4}{x_j^2} \right). \] In this paper, we develop the theory of Hardy spaces and BMO-type spaces associated with the Bessel operator $\Delta_\nu$. We then study the higher-order Riesz transforms associated with $\Delta_\nu$. First, we show that these transforms are Calder\'on-Zygmund operators. We further prove that they are bounded on the Hardy spaces and BMO-type spaces associated with $\Delta_\nu$.