For \(\beta \in (0, 2n)\) , let \(\mathcal {I}_\beta ^{(2n)}\) denote the Riesz operator on the Euclidean space \({\mathbb R}^{2n}\) . In this paper, for Muckenhoupt weights \(u\in A_p(\mathbb R^n)\) and \(v\in A_q(\mathbb R^n)\) with \(p,q\in (1,\infty )\) , the authors introduce the Riesz capacity \({\mathscr {R}}_\beta ^{p,\,q,\,u,\,v}\) , associated with the weighted mixed-norm Lebesgue spaces \(L_v^q(L_u^p)({\mathbb R}^{2n})\) , and establish the corresponding capacitary inequalities. The approach taken is mainly based on a new characterization of weighted mixed-norm Lebesgue spaces via using an operator \(T_\beta ^b\) , where \(b\ge 1\) is close to 1. This operator is defined by using either the Taylor remainder (when \(\beta \) is a non-integer) or the high order difference (when \(\beta \) is an integer) of the kernel of \(\mathcal {I}_\beta ^{(2n)}\) . To establish this characterization, the authors demonstrate that \(T_\beta ^b\) behaves like a Littlewood-Paley type operator, by deriving delicate and highly non-trivial off-diagonal estimates for its vector-valued kernel.
