that the limits exist separetely as $R_i((\cdot)^{\beta}1_Qf_2b(x,\cdot))(x),$ and where in the second to last esimate we used the assumption that $Q$ is centered at the origin. Dividing by ${|Q|}^2$ gives the claim.

*Case 2, $d\geq2$*: In the previous estimate we saw that the key issue with the lower bound for $C_b R_i$’s is the following: We introduce $1$ as $\sum_{i=1}^d(x_i-y_i)^2{|x-y|}^{-2}$, and would like to view this as $(x_i-y_i){|x-y|}^{-d-1}$ times functions that depend only on $x$ and only on $y$. As we saw: $$\label{eq:formula}
    1=\sum_{i=1}^d\frac{(x_i-y_i)^2}{{|x-y|}^{2}}
    =\sum_{i=1}^d\frac{(x_i-y_i)}{{|x-y|}^{d+1}}(x_i-y_i){|x-y|}^{d-1},$$ and the problem becomes about expanding ${|x-y|}^{d-1}$ when $d$ is even, hence $d$ was odd.

Consider the function $z_i{|z|}^{d-1}$ of $z\in{\mathbb{R}}^d$. By induction, we check that $\partial^\alpha(z_i{|z|}^{d-1})$ is a linear combination of terms of the form $z^\beta{|z|}^{d-{|\alpha|}-{|\beta|}}$, where ${|\beta|}\leq{|\alpha|}+1$. In particular, when ${|\alpha|}=d+1$, then $\partial^\alpha(z_i{|z|}^{d-1})$ is a linear combination of terms of the form $z^\beta{|z|}^{-1-{|\beta|}}$. In particular, ${|\partial^\alpha(z_i{|z|}^{d-1})|}\lesssim{|z|}^{-1}\in L^1_{{\operatorname{loc}}}({\mathbb{R}}^d)$ for $d\geq 2$.

Now let $\phi\in C_c^\infty({\mathbb{R}}^d)$ be a smooth bump such that $\phi\equiv 1$ in $Q(0,\frac14)$, and $\phi$ is supported in $Q(0,\frac12)$ (cube of centre $0$ and “radius” $\frac12$, hence sidelength $1$). We consider the function $\phi_i(z)=\phi(z)z_i{|z|}^{d-1}$. By the previous computation and product rule, this satisfies $${|\partial^\alpha\phi_i|}\lesssim{|z|}^{-1} 1_{Q(0,\frac12)} \in L^1({\mathbb{R}}^d)$$ for ${|\alpha|}=d+1$ and $d\geq 2$.

