New Stein–Weiss and weighted BMO results for Riesz–Bessel transforms

This work presents new Stein–Weiss type inequalities for higher-order Riesz–Bessel transforms constructed via the generalized translation operator associated with the Laplace–Bessel framework. We identify precise conditions under which these transforms map weighted Lebesgue spaces into weighted Lebesgue spaces, including weak-type endpoint estimates. In addition, the corresponding modified transforms are shown to act boundedly on suitable weighted BMO spaces. These results extend classical Stein–Weiss inequalities and reveal new interactions between higher-order transforms and weighted BMO structures, highlighting distinctive features of the Riesz–Bessel setting.