we have that $${\Big|{\dashint_{Q} {{\dashint_{Q} {\prod_{i=1}^{2}(b_i(x)-b_i(y))f_2(y)f_1(x) {\,\mathrm{d}}y {\,\mathrm{d}}x}}}}\Big|_{}} \leq C_d\sum_{i=1}^{k}{\big\|C_bR_i\big\|_{L^p\to L^p}} \left( {\dashint_{Q} {{|f_1|}^p}} \right) ^{1/p}\left( {\dashint_{Q} {{|f_2|}^{p'}}} \right) ^{1/p'}.$$

*Case 1, $d$ is odd*: Let $d = 2k+1$ for some $k\in{\mathbb{N}}$. By composing back and forth with the translation $x\mapsto x-c_Q$, we may assume that the cube $Q$ is centred at the origin. We begin with introducing $1$ as $$1=\sum_{i=1}^d\frac{(x_i-y_i)^2}{{|x-y|}^{2}}
    =\sum_{i=1}^d\frac{(x_i-y_i)}{{|x-y|}^{d+1}}(x_i-y_i){|x-y|}^{d-1},$$ denote $b(x,y) = \prod_{i=1}^{2}(b_i(x)-b_i(y)),$ and then proceed with: $$\begin{split}
    & {\Big|\int_Q\int_Qb(x,y)f_2(y)f_1(x){\,\mathrm{d}}y {\,\mathrm{d}}x\Big|_{}} = {\Big|\int_Q\lim_{\varepsilon\to 0}\int_{{|x-y|}>\varepsilon}b(x,y)f_2(y)1_Q(y)f_1(x){\,\mathrm{d}}y {\,\mathrm{d}}x\Big|_{}} \\   
    &= {\Big|\int_Q\lim_{\varepsilon\to 0}\int_{{|x-y|}>\varepsilon}b(x,y) \sum_{i=1}^d\frac{(x_i-y_i)^2}{{|x-y|}^2} f_2(y)1_Q(y)f_1(x){\,\mathrm{d}}y {\,\mathrm{d}}x\Big|_{}} \\
    &= {\Big|\int_Q\lim_{\varepsilon\to 0}\int_{{|x-y|}>\varepsilon}b(x,y) \sum_{i=1}^d\frac{Y_i(x-y)^2}{{|x-y|}^{2d}} f_2(y)1_Q(y)f_1(x){\,\mathrm{d}}y {\,\mathrm{d}}x\Big|_{}},  \\
    \end{split}$$ where $Y_i(x) = x_i{|x|}^{d-1}.$ We momentarily force the expression into this form in order to contrast it with the similar argument emplying spherical harmonics given in [@CRW].

For a given $\alpha = (\alpha_1,\dots,\alpha_d)\in{\mathbb{N}}^d$ and $x\in{\mathbb{R}}^d,$ let $x^{\alpha} = \prod_{i=1}^dx_i^{\alpha_i}.$ We continue with $$\begin{aligned}
        & {\Big|\int_Q\lim_{\varepsilon\to 0}\int_{{|x-y|}>\varepsilon}b(x,y) \sum_{i=1}^d\frac{Y_i(x-y)^2}{{|x-y|}^{2d}} f_2(y)1_Q(y)f_1(x){\,\mathrm{d}}y {\,\mathrm{d}}x\Big|_{}}  \\
                &= {\Big|\int_Q\lim_{\varepsilon\to 0}\int_{{|x-y|}>\varepsilon}b(x,y) \sum_{i=1}^d\frac{x_i-y_i}{{|x-y|}^{d+1}}(x_i-y_i)\Big(\sum_{j=1}^d(x_j-y_j)^2\Big)^{k} f_2(y)1_Q(y)f_1(x){\,\mathrm{d}}y {\,\mathrm{d}}x\Big|_{}}  \\
        &= {\Big|\int_Q\lim_{\varepsilon\to 0}\int_{{|x-y|}>\varepsilon}\sum_{i=1}^d\sum_{\alpha+ \beta= d}b(x,y)\frac{x_i-y_i}{{|x-y|}^{d+1}} a_{\alpha,\beta}f_1(x)x^{\alpha}y^{\beta}  f_2(y)1_Q(y){\,\mathrm{d}}y {\,\mathrm{d}}x\Big|_{}}  \\
        & \overset{\ast}{=} {\Big|\sum_{i=1}^d\sum_{\alpha+ \beta= d}a_{\alpha,\beta}\int_Qf_1(x)x^{\alpha}\lim_{\varepsilon\to 0}\int_{{|x-y|}>\varepsilon}b(x,y) \frac{x_i-y_i}{{|x-y|}^{d+1}}y^{\beta} f_2(y)1_Q(y){\,\mathrm{d}}y {\,\mathrm{d}}x\Big|_{}}  \\
                &\leq \sum_{i=1}^d\sum_{\alpha+ \beta= d}{|a_{\alpha,\beta}|}{\big\|(\cdot)^{\alpha}f_1\big\|_{L^{p'}(Q)}}{\big\|C_bR_i((\cdot)^{\beta}f_21_Q)\big\|_{L^p(Q)}} \\
        &\leq \sum_{i=1}^d\sum_{\alpha+ \beta= d}{|a_{\alpha,\beta}|}{\big\|(\cdot)^{\alpha}\big\|_{L^{\infty}(Q)}}{\big\|f_1\big\|_{L^{p'}(Q)}}{\big\|(\cdot)^{\beta}\big\|_{L^{\infty}(Q)}}{\big\|C_bR_i\big\|_{L^p\to L^p}}{\big\|f_2\big\|_{L^p(Q)}} \\
        &\leq C_d\sum_{i=1}^d\sum_{\alpha + \beta = d}{|a_{\alpha\beta}|} {|Q|}{\big\|C_bR_i\big\|_{L^p\to L^p}}{\big\|f_1\big\|_{L^{p'}(Q)}}{\big\|f_2\big\|_{L^p(Q)}}\\
        &\leq C_d{|Q|}\sum_i^d{\big\|C_bR_i\big\|_{L^p\to L^p}}{\big\|f_1\big\|_{L^{p'}(Q)}}{\big\|f_2\big\|_{L^p(Q)}},
        \end{aligned}$$ where at $\ast$ we used the fact 