\rp q} {\left\VertW(t)[\chi_{\lambda J}F]\right\Vert}_{L^{q}(\lambda I; L^{r}_xL^{p}_v)} \qquad\\
  \lesssim \lambda^{\beta(Q,r,\tilde q , \tilde r)}{\left\VertF\right\Vert}_{L^{\tilde{q}'}(\lambda J; L^{\tilde{r}'}_xL^{\tilde{p}'}_v)}
  \lesssim  \lambda^{\beta(Q,r,\tilde Q , \tilde r)}{\left\VertF\right\Vert}_{L^{\tilde{Q}'}(\lambda J; L^{\tilde{r}'}_xL^{\tilde{p}'}_v)}. \qedhere
 \end{split}$$

Let us define the range of validity of the local estimates as the set $\mathcal{E}$ in ${\mathbb R}^6$. Each point in $\mathcal{E}$ corresponds to a 6-tuple of exponents $(1/q, 1/r, 1/p, 1/\tilde q, 1/ \tilde r, 1/ \tilde p)$. Below we find the convex hull $\mathcal{E}^*$ ($\mathcal{E}^* \subseteq \mathcal{E}$) of the points in ${\mathbb R}^6$ that correspond to the estimates in the three lemmas above. We shall call any point or collection of points in $\mathcal{E}$ [*acceptable*]{}.

\[lem: local inhom estim\] Estimate holds whenever the exponent triplets $\trip$, $\tript$ satisfy the following conditions $$\begin{aligned}
0 \leq \rp q, \rp {\tilde{q}}, \rp p, \rp {\tilde{p}}, \rp r, \rp {\tilde{r}} \leq 1,\quad
\rp r \leq \rp p, &\quad \rp {\tilde r} \leq \rp {\tilde p}, \quad \operatorname{HM}(p,r)=\operatorname{HM}(\tilde p', \tilde r'),\\
\qconds, &\quad \qcondts, \qquad \label{cond: loc inhom 2}\\
\frac {n -1}{p'} < \frac {n}{\tilde{r}}, &\quad \frac {n-1}{\tilde{p}'} < \frac {n}{r},\qquad\qquad\qquad\end{aligned}$$ or if the point $(1/q, 1/r, 1/p, 1/\tilde q, 1/ \tilde r, 1/ \tilde p)$ lies inside one of the cubes in ${\mathbb R}^6$ below $$\label{omitted points 