The author proves the following analogue to a well-known result of Hardy and Littlewood [\textit{G. H. Hardy, J. E. Littlewood} and \textit{G. Pólya} [Inequalities. 2nd ed. Cambridge: At the University Press (1952; Zbl 0047.05302), Theorem 382]. Let \(p, q, r, s, t\) be positive numbers such that \(q>1\), \(1/p+1/q>1\), and either (i) \(11\). If \(u=(2-1/p-1/q)s+(1-1/r)t\) then \[ \begin{multlined} \int_{-\infty}^{+\infty} g(x)\,dx\;\int_{-\infty}^{+\infty} \,dz\;\int_{-\infty}^{+\infty} \frac{\psi(y,z)}{ [| x-y|^{2/s}+| z|^{2/s}]^{u/2} }\,dy\\ \leq C(p, q, r, s, t) \left(\int_{-\infty}^{+\infty} | g(x)|^q\,dx\right)^{1/q}\left(\int_{-\infty}^{+\infty}\,dz\left(\int_{-\infty}^{+\infty} | \psi(y,z)|^p\,dy\right)^{r/p}\right)^{1/r}.\end{multlined} \]