scaled-out interval $\lambda J$. Note that $$\bbeta=\frac 1 q + \frac 1 {\tilde{q}} - \frac n 2 \left(\rp p - \frac 1 r + \rp {\tilde p} - \rp {\tilde{r}}\right),$$ due to the scaling condition $\operatorname{HM}(p,r)=\operatorname{HM}(\tilde p', \tilde r')$. We set $$\beta(q,r,\tilde q, \tilde r) = \bbeta.$$

\[lem: local estim 1\] Estimate holds for any two non-endpoint KT-admissible triplets $\trip$ and $\tript$ with $a=\tilde a'$.

The proof follows trivially from theorem \[thm: Strich admiss\] due to the fact that $\beta(q,r,\tilde q, \tilde r)=0$ under the hypothesis of the lemma.

\[lem: local estim 2\] Estimate holds with $\trip=(\infty, r, p)$ and $\tript=(\infty, p', r')$, where $1 \leq p \leq r \leq \infty$.

Due to the decay estimate (\[DispEst\]) we have that $$\begin{aligned}
  \sup_{t \in \lambda I} {\left\VertW(t)[\chi_{\lambda J}F]\right\Vert}_{ L^{r}_xL^{p}_v} &\lesssim \sup_{t \in \lambda I}\int_{\lambda J} \frac {{\left\VertF(\tau)\right\Vert}_{ L^{p}_xL^{r}_v}}             {{\left\vertt-\tau\right\vert}^\betta}d\tau \\
                                                                          &\lesssim  \lambda^{\beta(\infty, r, \infty, p')}{\left\VertF\right\Vert}_{L^1(\lambda J; L^{p}_xL^{r}_v)}.\end{aligned}$$

\[lem: local estim 3\] Whenever $\trip$ and $\tript$ are exponent triplets for which estimate holds, we have that also holds with $(Q,r,p)$ and $(\tilde Q, \tilde r, \tilde p)$, where $1\leq Q \leq q$, $1 \leq \tilde{Q} \leq \tilde{q}$.

A trivial application of Hölder’s inequality $$\begin{split}
  {\left\VertW(t)[\chi_{\lambda J}F]\right\Vert}_{L^{Q}(\lambda I; L^{r}_xL^{p}_v)} \lesssim
  \lambda^{\rp Q - 