Real-Variable Characterizations Of Hardy Spaces Associated With Bessel Operators

Let $\lambda >0$, $p\in ((2\text{lz}+1)\mathrm{/}(2\text{lz}+2),1]$, and ${\mathrm{△}}_{\lambda }\equiv -\frac{d^2}{d{x}^2}-\frac{2\lambda }{x}\frac{d}{dx}$ be the Bessel operator. In this paper, the authors establish the characterizations of atomic Hardy spaces $H^p((0,\mathrm{\infty }),d{m}_{\lambda })$ associated with ${\mathrm{△}}_{\lambda }$ in terms of the radial maximal function, the nontangential maximal function, the grand maximal function, the Littlewood-Paley $g$-function and the Lusin-area function, where $d{m}_{\lambda }(x)\equiv x^{2\lambda }dx$. As an application, the authors further obtain the Riesz transform characterization of these Hardy spaces.