for $\tilde R$ {#sec-R}
------------------------

In this subsection, we prove a key technical estimate that we have not seen elsewhere in the literature (though Lemma \[lem-R\] has a similar flavour to estimates in [@Terhesiu-app]).

We have the estimate $\mu(E>t)=O(\ell(t)t^{-\beta})$ for various functions $E:Y\to{{\mathbb R}}$ related to $\varphi$ including the locally constant functions $E(a)=\inf(1_a\varphi)$ and hence $E(a)=|1_a\varphi|_\infty$ and $E(a)=|1_a\varphi|_{\theta_0}$ by condition (A1).

We use the following resummation argument extensively.

\[prop-resum\] Let $g,E:\alpha\to{{\mathbb R}}$ be such that $\mu(E>t)=O(\ell(t)t^{-\beta})$, $g$ is bounded, and $g(a)\le GE(a)$. Then $\sum_{a\in\alpha} \mu(a) g(a) \ll \ell(1/G) G^\beta$.

Write $$\begin{aligned}
\sum_{a\in\alpha} \mu(a) g(a) 
& \le G\sum_{a:E(a)\le L} \mu(a)E(a)+
\sum_{a:E(a)>L} \mu(a)C
 =GA+O(\ell(L) L^{-\beta}),\end{aligned}$$ where $$\begin{aligned}
A & = \sum_{a:E(a)\le L} \mu(a)E(a)  \le
\sum_{j=1}^L \sum_{a:E(a)\in(j-1,j]} \mu(a)j 
=\sum_{j=1}^L \sum_{a:E(a)> j-1} \mu(a)j- 
\sum_{j=0}^L \sum_{a:E(a)> j} \mu(a)j 
\\ &
 = 
\sum_{j=0}^{L-1} \sum_{a:E(a)> j} \mu(a)(j+1)- 
\sum_{j=0}^L \sum_{a:E(a)> j} \mu(a)j 
 \le \sum_{j=0}^{L-1} \sum_{a:E(a)> j} \mu(a)
\\ & = \sum_{j=0}^{L-1} \mu(E>j) \ll \ell(L)  L^{1-\beta}.\end{aligned}$$ Taking $L\approx 1/G$ yields the result.

\[lem-R\] Let $\epsilon>0$. There exists $\theta\in(0,1)$, $C>0$ such that $$\|\tilde R(ib_1)-\tilde R(ib_2)\|_\theta\le  C\bigl\{\ell(|b_1-b_2|^{-1})|b_1-b_2|^\beta+
\ell(|b_1-b_2|^{\epsilon/\beta-1}|b_2|^{-\epsilon/\beta})|b_2|^\epsilon|b_1-b_2|^{\beta-\epsilon}\bigr\}.$$

By , it suffices to prove the result for $\hat R_0(ib)$. We show that $$\begin{aligned}
|\hat R_0(ib_1)-\hat R_0(ib_2)v|_{\theta} & \le C\bigl\{\ell(|b_1-b_2|^{-1})|b_1-b_2|^\beta \\ & \qquad \qquad +\ell(|b_1-b_2|^{\epsilon/\beta-1}|b_2|^{-\epsilon/\beta})|b_2|^\epsilon|b_1-b_2|^{\beta-\epsilon}\bigr\}\|v\|_\theta.\end{aligned}$$ A simpler argument which we omit shows 