In this paper, we work in the setting of Bessel operators and Bessel Laplace equations studied by Weinstein, Huber, and the harmonic function theory in this setting introduced by Muckenhoupt–Stein, especially the generalised Cauchy–Riemann equations and the conjugate harmonic functions. We provide the equivalent characterizations of product Hardy spaces associated with Bessel operators in terms of the Bessel Riesz transforms, non-tangential and radial maximal functions defined via Poisson and heat semigroups, based on the atomic decomposition, the generalised Cauchy–Riemann equations, the extension of Merryfield’s result which connects the product non-tangential maximal function and area function, and on the grand maximal function technique which connects the product non-tangential and radial maximal function. We then obtain directly the decomposition of the product BMO space associated with Bessel operators.

在本文中，我们将研究由Weinstein，Huber研究的Bessel算子和Bessel Laplace方程，以及Muckenhoupt-Stein在此背景下引入的谐波函数理论，特别是广义的Cauchy-Riemann方程和共轭谐波函数。我们提供了与Bessel算子相关的乘积Hardy空间的等价刻画，包括Bessel Riesz变换，通过泊松和热半群定义的非切向和径向最大函数，基于原子分解，广义Cauchy-Riemann方程，扩展关于将乘积非切向最大函数和面积函数联系起来的Merryfield结果，以及关于将乘积非切向和径向最大函数联系起来的大极大函数技术的说明。