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

Let $\nu =({\nu }_1,\dots ,{\nu }_n)\in (-1\mathrm{/}2,\mathrm{\infty }{)}^n$, with $n\ge 1$, and let ${\mathrm{\Delta }}_{\nu }$ be the multivariate Bessel operator defined by $${\mathrm{\Delta }}_{\nu }=-\sum _{j=1}^n(\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}_j^2}-\frac{{\nu }_j^2-1\mathrm{/}4}{x_j^2})\mathrm{.}$$ In this paper, we develop the theory of Hardy spaces and BMO-type spaces associated with the Bessel operator ${\mathrm{\Delta }}_{\nu }$. We then study the higher-order Riesz transforms associated with ${\mathrm{\Delta }}_{\nu }$. First, we show that these transforms are Calderón-Zygmund operators. We further prove that they are bounded on the Hardy spaces and BMO-type spaces associated with ${\mathrm{\Delta }}_{\nu }$.